30-Day Revision Notes - Physical Sciences

Format per day: A. Definitions & Postulates · B. Key Equations / Rules · C. High-Yield Points · D. Comparison Table. Notation: ℏ = h/2π; ∇ operators standard; SI units unless stated.

MODULE 1 — MATHEMATICAL METHODS & CLASSICAL MECHANICS

DAY 1 — Matrices, Eigenvalues, Cayley–Hamilton, Vector Identities

A. Core Definitions

• Hermitian: A† = A (eigenvalues real). Anti-Hermitian: A† = −A (eigenvalues imaginary/zero). Unitary: A†A = I (|λ| = 1, preserves norm). Orthogonal: AᵀA = I (real unitary). Normal: AA† = A†A (diagonalisable by a unitary matrix; Hermitian, unitary, anti-Hermitian are all normal).
• Eigenproblem: A|v⟩ = λ|v⟩; characteristic equation det(A − λI) = 0.
• Similarity transform: A′ = S⁻¹AS preserves eigenvalues, trace, determinant, rank.
• Rank: number of linearly independent rows (= columns). Nullity: n − rank.
• Cayley–Hamilton theorem: every square matrix satisfies its own characteristic equation: p(A) = 0.

B. Key Equations / Rules

• tr A = Σλᵢ ; det A = Πλᵢ.
• 2×2: λ² − (tr A)λ + det A = 0 ⇒ A² − (tr A)A + (det A)I = 0 ⇒ A⁻¹ = [(tr A)I − A]/det A.
• 3×3 characteristic: λ³ − (tr A)λ² + (sum of principal 2×2 minors)λ − det A = 0.
• Eigenvalues of Aⁿ: λⁿ; of A⁻¹: 1/λ; of A + kI: λ + k; of Aᵀ: λ (same).
• Diagonalisation: P⁻¹AP = D, columns of P = eigenvectors; f(A) = P f(D) P⁻¹. e^A has eigenvalues e^λ; det e^A = e^{tr A}.
• Commuting diagonalisable matrices share a common eigenbasis.
• Pauli matrices: σᵢσⱼ = δᵢⱼI + iεᵢⱼₖσₖ ; [σᵢ,σⱼ] = 2iεᵢⱼₖσₖ ; eigenvalues ±1; tr σᵢ = 0.
• Linear system Ax = b: unique solution if rank A = rank[A|b] = n; infinite if rank A = rank[A|b] < n; none if ranks differ.
• Degenerate eigenvalue with algebraic multiplicity m has geometric multiplicity ≤ m; diagonalisable iff equal for all λ.

Vector identities (A, B, C vectors; φ scalar): - A·(B×C) = B·(C×A) = C·(A×B) (cyclic); A×(B×C) = B(A·C) − C(A·B) (BAC–CAB). - (A×B)·(C×D) = (A·C)(B·D) − (A·D)(B·C). - ∇×(∇φ) = 0 ; ∇·(∇×A) = 0. - ∇×(∇×A) = ∇(∇·A) − ∇²A. - ∇·(φA) = φ∇·A + A·∇φ ; ∇×(φA) = φ∇×A + ∇φ×A. - ∇·(A×B) = B·(∇×A) − A·(∇×B). - ∇(A·B) = A×(∇×B) + B×(∇×A) + (A·∇)B + (B·∇)A. - ∇×(A×B) = A(∇·B) − B(∇·A) + (B·∇)A − (A·∇)B. - Position vector r (|r| = r): ∇·r = 3 ; ∇×r = 0 ; ∇rⁿ = n rⁿ⁻² r ; ∇²(1/r) = −4πδ³(r) ; ∇·(r̂/r²) = 4πδ³(r). - Integral theorems: Gauss ∮A·dS = ∫∇·A dV ; Stokes ∮A·dl = ∫(∇×A)·dS ; Green ∮(φ∇ψ − ψ∇φ)·dS = ∫(φ∇²ψ − ψ∇²φ)dV.

C. High-Yield Points

1. Cayley–Hamilton gives A⁻¹ only if det A ≠ 0 (constant term of p(λ) ≠ 0); it also reduces high powers Aⁿ to polynomials of degree ≤ n−1.
2. A real symmetric matrix always has real eigenvalues and orthogonal eigenvectors for distinct eigenvalues; a non-symmetric real matrix may have complex eigenvalues (e.g., rotation matrix, λ = e^{±iθ}).
3. Hermitian ≠ symmetric for complex matrices; a complex symmetric matrix need not have real eigenvalues.
4. Trace is invariant under cyclic permutation only: tr(ABC) = tr(BCA) ≠ tr(BAC) in general. Commutator trace: tr[A,B] = 0 ⇒ [x,p] = iℏ cannot be realised by finite matrices.
5. Defective matrix (e.g., [[1,1],[0,1]]) satisfies Cayley–Hamilton but is not diagonalisable.
6. A nilpotent matrix has all eigenvalues 0; a projector has eigenvalues 0 or 1; a traceless 2×2 Hermitian matrix has eigenvalues ±√(−det).
7. Curl-free ⇒ gradient of scalar (simply connected domain); divergence-free ⇒ curl of a vector potential. ∇×(r f(r)) = 0 for any f(r).

D. Comparison Table

DAY 2 — ODEs (1st & 2nd Order) and Special Functions

A. Core Definitions

• Exact ODE: M dx + N dy = 0 with ∂M/∂y = ∂N/∂x. Integrating factor μ(x) = exp∫p dx for y′ + p(x)y = q(x).
• Wronskian: W(y₁,y₂) = y₁y₂′ − y₂y₁′; W ≠ 0 ⇔ linearly independent. Abel: W = C exp(−∫P dx) for y″ + Py′ + Qy = 0.
• Sturm–Liouville problem: d/dx[p(x)y′] + [q(x) + λw(x)]y = 0; eigenfunctions orthogonal with weight w(x): ∫ yₘyₙ w dx = 0 (m ≠ n).
• Ordinary point / regular singular point: Frobenius series y = Σ aₙ x^{n+s} valid at regular singular points; indicial equation gives s.

B. Key Equations / Rules

• Linear first order: y = (1/μ)[∫μq dx + C]. Bernoulli: y′ + py = qyⁿ ⇒ v = y^{1−n} linear.
• Constant coefficients y″ + ay′ + by = 0: roots m; distinct real → c₁e^{m₁x} + c₂e^{m₂x}; repeated → (c₁ + c₂x)e^{mx}; complex α ± iβ → e^{αx}(c₁cos βx + c₂ sin βx).
• Particular solution: undetermined coefficients (resonance ⇒ multiply by x); variation of parameters y_p = −y₁∫y₂f/W + y₂∫y₁f/W.
• Euler–Cauchy: x²y″ + axy′ + by = 0 ⇒ y = x^m.
• Damped oscillator: ẍ + 2γẋ + ω₀²x = 0; underdamped γ < ω₀ (ω = √(ω₀²−γ²)), critical γ = ω₀, overdamped γ > ω₀.

Legendre (1−x²)y″ − 2xy′ + l(l+1)y = 0, x ∈ [−1,1]: - Rodrigues: Pₗ(x) = (1/2ˡ l!) dˡ/dxˡ (x²−1)ˡ. P₀ = 1, P₁ = x, P₂ = ½(3x²−1), P₃ = ½(5x³−3x). - Orthogonality: ∫₋₁¹ PₘPₙ dx = 2δₘₙ/(2n+1). Pₙ(1) = 1, Pₙ(−x) = (−1)ⁿPₙ(x). Generating function: (1−2xt+t²)^{-1/2} = ΣPₙtⁿ. - Recurrence: (n+1)Pₙ₊₁ = (2n+1)xPₙ − nPₙ₋₁. - Associated: Pₗᵐ(x) = (1−x²)^{m/2} dᵐPₗ/dxᵐ ; orthogonality ∫PₗᵐPₗ′ᵐ dx = 2(l+m)!/[(2l+1)(l−m)!] δₗₗ′.

Hermite y″ − 2xy′ + 2ny = 0 (weight e^{−x²}, x ∈ (−∞,∞)): - Rodrigues: Hₙ(x) = (−1)ⁿ eˣ² dⁿ/dxⁿ e^{−x²}. H₀ = 1, H₁ = 2x, H₂ = 4x²−2, H₃ = 8x³−12x. - ∫Hₘ Hₙ e^{−x²}dx = √π 2ⁿ n! δₘₙ. Recurrence: Hₙ₊₁ = 2xHₙ − 2nHₙ₋₁; Hₙ′ = 2nHₙ₋₁. Parity (−1)ⁿ.

Laguerre xy″ + (1−x)y′ + ny = 0 (weight e^{−x}, x ∈ [0,∞)): Lₙ = (eˣ/n!) dⁿ/dxⁿ(xⁿe^{−x}); L₀ = 1, L₁ = 1−x, L₂ = 1 − 2x + x²/2; ∫LₘLₙe^{−x}dx = δₘₙ.

Bessel x²y″ + xy′ + (x²−ν²)y = 0: Jν(x) = Σ (−1)ᵏ/[k!Γ(k+ν+1)] (x/2)^{2k+ν}. Non-integer ν: Jν, J₋ν independent; integer n: J₋ₙ = (−1)ⁿJₙ, use Yₙ. Recurrence: (2ν/x)Jν = Jν₋₁ + Jν₊₁; Jν′ = ½(Jν₋₁ − Jν₊₁); d/dx[xᵛJν] = xᵛJν₋₁. J₀′ = −J₁. J₁/₂ = √(2/πx) sin x. Orthogonality ∫₀¹ xJν(αₘx)Jν(αₙx)dx = ½[Jν₊₁(αₙ)]²δₘₙ.

C. High-Yield Points

1. Legendre equation: finite polynomial solutions only when l is a non-negative integer; second solution Qₗ diverges at x = ±1.
2. Hermite polynomials have n real zeros and weight e^{−x²} — quantum oscillator wavefunction ψₙ ∝ Hₙ(√(mω/ℏ) x)e^{−mωx²/2ℏ}.
3. Frobenius: indicial roots differing by an integer may force a log term in the second solution (Bessel integer order).
4. Wronskian vanishing identically does not always imply dependence unless both are solutions of the same linear ODE.
5. Bessel J_{1/2}, J_{−1/2} are elementary (sin/cos); J₀(0) = 1, Jₙ(0) = 0 for n ≥ 1; Neumann Yₙ diverges at 0.
6. Total number of zeros of Pₙ in (−1,1) is n; Pₗ(cosθ) used for azimuthal-symmetric Laplace solutions: V = Σ(Aₗrˡ + Bₗr^{−(l+1)})Pₗ(cosθ).
7. First-order ODE y′ = f(ax+by) → substitution v = ax+by; homogeneous ODE y′ = f(y/x) → v = y/x.

D. Comparison Table

DAY 3 — Complex Variables, Residues, Laplace & Fourier Transforms

A. Core Definitions

• Analytic (holomorphic) function: complex-differentiable in a neighbourhood. Entire: analytic on all ℂ.
• Singularities: removable (lim finite), pole of order m (Laurent principal part finite), essential (infinite principal part, e.g., e^{1/z} at 0). Residue: coefficient a₋₁ of (z−z₀)⁻¹ in the Laurent series.
• Fourier transform: F(k) = ∫f(x)e^{−ikx}dx ; Laplace: F(s) = ∫₀^∞ f(t)e^{−st}dt.

B. Key Equations / Rules

• Cauchy–Riemann (f = u + iv): ∂u/∂x = ∂v/∂y, ∂u/∂y = −∂v/∂x. Polar: ∂u/∂r = (1/r)∂v/∂θ, ∂v/∂r = −(1/r)∂u/∂θ. Then ∇²u = ∇²v = 0 (harmonic conjugates), f′ = ∂u/∂x + i∂v/∂x. Sufficient: partial derivatives continuous + CR.
• Cauchy theorem: ∮f dz = 0 (f analytic inside C). Integral formula: f(z₀) = (1/2πi)∮f(z)/(z−z₀)dz ; f⁽ⁿ⁾(z₀) = (n!/2πi)∮f/(z−z₀)ⁿ⁺¹dz.
• Residue theorem: ∮f dz = 2πi Σ Res.
• Simple pole: Res = lim (z−z₀)f(z); f = g/h with h′(z₀) ≠ 0: Res = g(z₀)/h′(z₀). Pole order m: Res = (1/(m−1)!) lim d^{m−1}/dz^{m−1}[(z−z₀)ᵐf].
• Real integrals: ∫₀^{2π}R(cosθ,sinθ)dθ with z = e^{iθ}; ∫₋∞^∞ P/Q dx = 2πiΣRes(upper half-plane) (deg Q ≥ deg P + 2); Jordan's lemma for ∫f(x)e^{iax}dx (a > 0), close in UHP. Standard: ∫₋∞^∞ dx/(1+x²) = π ; ∫₋∞^∞ sin x/x dx = π ; ∫₀^∞ sin x/x dx = π/2.
• Taylor series valid within circle to nearest singularity; Laurent valid in annulus.
• Laplace table: L[1] = 1/s ; L[tⁿ] = n!/sⁿ⁺¹ ; L[e^{at}] = 1/(s−a) ; L[sin ωt] = ω/(s²+ω²) ; L[cos ωt] = s/(s²+ω²) ; L[sinh at] = a/(s²−a²) ; L[cosh at] = s/(s²−a²).
• Properties: L[f′] = sF − f(0); L[f″] = s²F − sf(0) − f′(0); L[e^{at}f] = F(s−a) (shift); L[f(t−a)u(t−a)] = e^{−as}F(s); L[tf] = −F′(s); L[∫f] = F/s; convolution L[f*g] = FG; L[δ(t−a)] = e^{−as}.
• Fourier: F[f′] = ikF ; F[e^{−ax²}] = √(π/a)e^{−k²/4a} (Gaussian → Gaussian); F[δ] = 1; F[1] = 2πδ(k); Parseval ∫|f|²dx = (1/2π)∫|F|²dk; convolution F[f*g] = FG; shift F[f(x−a)] = e^{−ika}F.
• Fourier series on [−L,L]: aₙ = (1/L)∫f cos(nπx/L), bₙ = (1/L)∫f sin(nπx/L); even f → cosine only, odd → sine only; at jump discontinuity series converges to midpoint (Dirichlet); Gibbs overshoot ≈ 9%.

C. High-Yield Points

1. CR equations are necessary, not sufficient for differentiability (|z|², z̄, Re z fail except isolated points; f = |z|² differentiable only at z = 0).
2. A non-constant analytic function cannot be purely real or have constant modulus (Liouville: bounded entire ⇒ constant).
3. Residue at essential singularity: for e^{1/z} Res = 1 at z = 0; sin z/z has a removable singularity (Res = 0). For 1/sin z poles at nπ, Res = (−1)ⁿ.
4. Branch cuts: √z, ln z are multivalued; Res theorem applies only after cut excluded.
5. Laplace transform requires f of exponential order; initial value theorem lim_{s→∞}sF = f(0⁺); final value lim_{s→0}sF = f(∞) only if poles in left half-plane.
6. Narrower f(x) ⇒ broader F(k): Δx·Δk ≥ ½ (Gaussian saturates).
7. Contour integral ∮dz/(z−z₀)ⁿ = 2πi for n = 1, zero for n ≠ 1 (integer) — basis of the residue method.

D. Comparison Table

DAY 4 — Lagrangian Mechanics

A. Core Definitions & Postulates

• Generalized coordinates qᵢ: independent coordinates equal in number to degrees of freedom f = 3N − k (k holonomic constraints).
• Holonomic constraint: expressible as g(r₁,…,t) = 0; non-holonomic: inequalities or non-integrable velocity constraints (rolling without slipping on a plane is non-holonomic for sphere; a hoop rolling in a line is holonomic). Scleronomic: time-independent; rheonomic: time-dependent.
• Hamilton's principle: δ∫_{t₁}^{t₂} L dt = 0, L = T − V.
• Cyclic (ignorable) coordinate: one absent from L.

B. Key Equations / Rules

• Euler–Lagrange: d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = Qᵢ (non-conservative generalized force; Qᵢ = 0 for conservative).
• Generalized momentum pᵢ = ∂L/∂q̇ᵢ. Cyclic qᵢ ⇒ pᵢ = const. Energy function h = Σpᵢq̇ᵢ − L; dh/dt = −∂L/∂t. h = T + V if T is quadratic homogeneous in q̇ and V independent of velocity and constraints scleronomic.
• Noether conservation theorems: time-translation → energy; spatial translation → linear momentum; rotation → angular momentum.
• Equivalent Lagrangians: L′ = L + dF(q,t)/dt (same equations of motion).
• Charged particle: L = ½mv² − qφ + qA·v ; canonical p = mv + qA.
• Variational calculus: stationary ∫F(y,y′,x)dx ⇒ ∂F/∂y − d/dx(∂F/∂y′) = 0; special case F independent of x ⇒ y′∂F/∂y′ − F = const (Beltrami).
• Standard Lagrangians: pendulum L = ½ml²θ̇² + mgl cosθ ; Atwood: a = (m₁−m₂)g/(m₁+m₂); spherical coordinates T = ½m(ṙ² + r²θ̇² + r²sin²θ φ̇²); plane polar T = ½m(ṙ² + r²θ̇²).
• Rayleigh dissipation function: Qᵢ = −∂ℱ/∂q̇ᵢ, ℱ = ½bẋ².
• Lagrange multipliers: d/dt(∂L/∂q̇) − ∂L/∂q = Σλₖ∂gₖ/∂q (gives constraint forces).
• Brachistochrone → cycloid; geodesic on a plane → straight line; minimal surface of revolution (catenoid) from Beltrami identity.

C. High-Yield Points

1. Energy is conserved (h = const) whenever L has no explicit time dependence, but h ≠ T + V when constraints are time-dependent (e.g., bead on a rotating wire, h = T₂ − T₀ + V).
2. Lagrangian is not unique (gauge freedom L → L + dF/dt); canonical momenta change under such transformations.
3. Lagrange's equations in this form apply to holonomic systems with forces from a potential (possibly velocity-dependent, generalized potential U(q,q̇) with Q = −∂U/∂q + d/dt ∂U/∂q̇).
4. Number of independent EL equations = degrees of freedom; N particles with k constraints → 3N − k.
5. Cyclic coordinate in Lagrangian is cyclic in the Hamiltonian too (since ∂H/∂q = −∂L/∂q); canonical momentum p is conserved — e.g., p_φ = mr²sin²θ φ̇ for central potentials.
6. Constraint forces do no virtual work (D'Alembert's principle: Σ(Fᵢ − ṗᵢ)·δrᵢ = 0).
7. Free particle in rotating frame: L = ½m[(ẋ − Ωy)² + (ẏ + Ωx)²] ⇒ centrifugal mΩ²r and Coriolis 2mΩ×v.

D. Comparison Table

DAY 5 — Hamiltonian Formalism, Poisson Brackets, Small Oscillations

A. Core Definitions

• Hamiltonian: H(q,p,t) = Σpᵢq̇ᵢ − L (Legendre transform). Phase space: (qᵢ,pᵢ), dimension 2f.
• Poisson bracket: {A,B} = Σ(∂A/∂qᵢ ∂B/∂pᵢ − ∂A/∂pᵢ ∂B/∂qᵢ).
• Canonical transformation: preserves the form of Hamilton's equations; preserves fundamental brackets.
• Normal modes: collective oscillations with a single frequency.

B. Key Equations / Rules

• Hamilton's equations: q̇ᵢ = ∂H/∂pᵢ ; ṗᵢ = −∂H/∂qᵢ ; dH/dt = ∂H/∂t.
• dA/dt = {A,H} + ∂A/∂t. Conserved if {A,H} = 0 and no explicit t.
• Fundamental brackets: {qᵢ,pⱼ} = δᵢⱼ ; {qᵢ,qⱼ} = {pᵢ,pⱼ} = 0 (Quantum: {,} → [,]/iℏ).
• Properties: antisymmetric; bilinear; Jacobi {A,{B,C}} + {B,{C,A}} + {C,{A,B}} = 0; {A,BC} = {A,B}C + B{A,C}.
• {Lᵢ,Lⱼ} = εᵢⱼₖLₖ ; {L²,Lᵢ} = 0.
• Liouville's theorem: phase-space density constant along flow (dρ/dt = 0), volume preserved.
• Generating functions: F₁(q,Q,t): p = ∂F₁/∂q, P = −∂F₁/∂Q; F₂(q,P,t): p = ∂F₂/∂q, Q = ∂F₂/∂P; K = H + ∂F/∂t. Identity: F₂ = ΣqᵢPᵢ.
• Hamilton–Jacobi: H(q,∂S/∂q,t) + ∂S/∂t = 0; time-independent: H(q,∂W/∂q) = E.
• Action–angle: J = ∮p dq (Bohr–Sommerfeld: J = nh).
• Harmonic oscillator: H = p²/2m + ½mω²q².

Small oscillations: L = ½Σ Tᵢⱼq̇ᵢq̇ⱼ − ½Σ Vᵢⱼqᵢqⱼ, Vᵢⱼ = ∂²V/∂qᵢ∂qⱼ at equilibrium. - Secular (characteristic) equation: det(V − ω²T) = 0. Solutions ω² = eigenvalues; eigenvectors = normal modes; normal coordinates diagonalise both T and V. - Two equal coupled pendula (coupling spring k): ω₁ = √(g/l) (in phase), ω₂ = √(g/l + 2k/m) (antiphase); beat frequency = ω₂ − ω₁. - Two masses, three springs (k, κ, k): ω² = k/m, (k + 2κ)/m. - Linear triatomic CO₂-type (m, M, m): ω₁ = 0 (translation), ω₂ = √(k/m) (symmetric stretch), ω₃ = √[(k/m)(1 + 2m/M)] (antisymmetric). - Stability: V″ > 0 stable; V″ < 0 unstable.

C. High-Yield Points

1. {A,H} = 0 does not make A conserved if A depends explicitly on t.
2. Poisson bracket is preserved only by canonical transformations; check {Q,P}_{q,p} = 1.
3. Number of normal modes = degrees of freedom; zero-frequency modes correspond to free translation/rotation (zero eigenvalue of V).
4. Phase-space trajectories never cross (for autonomous system) – each point has unique velocity; closed orbit for oscillator = ellipse of area 2πE/ω.
5. Hamiltonian for time-dependent constraint ≠ total energy; H is conserved but not equal to E.
6. For velocity-dependent charged particle: H = (p − qA)²/2m + qφ.
7. Liouville ≠ conservation of entropy of coarse-grained distribution; it expresses incompressibility of flow; Hamiltonian flow is symplectic.

D. Comparison Table

DAY 6 — Central Forces, Orbits, Special Relativity

A. Core Definitions

• Central force: F = f(r)r̂; angular momentum L = r × p conserved ⇒ motion in a plane; areal velocity dA/dt = L/2μ (Kepler II).
• Reduced mass: μ = m₁m₂/(m₁+m₂).
• Inertial frame invariance (SR): laws of physics identical in all inertial frames; speed of light c is invariant.
• Proper time dτ = dt/γ ; interval s² = c²t² − x² − y² − z² invariant.

B. Key Equations / Rules

• Effective potential: V_eff = V(r) + L²/2μr²; ½μṙ² + V_eff = E.
• Orbit equation (u = 1/r): d²u/dθ² + u = −(μ/L²u²)F(1/u).
• Inverse-square F = −k/r²: r = ℓ/(1 + e cos θ), ℓ = L²/μk, e = √(1 + 2EL²/μk²). e = 0 circle (E = −μk²/2L²), 0 < e < 1 ellipse (E < 0), e = 1 parabola (E = 0), e > 1 hyperbola (E > 0).
• Ellipse: E = −k/2a; a = ℓ/(1−e²); T² = 4π²μa³/k (Kepler III; T² ∝ a³). Circular speed v = √(k/μr); escape speed v_esc = √2 v_circ = √(2GM/R). Virial theorem for r^n potentials: 2⟨T⟩ = n⟨V⟩ ⇒ gravity ⟨T⟩ = −½⟨V⟩, harmonic ⟨T⟩ = ⟨V⟩.
• Bertrand's theorem: only V ∝ −1/r and V ∝ r² give closed orbits for all bound orbits. Precession for perturbation ∝ 1/r³ (GR): Δφ = 6πGM/[c²a(1−e²)] per orbit (Mercury 43″/century).
• Turning points: E = V_eff. Stable circular orbit: V_eff′ = 0, V_eff″ > 0. For F ∝ r⁻ⁿ stable circular orbits exist only for n < 3.
• Rutherford scattering: dσ/dΩ = (Z₁Z₂e²/4E·4πε₀)² / sin⁴(θ/2). Impact parameter b = (k/2E)cot(θ/2).
• Lab–CM: tan θ_lab = sin θ_cm/(cos θ_cm + m₁/m₂); maximum lab angle for m₁ > m₂ is sin θ_max = m₂/m₁.

Special relativity: - γ = 1/√(1 − β²), β = v/c. Lorentz boost along x: x′ = γ(x − vt), t′ = γ(t − vx/c²), y′ = y, z′ = z. - Time dilation Δt = γΔτ ; length contraction L = L₀/γ ; velocity addition u′ = (u − v)/(1 − uv/c²). - Relativistic Doppler (source receding, longitudinal): f_obs = f₀√[(1 − β)/(1 + β)]; transverse f_obs = f₀/γ. - Momentum p = γmv ; E = γmc² ; E² = p²c² + m²c⁴ ; kinetic energy K = (γ−1)mc². Low-speed K ≈ ½mv² + (3/8)mv⁴/c². - 4-vectors: x^μ = (ct, x), p^μ = (E/c, p), p·p = m²c² invariant. Photon: E = pc. - Threshold kinetic energy (beam m₁ on fixed target m₂): T_th = [(Σm_final)² − (m₁ + m₂)²]c²/2m₂ (e.g., p + p → p + p + p + p̄: T_th = 6m_pc² = 5.63 GeV). - Compton shift Δλ = (h/mc)(1 − cos θ) = 2.43 pm (1 − cos θ). - Force: F = dp/dt = γ³m a (longitudinal), γm a (transverse). - Mass-energy: Δm c² = Q; 1 u = 931.5 MeV/c². - Twin paradox: travelling twin ages less (acceleration breaks symmetry). Rapidity additive: tanh φ = β.

C. High-Yield Points

1. For a central force, L conserved ⇒ planar orbit; energy conserved only for conservative force — central forces are conservative if f depends on r only.
2. Closed orbits ⇒ Bertrand (1/r and r² only). Orbits under 1/r³ spiral in or out unless E = 0 critical.
3. Elliptical orbit period depends on semimajor axis a only, not eccentricity.
4. Hyperbolic path (E > 0) and parabolic (E = 0) are unbound; perihelion speed maximum at r_min.
5. Simultaneity is relative; length contraction is along motion only; transverse dimensions unchanged. Proper time is the minimum elapsed time between events.
6. Rest mass is invariant; "relativistic mass" γm is frame dependent. In fission/fusion Q = (Σm_i − Σm_f)c².
7. Lorentz invariants: E² − p²c², s = (p₁ + p₂)², c²t² − r², ω² − k²c² (photon: 0). CM energy for collider of equal beams = 2E; for fixed target √(2mc²E + 2m²c⁴) ≈ √(2mc²E) at high E.

D. Comparison Table

MODULE 2 — ELECTRODYNAMICS & OPTICS

DAY 7 — Gauss's Law, Laplace/Poisson, Image Charges, Dipoles

A. Core Definitions

• Gauss's law: ∮E·dA = Q_enc/ε₀ ⇔ ∇·E = ρ/ε₀. Electrostatic field: ∇×E = 0 ⇒ E = −∇V.
• Poisson: ∇²V = −ρ/ε₀; Laplace: ∇²V = 0 (no charge).
• Uniqueness theorem: solution of Poisson's equation is unique given V (Dirichlet) or ∂V/∂n (Neumann) on the boundary.
• Dipole moment: p = Σqᵢrᵢ (origin-independent if total charge zero).

B. Key Equations / Rules

• Coulomb: F = q₁q₂/(4πε₀r²). Energy of configuration U = ½∫ρV dτ = (ε₀/2)∫E² dτ.
• Fields: infinite line λ/(2πε₀s); infinite sheet σ/2ε₀; conductor surface σ/ε₀ (normal); uniform sphere inside E = ρr/3ε₀ (∝ r), outside Q/4πε₀r²; spherical shell: zero inside.
• Self-energy: uniform sphere (3/5)Q²/4πε₀R; shell Q²/8πε₀R. Capacitance: sphere 4πε₀R; parallel plate ε₀A/d; coaxial 2πε₀L/ln(b/a); concentric spheres 4πε₀ab/(b−a).
• Laplace solutions: Cartesian (separation) V = X(x)Y(y)Z(z); spherical azimuthal symmetry V = Σ(Aₗrˡ + Bₗ/rˡ⁺¹)Pₗ(cosθ); cylindrical V = Σ(Aₛsⁿ + B/sⁿ)(C cos nφ + D sin nφ), and (A ln s + B) for n = 0.
• Mean value property: V at a point = average over any sphere centred there; no local extrema of V (Earnshaw's theorem: no stable electrostatic equilibrium).
• Boundary conditions at conductor surface: E_∥ = 0, E_⊥ = σ/ε₀; V constant.
• Image charge (point charge q at distance d from grounded infinite plane): image −q at −d; force F = −q²/16πε₀d²; induced charge total −q; σ(ρ) = −qd/[2π(ρ²+d²)^{3/2}]; work to remove charge to infinity W = q²/16πε₀d (half of naïve value).
• Grounded conducting sphere radius R, charge q at distance a > R: image q′ = −qR/a at b = R²/a; force F = (1/4πε₀) q²Ra/(a²−R²)². Isolated neutral sphere: add +qR/a at centre. Sphere at potential V₀: add charge 4πε₀RV₀ at centre.
• Multipole expansion: V = (1/4πε₀)Σ(1/rⁿ⁺¹)∫(r′)ⁿPₙ(cosθ′)ρ dτ′. Monopole 1/r, dipole 1/r², quadrupole 1/r³.
• Electric dipole: V = p cosθ/(4πε₀r²); E_r = 2p cosθ/(4πε₀r³), E_θ = p sinθ/(4πε₀r³); E = [3(p·r̂)r̂ − p]/(4πε₀r³) (+ −p δ³/3ε₀ contact term). Torque τ = p×E; energy U = −p·E; force F = (p·∇)E.
• Dipole–dipole U = [p₁·p₂ − 3(p₁·r̂)(p₂·r̂)]/(4πε₀r³).
• Dielectric: P = ε₀χₑE, D = ε₀E + P = εE; bound charges σ_b = P·n̂, ρ_b = −∇·P; ∇·D = ρ_free.
• Uniformly polarized sphere: E_in = −P/3ε₀. Dielectric sphere in uniform field E₀: E_in = 3E₀/(ε_r + 2).

C. High-Yield Points

1. Gauss's law always holds; it is useful only with spherical, cylindrical, or planar symmetry.
2. Inside a conductor cavity with no charge, E = 0 (shielding); cavity with charge q induces −q on cavity wall and +q on the outer surface.
3. Image method valid only in the region outside the actual image position; the field inside the conductor is zero, not that of the image system. Energy is half of the real-pair energy.
4. Dipole field falls as 1/r³ (in-plane axial field is twice the equatorial: E_axial = 2p/4πε₀r³, E_equatorial = p/4πε₀r³ opposite direction).
5. Net force on a dipole in a uniform field is zero, torque not; in a non-uniform field both nonzero.
6. Electric field is discontinuous across surface charge by σ/ε₀, but V is continuous.
7. For a linear dielectric the energy is W = ½∫D·E dτ; for the quadrupole potential the first nonvanishing multipole is independent of origin.

D. Comparison Table

DAY 8 — Biot–Savart, Ampère, Vector Potential, Boundary Conditions

A. Core Definitions

• Biot–Savart: B(r) = (μ₀/4π)∮ I dl′×R̂/R². Ampère's law: ∮B·dl = μ₀I_enc ⇔ ∇×B = μ₀J. No monopoles: ∇·B = 0 ⇒ B = ∇×A.
• Magnetization M; H = B/μ₀ − M; ∇×H = J_free; bound currents J_b = ∇×M, K_b = M×n̂.
• Susceptibility: M = χ_mH; B = μ₀(1+χ_m)H = μH.

B. Key Equations / Rules

• Straight wire B = μ₀I/2πs; finite wire B = (μ₀I/4πs)(sinθ₂ − sinθ₁); circular loop axis B = μ₀IR²/2(R²+z²)^{3/2}; centre μ₀I/2R; solenoid (ideal) B = μ₀nI inside, 0 outside; toroid B = μ₀NI/2πr; infinite sheet K: B = μ₀K/2.
• Lorentz force F = q(E + v×B); force on wire F = I∫dl×B; parallel wires force per length μ₀I₁I₂/2πd (attractive if parallel currents).
• Magnetic dipole m = IA (n̂); B = (μ₀/4πr³)[3(m·r̂)r̂ − m]; A = μ₀ m×r̂/4πr²; torque m×B; energy U = −m·B.
• Vector potential: ∇²A = −μ₀J (Coulomb gauge); A = (μ₀/4π)∫J/R dτ′; A for uniform B: A = ½B×r; A for solenoid: A_φ = Φ/2πs (outside) — nonzero outside though B = 0 (Aharonov–Bohm). Flux Φ = ∮A·dl.
• Continuity: ∇·J = −∂ρ/∂t.
• Boundary conditions (interface with unit normal n̂ from medium 1 to 2):
• D_⊥: D₂·n̂ − D₁·n̂ = σ_f.
• E_∥ continuous: E₂∥ = E₁∥ (n̂×(E₂−E₁) = 0).
• B_⊥ continuous: B₂·n̂ = B₁·n̂.
• H: n̂×(H₂ − H₁) = K_f (H_∥ continuous if no free surface current).
• Linear dielectrics: ε₂E₂⊥ − ε₁E₁⊥ = σ_f; potential V continuous across the interface.
• Refraction of field lines: tanθ₂/tanθ₁ = ε₂/ε₁ (E) or μ₂/μ₁ (B).
• Faraday: ε = −dΦ/dt; Lenz. Self-inductance: L = Φ/I, solenoid μ₀n²·(volume); energy U = ½LI² = (1/2μ₀)∫B²dτ. Mutual inductance M₁₂ = M₂₁ (Neumann).
• Larmor/cyclotron: ω_c = qB/m, r = mv/qB; Hall field E = vB.
• Magnetic field energy density B²/2μ₀; for linear media ½B·H.

C. High-Yield Points

1. Ampère's law (original) fails for time-varying fields (charging capacitor) ⇒ Maxwell displacement current J_d = ε₀∂E/∂t.
2. A is defined only up to gradient (A → A + ∇λ); B physical; Aharonov–Bohm phase = (q/ℏ)∮A·dl shows A is "more fundamental" in QM.
3. B_⊥ always continuous, E_∥ always continuous; surface current produces jump in tangential B of μ₀K, surface charge a jump in normal E of σ/ε₀.
4. Magnetic force does no work; it only changes direction; a charged particle in uniform B follows helix with pitch 2πmv_∥/qB.
5. Inside an ideal solenoid B uniform; outside zero; toroid B ∝ 1/r.
6. Diamagnets χ_m < 0 (~−10⁻⁵), paramagnets 0 < χ_m ≪ 1, ferromagnets χ_m ≫ 1 with hysteresis and Curie temperature.
7. Superconductor is perfect diamagnet χ_m = −1 (B = 0 inside).

D. Comparison Table

DAY 9 — Maxwell's Equations, Gauge, Poynting

A. Core Definitions

• Maxwell (vacuum): ∇·E = ρ/ε₀; ∇·B = 0; ∇×E = −∂B/∂t; ∇×B = μ₀J + μ₀ε₀∂E/∂t.
• In matter: ∇·D = ρ_f; ∇·B = 0; ∇×E = −∂B/∂t; ∇×H = J_f + ∂D/∂t.
• Potentials: B = ∇×A; E = −∇φ − ∂A/∂t. Gauge transformation: A → A + ∇λ, φ → φ − ∂λ/∂t.

B. Key Equations / Rules

• Speed c = 1/√(μ₀ε₀). In matter v = 1/√(με), n = √(μ_rε_r).
• Coulomb gauge ∇·A = 0: ∇²φ = −ρ/ε₀ (instantaneous Coulomb potential); □-type equation ∇²A − μ₀ε₀∂²A/∂t² = −μ₀J + μ₀ε₀∇(∂φ/∂t). Transverse radiation easily seen; used for radiation.
• Lorenz gauge ∇·A + μ₀ε₀∂φ/∂t = 0: □²φ = −ρ/ε₀ and □²A = −μ₀J, with □² = ∇² − (1/c²)∂²/∂t². Manifestly Lorentz covariant: A^μ = (φ/c, A), □A^μ = −μ₀J^μ.
• Retarded solutions: φ(r,t) = (1/4πε₀)∫ρ(r′,t_r)/R dτ′, t_r = t − R/c.
• Continuity ∇·J + ∂ρ/∂t = 0 follows from Maxwell's equations.
• Poynting theorem: −dU_em/dt = ∮S·da + ∫J·E dτ; S = (1/μ₀)E×B (= E×H in matter); u = ½(ε₀E² + B²/μ₀). Poynting vector in a wire carrying DC: radial inflow of energy, P = EI = I²R.
• Momentum density g = S/c² = ε₀E×B; radiation pressure = I/c (absorber), 2I/c (perfect reflector). Maxwell stress tensor Tᵢⱼ = ε₀(EᵢEⱼ − ½δᵢⱼE²) + (1/μ₀)(BᵢBⱼ − ½δᵢⱼB²).
• Angular momentum density r × g.
• Lorentz transformations of fields: E′∥ = E∥; B′∥ = B∥; E′⊥ = γ(E + v×B)⊥; B′⊥ = γ(B − v×E/c²)⊥. Invariants: E² − c²B² and E·B.
• Field tensor Fμν; ∂μFμν = μ₀Jν; Maxwell's equations are Lorentz covariant (and not Galilean).
• Magnetic monopole generalization: ∇·B = μ₀ρ_m (not observed).
• Wave equation: ∇²E = μ₀ε₀∂²E/∂t² (vacuum, ρ = J = 0).

C. High-Yield Points

1. Four-potential gauge freedom: φ, A individually are not physical; only E, B. Gauge transformations that preserve Lorenz gauge satisfy □²λ = 0 (residual gauge).
2. Coulomb gauge does not cause superluminal signals: the total E = −∇φ − ∂A/∂t is still causal because the instantaneous part is cancelled by the longitudinal part of ∂A/∂t.
3. Displacement current makes ∇·(∇×B) = 0 consistent with charge conservation.
4. S has units W/m²; averaged ⟨S⟩ = ½E₀²/(μ₀c) = ½cε₀E₀² for a plane wave; E₀ = cB₀ in vacuum.
5. Electromagnetic field carries momentum ⇒ force on absorbing surface I/c; Solar radiation pressure on a perfect absorber at 1 AU ≈ 4.5 µPa.
6. Maxwell's equations are invariant under Lorentz, not Galilean; they predicted c from ε₀ and μ₀ ⇒ ether problem ⇒ SR.
7. Under parity, E is a polar vector, B an axial vector; ∇·B = 0 ⇒ no magnetic monopole.

D. Comparison Table

DAY 10 — EM Waves in Vacuum, Conductors, Dielectrics

A. Core Definitions

• Plane wave: E = E₀ ei(k·r − ωt), B = (1/ω)k×E; transverse, E ⊥ B ⊥ k, |E| = c|B|.
• Skin depth: distance over which wave amplitude falls by 1/e in a conductor.
• Refractive index n = √(μ_rε_r) ≈ √ε_r; impedance Z = √(μ/ε); Z₀ = 376.7 Ω.

B. Key Equations / Rules

• Vacuum: ω = ck; energy density u = ε₀E² (equal electric & magnetic); intensity I = ½cε₀E₀².
• Dielectric: v = c/n; k = nω/c; wave impedance η = √(μ/ε) = E/H.
• Conductor (σ, ε, μ): k̃² = μεω² + iμσω. Good conductor (σ ≫ ωε): k ≈ (1+i)/δ with δ = √(2/μσω) (= 1/√(πfμσ)). Magnetic field lags E by 45°; E/H ratio magnitude = √(ωμ/σ); S decays e^{−2z/δ}. Copper at 1 MHz: δ ≈ 66 µm.
• Poor conductor/weakly absorbing: δ ≈ (2/σ)√(ε/μ).
• Plasma: ε_r = 1 − ω_p²/ω², ω_p² = ne²/ε₀m. For ω < ω_p wave evanescent (reflected); ω > ω_p propagates; ω² = ω_p² + c²k². Ionosphere: ω_p ≈ 2π(9√n) Hz (n in m⁻³). Group velocity v_g = c√(1 − ω_p²/ω²), phase velocity v_p = ω/k > c with v_p v_g = c².
• Dispersion: n(ω) = 1 + (Ne²/2ε₀m)Σ f_j/(ω_j² − ω² − iγ_jω); normal dispersion dn/dω > 0 away from resonances; anomalous near resonance (absorption).
• Fresnel (non-magnetic, incidence θᵢ, transmission θₜ, n₁ sinθᵢ = n₂ sinθₜ):
• r_⊥ = (n₁cosθᵢ − n₂cosθₜ)/(n₁cosθᵢ + n₂cosθₜ) = −sin(θᵢ−θₜ)/sin(θᵢ+θₜ); t_⊥ = 2n₁cosθᵢ/(n₁cosθᵢ + n₂cosθₜ).
• r_∥ = (n₂cosθᵢ − n₁cosθₜ)/(n₂cosθᵢ + n₁cosθₜ) = tan(θᵢ−θₜ)/tan(θᵢ+θₜ); t_∥ = 2n₁cosθᵢ/(n₂cosθᵢ + n₁cosθₜ).
• Reflectance R = |r|²; transmittance T = (n₂cosθₜ/n₁cosθᵢ)|t|²; R + T = 1.
• Normal incidence: r = (n₁ − n₂)/(n₁ + n₂), t = 2n₁/(n₁ + n₂); R = [(n₁−n₂)/(n₁+n₂)]². Air–glass (1.5): R = 4%.
• Brewster angle tanθ_B = n₂/n₁ (r_∥ = 0), θ_B + θₜ = 90°.
• Total internal reflection for n₁ > n₂ at θ_c = sin⁻¹(n₂/n₁); evanescent wave penetration depth d = λ/[2π√(n₁²sin²θ − n₂²)].
• Phase change on reflection: π for r < 0 (denser medium, perpendicular component), 0 for lighter medium.
• Normal incidence on a perfect conductor: r = −1; standing wave with E node at the surface.
• Antireflection coating: quarter-wave thickness t = λ/4n₂ with n₂ = √(n₁n₃).

C. High-Yield Points

1. In a conductor E and B are out of phase by 45° (B lags E), unlike vacuum where in phase.
2. Skin depth decreases with increasing frequency, σ, and μ — high-frequency currents flow on the surface.
3. At Brewster's angle only ⊥ (s) polarization is reflected; reflected light is perfectly linearly polarized with E ⊥ plane of incidence.
4. Phase velocity in plasma exceeds c; information travels at v_g < c; v_pv_g = c².
5. TIR is not total at the evanescent level — frustrated TIR transmits through a thin gap; no energy loss on average.
6. R + T = 1 holds for energy (intensity), not amplitudes; for the ⊥ component 1 + r_⊥ = t_⊥ at any angle (continuity of E_∥), but |r|² + |t|² ≠ 1 unless weighted by n cosθ.
7. Sky blue: Rayleigh scattering ∝ 1/λ⁴; polarization of scattered light maximum at 90° to the beam.

D. Comparison Table

DAY 11 — Polarization, Interference, Diffraction

A. Core Definitions

• Polarization: orientation of E vector. Linear, circular (equal amplitudes, ±90° phase), elliptical (general).
• Coherence: fixed phase relationship; temporal coherence length L_c = c τ_c ≈ λ²/Δλ; spatial coherence area ≈ (λ/θ_s)² (θ_s source angular size).
• Interference: superposition of coherent waves; diffraction: deviation from rectilinear propagation (Fresnel: near field; Fraunhofer: far field, plane wavefronts).

B. Key Equations / Rules

• Jones vectors (E_x, E_y): horizontal (1,0); vertical (0,1); +45° (1,1)/√2; right circular (1,−i)/√2 (for e^{i(kz−ωt)} convention); left circular (1,i)/√2.
• Jones matrices: horizontal polarizer [[1,0],[0,0]]; quarter-wave plate (fast axis horizontal) e^{−iπ/4}[[1,0],[0,i]]; half-wave plate [[1,0],[0,−1]] (rotates linear polarization by 2θ about the fast axis); rotation by θ: R(θ) = [[cos θ, −sin θ],[sin θ, cos θ]].
• QWP at 45° to linear light → circular; HWP converts RCP ↔ LCP. Plate thickness: QWP d = (m + ¼)λ/|nₒ − nₑ|; HWP d = (m + ½)λ/|nₒ − nₑ|.
• Malus' law: I = I₀cos²θ. Unpolarized through polarizer: I₀/2. Two crossed polarizers + intermediate at 45°: I = I₀/8.
• Stokes parameters S₀ = I, S₁ = I₀ − I₉₀, S₂, S₃; degree of polarization = √(S₁²+S₂²+S₃²)/S₀.
• Optical activity; birefringence (calcite): nₒ = 1.658, nₑ = 1.486. Nicol/Wollaston prisms.
• Young's double slit: path difference Δ = d sinθ ≈ yd/D; maxima dsinθ = mλ, fringe width β = λD/d; I = 4I₀cos²(δ/2), δ = 2πΔ/λ. Visibility V = (I_max − I_min)/(I_max + I_min).
• Thin film (normal incidence, n_f): reflection maxima 2n_ft = (m + ½)λ if one π phase jump (air–film–glass with n_air < n_f < n_g has two jumps ⇒ maxima at 2n_ft = mλ). Newton's rings: r_m² = mλR (reflected, dark centre); dark ring radius r_m = √(mλR); diameter ∝ √m; with liquid divide by n.
• Michelson: Δ = 2d cosθ; fringe shift N = 2Δd/λ. Fabry–Pérot: finesse F = π√R/(1−R); resolving power R = mF; free spectral range Δλ = λ²/2nt. Coherence length measured via fringe visibility.
• Single slit Fraunhofer: I = I₀(sinβ/β)², β = (πa sinθ)/λ; minima a sinθ = mλ (m ≠ 0); central maximum width 2λD/a; secondary maxima ≈ 4.5% (first: 4.7%) of central.
• N-slit grating: I = I₀(sinβ/β)²(sin Nα/ sinα)², α = (πd sinθ)/λ. Principal maxima d sinθ = mλ; (N − 1) minima and (N − 2) secondary maxima between adjacent principal maxima. Missing orders where d/a = integer ratio (m = j d/a).
• Grating: angular dispersion dθ/dλ = m/(d cosθ); resolving power R = λ/Δλ = mN; maximum order m_max < d/λ.
• Rayleigh criterion circular aperture: θ_min = 1.22λ/D; slit: λ/a. Prism R = t(dn/dλ).
• Circular aperture Airy pattern: I ∝ [2J₁(x)/x]²; first zero x = 3.83; 84% of power in the central disc.
• Fresnel half-period zones: radius rₙ = √(nλ·ab/(a+b)); zone plate focal length f = r₁²/λ; circular obstacle → bright spot of Poisson/Arago.
• Fresnel number F = a²/λL; Fraunhofer for F ≪ 1.
• Babinet's principle: complementary screens produce identical diffraction patterns (except forward direction).
• Bragg: 2d sinθ = mλ (X-rays).

C. High-Yield Points

1. A QWP converts circular ↔ linear, but only 45° orientation to its axes gives circular from linear light; otherwise elliptical.
2. Unpolarized light is not a Jones-vector state — incoherent mixture (use Stokes/coherency matrix).
3. Interference redistributes energy; the average over the pattern equals sum of individual intensities (energy conserved). Incoherent sources show no fringes.
4. In a grating, increasing slit number N sharpens the principal maxima but doesn't move them; spacing d sets positions, slit width a sets envelope.
5. Reflection from denser medium gives π phase shift ⇒ Newton's rings centre is dark in reflected light, bright in transmitted.
6. Fraunhofer single-slit: increasing slit width narrows the pattern (angular width ∝ λ/a).
7. Order m maximum limited by sinθ ≤ 1 ⇒ m < d/λ; a grating of 5000 lines/cm gives (d = 2 µm; visible 600 nm) ≤ 3 orders.

D. Comparison Table

DAY 12 — Waveguides, Retarded Potentials, Dipole Radiation

A. Core Definitions

• Waveguide mode types: TE (E_z = 0), TM (B_z = 0), TEM (both zero; needs two conductors, e.g., coax, no cutoff).
• Cutoff frequency: below which a mode is evanescent.
• Retarded potential: potential at (r,t) determined by sources at earlier time t_r = t − |r − r′|/c.
• Liénard–Wiechert potentials for a point charge.

B. Key Equations / Rules

• Rectangular guide (a > b), mode (m,n): cutoff f_c = (c/2)√[(m/a)² + (n/b)²] (filled: replace c with c/n). TE_mn: m,n ≥ 0 not both zero; TM_mn: m,n ≥ 1. Dominant mode TE₁₀: f_c = c/2a, λ_c = 2a.
• Guide wavelength λ_g = λ₀/√(1 − (f_c/f)²); k_g = √(ω²/c² − k_c²); v_p = c/√(1 − (f_c/f)²) > c; v_g = c√(1 − (f_c/f)²) < c; v_pv_g = c². Wave impedance TE: Z_TE = Z₀/√(1 − (f_c/f)²); TM: Z_TM = Z₀√(1 − (f_c/f)²).
• Circular guide: TE₁₁ dominant, f_c = 1.841c/2πa; TM₀₁ f_c = 2.405c/2πa.
• Cavity resonator (a×b×d): f = (c/2)√[(m/a)² + (n/b)² + (p/d)²]; Q = ω × stored energy/power loss.
• Transmission line: Z₀ = √(L/C) (lossless), v = 1/√(LC); reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀); VSWR = (1+|Γ|)/(1−|Γ|); matched Γ = 0; short Γ = −1; open Γ = +1; quarter-wave transformer Z_in = Z₀²/Z_L.
• Retarded potentials: φ(r,t) = (1/4πε₀)∫ρ(r′,t_r)/R dτ′ ; A = (μ₀/4π)∫J(r′,t_r)/R dτ′.
• Liénard–Wiechert: φ = (q/4πε₀)·1/[R(1 − n̂·β)]_ret; A = (μ₀qv)/[4πR(1 − n̂·β)]_ret.
• Electric dipole radiation (p(t) = p₀cos ωt, far field kr ≫ 1, d ≪ λ): E_θ = −μ₀p₀ω²sinθ/(4πr)·cos[ω(t − r/c)]; B_φ = E_θ/c; ⟨S⟩ = (μ₀p₀²ω⁴/32π²c)(sin²θ/r²) r̂.
• Total power ⟨P⟩ = μ₀p₀²ω⁴/12πc = p₀²ω⁴/(12πε₀c³). Pattern sin²θ (null along dipole axis); P ∝ ω⁴ ∝ 1/λ⁴.
• Magnetic dipole: ⟨P⟩ = μ₀m₀²ω⁴/12πc³. Electric quadrupole ∝ ω⁶.
• Larmor formula: P = q²a²/(6πε₀c³) (non-relativistic). Relativistic (Liénard): P = (q²γ⁶/6πε₀c³)[a² − |v×a|²/c²]. Linear acceleration P ∝ γ⁶a²; circular motion P = q²a²γ⁴/(6πε₀c³) (synchrotron): energy loss per turn ΔE = q²γ⁴β³/(3ε₀R) (∝ E⁴/m⁴R).
• Half-wave antenna: R_rad ≈ 73 Ω; short dipole pattern sin²θ; half-wave pattern [cos((π/2)cosθ)/sinθ]².
• Radiation reaction (Abraham–Lorentz): F_rad = (μ₀q²/6πc)(da/dt) (non-relativistic).
• Cherenkov: cosθ = 1/(nβ), threshold β > 1/n; Bremsstrahlung from sudden deceleration; synchrotron radiation beamed in cone angle 1/γ.
• Radiation resistance P = ½I₀²R_rad.

C. High-Yield Points

1. Waveguide cannot propagate below cutoff; the dominant mode in a rectangular guide is TE₁₀; no TEM mode in a single-conductor hollow guide (coaxial lines support TEM).
2. v_phase > c is allowed since it carries no information; v_group < c; v_pv_g = c² for hollow guides.
3. TE₀₀ and TM₀ₙ, TMₘ₀ do not exist in rectangular guides; TE₁₀ and TE₀₁ exist.
4. Radiation requires accelerated charge; uniformly moving charge doesn't radiate in vacuum; static charges/dc currents produce only 1/r² fields — radiation fields fall as 1/r.
5. Dipole radiation: no radiation along the axis; maximum at θ = 90°; power ∝ ω⁴ (explains blue sky with Rayleigh scattering).
6. For synchrotron, power ∝ γ⁴ and inversely ∝ m⁴ ⇒ electrons radiate far more than protons at same energy.
7. Retarded time derives from finite speed of EM propagation; instantaneous (Coulomb gauge φ) is not signal.

D. Comparison Table

MODULE 3 — QUANTUM MECHANICS & ATOMIC/MOLECULAR PHYSICS

DAY 13 — Wave-function, Wells, Steps, Barriers

A. Core Definitions & Postulates

1. State = ray in Hilbert space; wavefunction ψ(r,t) with ∫|ψ|²dτ = 1 (|ψ|² = probability density).
2. Observables ↔ Hermitian operators; measurement outcomes are eigenvalues; expectation ⟨A⟩ = ⟨ψ|A|ψ⟩.
3. Time evolution: iℏ∂ψ/∂t = Ĥψ. Time-independent: Ĥψ = Eψ, ψ(r,t) = ψ(r)e^{−iEt/ℏ}.
4. Collapse onto eigenstate after measurement; probability |⟨aₙ|ψ⟩|². - Acceptable wavefunction: single-valued, finite, continuous; ψ′ continuous wherever V is finite (for V = −gδ(x): Δψ′ = −(2mg/ℏ²)ψ(0)). - Probability current j = (ℏ/2mi)(ψ∇ψ − ψ∇ψ); continuity ∂ρ/∂t + ∇·j = 0.

B. Key Equations / Rules

• Momentum operator p̂ = −iℏ∇; [x,p] = iℏ; ⟨p²⟩ etc. Ehrenfest: d⟨x⟩/dt = ⟨p⟩/m; d⟨p⟩/dt = −⟨∇V⟩.
• Infinite well (0 < x < L): ψₙ = √(2/L) sin(nπx/L); Eₙ = n²π²ℏ²/2mL² = n²h²/8mL². n = 1,2,…; nodes n − 1; ⟨x⟩ = L/2; ⟨x²⟩ = L²(1/3 − 1/2n²π²); ⟨p²⟩ = (nπℏ/L)²; ΔxΔp = (ℏ/2)√(n²π²/3 − 2) > ℏ/2. Well from −a to +a (width 2a): odd/even: ψ = (1/√a)cos(nπx/2a) (n odd), sin (n even).
• 3D box: E = (π²ℏ²/2m)(nₓ²/a² + n_y²/b² + n_z²/c²); cube degeneracy of (1,1,2) = 3.
• Finite square well (depth V₀, width 2a, |x| < a): even states: k tan(ka) = κ; odd: −k cot(ka) = κ with k = √(2mE)/ℏ, κ = √(2m(V₀ − E))/ℏ; define z₀ = a√(2mV₀)/ℏ, z² + (κa)² = z₀². Number of bound states = ⌈2z₀/π⌉ (= 1 + int(2z₀/π)); at least one bound state (even) always exists in 1D (attractive); 3D needs threshold.
• Delta potential V = −αδ(x): bound state ψ = √κ e^{−κ|x|}, κ = mα/ℏ², E = −mα²/2ℏ²; only one bound state. Scattering: T = 1/(1 + mα²/2ℏ²E), R = 1 − T.
• Potential step (V₀ for x > 0): E > V₀: R = [(k₁ − k₂)/(k₁ + k₂)]², T = 4k₁k₂/(k₁ + k₂)²; E < V₀: total reflection with evanescent decay length 1/κ, κ = √(2m(V₀−E))/ℏ.
• Rectangular barrier (height V₀, width a), E < V₀: T = [1 + V₀² sinh²(κa)/4E(V₀ − E)]⁻¹ ≈ 16E(V₀−E)/V₀² · e^{−2κa} for κa ≫ 1. E > V₀: T = [1 + V₀² sin²(k′a)/4E(E − V₀)]⁻¹, T = 1 when k′a = nπ (resonant transmission). Gamow: T ≈ exp(−2∫√(2m(V−E))/ℏ dx).
• Free particle packet: Gaussian spreads: Δx(t) = Δx₀√(1 + (ℏt/2mΔx₀²)²); v_g = ℏk/m (phase velocity ω/k = ℏk/2m).
• Parity operator: even/odd eigenfunctions for symmetric V; ground state nodeless; nth state has n − 1 nodes (1D); nondegenerate bound states in 1D.
• Completeness: Σ|ψₙ⟩⟨ψₙ| = 1; expansion cₙ = ⟨ψₙ|ψ⟩; ⟨E⟩ = Σ|cₙ|²Eₙ.

C. High-Yield Points

1. Bound states in 1D are non-degenerate; 3D central problems may be degenerate. Energy eigenfunctions of 1D bound problems can be chosen real.
2. In the infinite well, E ∝ n²; spacing increases with n; in the harmonic oscillator, E ∝ (n+½); in hydrogen, E ∝ −1/n².
3. Tunnelling probability decays exponentially with barrier width and √(m(V₀−E)); explains α-decay, STM, tunnel diode.
4. At potential step E > V₀: classical particle would be fully transmitted, but QM gives nonzero reflection; T + R = 1 for flux (not for |ψ|² ratios) because k differs.
5. ψ must go to zero at an infinite wall (ψ′ discontinuous there); for finite V, both ψ and ψ′ continuous. For a δ potential, ψ′ jumps.
6. Wavefunction in the classically forbidden region decays exponentially, penetration depth 1/κ = ℏ/√(2m(V₀−E)).
7. Zero-point energy: ground state E₁ ≠ 0 for any confined particle (uncertainty). ⟨T⟩ in box ground state = E₁.

D. Comparison Table

DAY 14 — Harmonic Oscillator, Uncertainty, Dirac Notation

A. Core Definitions

• Commutator [A,B] = AB − BA. Compatible observables commute ⇒ simultaneous eigenstates.
• Dirac notation: ket |ψ⟩, bra ⟨ψ|, ⟨φ|ψ⟩ = ∫φ*ψ dτ; projector |a⟩⟨a|; completeness Σ|aᵢ⟩⟨aᵢ| = 1 (or ∫|x⟩⟨x|dx = 1).
• Hermitian adjoint: (AB)† = B†A†; ⟨φ|A|ψ⟩* = ⟨ψ|A†|φ⟩. Unitary U†U = 1 preserves inner products.

B. Key Equations / Rules

• 1D oscillator H = p²/2m + ½mω²x². Eₙ = (n + ½)ℏω; ψₙ(x) = (mω/πℏ)^{1/4}(2ⁿn!)^{−1/2}Hₙ(ξ)e^{−ξ²/2}, ξ = x√(mω/ℏ). ψ₀ = (mω/πℏ)^{1/4}e^{−mωx²/2ℏ}.
• Ladder operators: a = √(mω/2ℏ)(x + ip/mω); a† = √(mω/2ℏ)(x − ip/mω); [a,a†] = 1; H = ℏω(a†a + ½); N = a†a.
• a|n⟩ = √n |n−1⟩; a†|n⟩ = √(n+1)|n+1⟩; a|0⟩ = 0; |n⟩ = (a†)ⁿ|0⟩/√n!.
• x = √(ℏ/2mω)(a + a†); p = i√(mℏω/2)(a† − a).
• ⟨x²⟩ = (ℏ/2mω)(2n+1); ⟨p²⟩ = (mℏω/2)(2n+1); ⟨T⟩ = ⟨V⟩ = Eₙ/2 (virial); ΔxΔp = (n+½)ℏ; ground state minimum uncertainty ℏ/2.
• Matrix elements ⟨n|x|m⟩ ≠ 0 only for m = n ± 1 (selection rule Δn = ±1).
• Classical turning points xₙ = ±√((2n+1)ℏ/mω); for large n probability → classical 1/√(A²−x²).
• Coherent state |α⟩: a|α⟩ = α|α⟩, |α⟩ = e^{−|α|²/2}Σαⁿ/√n! |n⟩; Poisson distribution of n with mean |α|²; minimum uncertainty; ⟨x⟩ oscillates classically.
• 3D isotropic oscillator: E = (N + 3/2)ℏω, degeneracy (N+1)(N+2)/2.
• Uncertainty: ΔAΔB ≥ ½|⟨[A,B]⟩|. ΔxΔp ≥ ℏ/2; ΔEΔt ≥ ℏ/2 (t not an operator; Δt = characteristic time for change); ΔL_xΔL_y ≥ (ℏ/2)|⟨L_z⟩|. Natural linewidth ΔE ≈ ℏ/τ.
• Commutator algebra: [A,BC] = [A,B]C + B[A,C]; [x,pⁿ] = iℏn p^{n−1}; [x,f(p)] = iℏ f′(p); [L_i,L_j] = iℏε_{ijk}L_k; [L²,L_i] = 0; [L_z, x] = iℏy.
• Heisenberg picture: dA/dt = (i/ℏ)[H,A] + ∂A/∂t. Constant of motion if [A,H] = 0.
• Position/momentum representation: ⟨x|p⟩ = e^{ipx/ℏ}/√(2πℏ); φ(p) = (1/√(2πℏ))∫ψ(x)e^{−ipx/ℏ}dx; x̂ → iℏ∂/∂p.
• Spin-½ (Pauli): S = (ℏ/2)σ; [S_x,S_y] = iℏS_z; eigenvalues ±ℏ/2; |↑⟩ along n: cos(θ/2)|↑⟩ + e^{iφ}sin(θ/2)|↓⟩.
• Density matrix ρ: pure ρ² = ρ; tr ρ = 1; ⟨A⟩ = tr(ρA).
• Entanglement; Bell state (|↑↓⟩ − |↓↑⟩)/√2 non-separable.
• Time evolution: |ψ(t)⟩ = e^{−iHt/ℏ}|ψ(0)⟩; for H = (ℏω/2)σ_z a spin initially along +x precesses about z at angular frequency ω.

C. High-Yield Points

1. If [A,B] = 0 they share a complete set of eigenfunctions; if [A,B] = iC (C constant) no state is simultaneous eigenstate of both.
2. ΔEΔt is not derived from commutator; Δt is not a standard deviation of an operator.
3. Oscillator levels are equally spaced; zero-point energy ½ℏω; ⟨x⟩ = ⟨p⟩ = 0 in every energy eigenstate.
4. For ground state of oscillator Δx = √(ℏ/2mω), Δp = √(mℏω/2).
5. Hermitian operator eigenvalues real; eigenfunctions of different eigenvalue orthogonal; a product of two Hermitian operators is Hermitian iff they commute. AB + BA and i[A,B] are Hermitian.
6. Eigenvalues of a unitary operator lie on unit circle; of projector 0 or 1; of parity ±1.
7. Isotropic 3D oscillator degeneracy (N+1)(N+2)/2; anisotropic loses degeneracy. [x,p] = iℏ is impossible in finite dimension (trace argument).

D. Comparison Table

| Operator | Action on |n⟩ | Commutator | |---|---|---| | a (lowering) | √n |n−1⟩ | [a,a†] = 1 | | a† (raising) | √(n+1) |n+1⟩ | [N,a†] = a† | | N = a†a | n|n⟩ | [N,a] = −a | | H | ℏω(n+½) | [H,a†] = ℏωa† |

DAY 15 — Hydrogen Atom, Angular Momentum, Spin Addition

A. Core Definitions

• Orbital angular momentum L = r × p; operators L_x, L_y, L_z, L²: [L_i,L_j] = iℏε_{ijk}L_k.
• General angular momentum J: J²|j,m⟩ = ℏ²j(j+1)|j,m⟩; J_z|j,m⟩ = ℏm|j,m⟩; m = −j…j; j = 0, ½, 1, 3/2,… Ladder J± = J_x ± iJ_y: J±|j,m⟩ = ℏ√(j(j+1) − m(m±1))|j,m±1⟩. Orbital l integer only; spin allows half-integer.
• Spin s = ½ for electron; S = ℏσ/2; g_s ≈ 2.

B. Key Equations / Rules

• L² Y_lm = ℏ²l(l+1)Y_lm; L_z Y_lm = mℏY_lm. Y₀₀ = 1/√4π; Y₁₀ = √(3/4π)cosθ; Y₁,±₁ = ∓√(3/8π)sinθ e^{±iφ}; Y₂₀ = √(5/16π)(3cos²θ − 1). Orthonormal on sphere. Parity (−1)^l. Addition theorem: P_l(cosγ) = (4π/(2l+1))ΣY*_lm(1)Y_lm(2). L_z = −iℏ∂/∂φ; L² = −ℏ²[(1/sinθ)∂θ(sinθ∂θ) + (1/sin²θ)∂φ²].
• Hydrogen: V = −e²/(4πε₀r) (Z: −Ze²/..). ψ_nlm = R_nl(r)Y_lm(θ,φ).
• Eₙ = −(μe⁴/2(4πε₀)²ℏ²)(Z²/n²) = −13.6 Z²/n² eV; n = 1,2,…; l = 0…n−1; m = −l…l.
• Degeneracy n² (2n² with spin); a₀ = 4πε₀ℏ²/me² = 0.529 Å; Rydberg R_∞ = 109737 cm⁻¹; reduced mass μ correction (positronium E = −6.8/n² eV; muonium; deuterium isotope shift).
• R₁₀ = 2(Z/a₀)^{3/2}e^{−Zr/a₀}; R₂₀ = (Z/2a₀)^{3/2}(2 − Zr/a₀)e^{−Zr/2a₀}; R₂₁ = (Z/2a₀)^{3/2}(Zr/a₀√3)e^{−Zr/2a₀}.
• Radial nodes n − l − 1; angular nodes l. Radial equation effective potential ℏ²l(l+1)/2μr². R_nl ∝ r^l near origin; ψ(0) ≠ 0 only for s-states, |ψ_ns(0)|² = Z³/(πn³a₀³).
• ⟨r⟩ = (a₀/2Z)[3n² − l(l+1)]; ⟨1/r⟩ = Z/n²a₀; ⟨1/r²⟩ = Z²/[n³(l+½)a₀²]; ⟨1/r³⟩ = Z³/[n³l(l+½)(l+1)a₀³]. ⟨T⟩ = −Eₙ, ⟨V⟩ = 2Eₙ (virial).
• Most probable radius of 1s: a₀; ⟨r⟩₁ₛ = 1.5a₀.
• Selection rules (E1): Δl = ±1, Δm = 0, ±1, Δn arbitrary. Lyman: n→1 (UV); Balmer n→2 (visible: Hα 656.3 nm); Paschen n→3 (IR). 1/λ = R(1/n₁² − 1/n₂²).
• Accidental degeneracy in l (Coulomb symmetry SO(4); Runge–Lenz).
• Angular momentum addition: J = J₁ + J₂: j = |j₁−j₂|,…, j₁+j₂ (total states (2j₁+1)(2j₂+1)). Two spin-½: triplet S = 1 (symmetric; m = 1,0,−1: |↑↑⟩, (|↑↓⟩+|↓↑⟩)/√2, |↓↓⟩) and singlet S = 0 ((|↑↓⟩−|↓↑⟩)/√2, antisymmetric). Clebsch–Gordan coefficients: |j,m⟩ = ΣC|j₁m₁⟩|j₂m₂⟩. L·S = ½[J² − L² − S²] = (ℏ²/2)[j(j+1) − l(l+1) − s(s+1)].
• Spin-½ spinors, rotation by 2π gives −1 (SU(2) double cover); Stern–Gerlach splits beam into 2s+1 components.
• Magnetic moment μ_L = −(e/2m)L; μ_s = −g_s(e/2m)S; Bohr magneton μ_B = eℏ/2m = 9.274 × 10⁻²⁴ J/T = 5.788 × 10⁻⁵ eV/T.
• Bohr model: rₙ = n²a₀/Z, vₙ = Zαc/n (α = e²/4πε₀ℏc ≈ 1/137), Lₙ = nℏ.

C. High-Yield Points

1. For fixed n there are n² orbital states; n² × 2 including spin. Count of electrons per shell 2n².
2. j₁ ⊗ j₂ decomposition: dimension check — e.g., 1 ⊗ ½ = 3/2 ⊕ ½: 6 = 4 + 2. For l = 1, s = ½: j = 3/2, 1/2.
3. Quantum number l ≤ n−1 and hydrogen 2s and 2p degenerate (nonrelativistic) but not for alkali atoms (screening removes l-degeneracy).
4. Only s-states have nonzero probability density at nucleus (hyperfine contact term ∝ |ψ(0)|²).
5. Spherical harmonics have parity (−1)^l; hydrogenic orbitals ψ_nlm parity (−1)^l — basis for Laporte rule.
6. ⟨L_x⟩ = ⟨L_y⟩ = 0 in L_z eigenstates; ⟨L_x²⟩ = ⟨L_y²⟩ = ½ℏ²[l(l+1) − m²]; uncertainty ΔL_xΔL_y ≥ ½ℏ²|m|.
7. Singlet/triplet exchange symmetry: total wavefunction of two electrons antisymmetric ⇒ spatial symmetric ↔ spin singlet (para), spatial antisymmetric ↔ triplet (ortho; lower energy via exchange hole, Hund).

D. Comparison Table

DAY 16 — Perturbation Theory, Variational Method, WKB

A. Core Definitions

• Perturbation theory: H = H₀ + λH′; H₀ solved exactly; corrections expansions in λ.
• Variational principle: E₀ ≤ ⟨ψ_trial|H|ψ_trial⟩/⟨ψ_trial|ψ_trial⟩ for any trial state.
• WKB: semiclassical approximation valid when |dλ/dx| ≪ 1 (slowly varying potential).

B. Key Equations / Rules

• Non-degenerate: Eₙ⁽¹⁾ = ⟨n|H′|n⟩; |n⁽¹⁾⟩ = Σ_{m≠n}[⟨m|H′|n⟩/(Eₙ − Eₘ)]|m⟩; Eₙ⁽²⁾ = Σ_{m≠n}|⟨m|H′|n⟩|²/(Eₙ⁽⁰⁾ − Eₘ⁽⁰⁾). Second-order correction to ground state always negative. Validity: |⟨m|H′|n⟩| ≪ |Eₙ − Eₘ|.
• Degenerate: diagonalise H′ in the degenerate subspace: det(⟨i|H′|j⟩ − E⁽¹⁾δᵢⱼ) = 0 (“good” states are eigenstates of an operator commuting with both H₀ and H′).
• Examples: anharmonic oscillator H′ = λx⁴: E⁽¹⁾ₙ = λ(ℏ/2mω)²·3(2n² + 2n + 1). Infinite well with perturbation V₀ over half: E⁽¹⁾ = V₀/2. Stark effect: ground state of H atom — second order only (quadratic): ΔE = −(9/4)(4πε₀)a₀³ℰ²; n = 2 linear Stark splitting ΔE = ±3eℰa₀ (degenerate, 4-fold → 3 levels). Zeeman as perturbation H′ = (e/2m)(L + 2S)·B.
• Helium ground state: first-order E = −74.8 eV (⟨e²/r₁₂⟩ = (5/4)Z·13.6 eV = 34 eV); variational with effective charge Z_eff = Z − 5/16 = 1.6875 gives −77.5 eV (expt −79.0 eV).
• Variational: minimise E(α) = ⟨H⟩(α); Gaussian trial ψ = e^{−αx²} for harmonic oscillator gives exact result (α = mω/2ℏ). For ground state; excited states only if trial orthogonal to lower states. For hydrogen with e^{−αr}: exact at α = 1/a₀. Upper bound always.
• Time-dependent: c_f(t) ≈ (1/iℏ)∫⟨f|H′(t′)|i⟩e^{iω_{fi}t′}dt′; harmonic perturbation H′ = V cos ωt ⇒ transition rate (Fermi's golden rule): Γ_{i→f} = (2π/ℏ)|⟨f|H′|i⟩|²ρ(E_f) (energy conserving). Sudden approximation: state unchanged; adiabatic theorem: system stays in the instantaneous eigenstate (if gap persists).
• WKB: ψ(x) ≈ [C/√p(x)] exp(± (i/ℏ)∫p dx), p = √(2m(E−V)). Connection formula at turning points (linear).
• Bohr–Sommerfeld quantisation: ∮p dx = (n + ½)h (two soft turning points); for one hard wall (e.g., l = 0 radial/ half oscillator) ∮ = (n + ¾)h; two hard walls: ∮p dx = nh. Harmonic oscillator exactly gives (n+½)ℏω.
• Tunnelling: T ≈ exp[−(2/ℏ)∫ₓ₁ˣ² √(2m(V − E))dx]; Gamow α-decay: T ≈ e^{−2G}, G = (1/ℏ)∫√(2m(V−E))dr from nuclear radius R to classical turning point b = Z₁Z₂e²/4πε₀E.
• Validity breaks at turning points (p → 0) — need Airy connection formulas.
• Spin-orbit as perturbation: H′ = (1/2m²c²)(1/r)(dV/dr)L·S; relativistic kinetic correction H′ = −p⁴/8m³c².

C. High-Yield Points

1. First-order energy correction is just expectation value of H′ in the unperturbed state; second order always lowers the ground state energy.
2. Degenerate theory applies when H′ connects states of the same H₀ energy; choose basis diagonalising H′ (e.g., |j,m⟩ for spin-orbit rather than |l,m_l,m_s⟩).
3. Variational energy is an upper bound to E₀ — never below; accuracy improves with better trial functions; error in energy is second order in error in the wavefunction.
4. WKB quantisation ∮p dx = (n+½)h is exact for harmonic oscillator and hydrogen; for infinite well (hard walls) ∮ = nh (no ½).
5. Fermi's golden rule requires a continuum (or broadened) final states and weak perturbation; rate constant in time (long times).
6. Linear Stark effect appears only for degenerate states with permanent dipole-like mixing (H n = 2); ground state has only quadratic Stark.
7. Perturbation expansion in x⁴ (positive anharmonic) is asymptotic, not convergent.

D. Comparison Table

DAY 17 — Fine Structure, Coupling Schemes, Term Symbols, Zeeman

A. Core Definitions

• Fine structure: splitting from spin–orbit coupling + relativistic kinetic correction + Darwin term; scale ~α² × gross energy (~10⁻⁴ eV).
• LS (Russell–Saunders) coupling: L = ΣLᵢ, S = ΣSᵢ, J = L + S (valid for light atoms, Z ≲ 30). jj coupling: jᵢ = lᵢ + sᵢ, J = Σjᵢ (heavy atoms).
• Term symbol: ²ˢ⁺¹L_J; L = S, P, D, F, G… for 0,1,2,3,4. Multiplicity 2S+1. Hund's rules (ground term): (1) max S, (2) max L (consistent with S), (3) J = |L−S| if shell less than half-filled, J = L+S if more than half-filled (half-filled L = 0, J = S).

B. Key Equations / Rules

• Fine structure of hydrogen: E_nj = Eₙ[1 + (α²Z²/n²)(n/(j+½) − 3/4)]; ΔE depends on n and j only (2s½ and 2p½ degenerate; Lamb shift lifts it ≈ 1057 MHz). Fine-structure constant α = e²/4πε₀ℏc = 1/137.036.
• Spin–orbit energy: ΔE_so = (ℏ²/2)ξ[j(j+1) − l(l+1) − s(s+1)], ξ ∝ Z⁴/n³l(l+½)(l+1) (hydrogenic). Hydrogen n = 2 (2p₃/₂–2p₁/₂): ΔE = α²(13.6 eV)/16 ≈ 4.5 × 10⁻⁵ eV (≈ 10.9 GHz). Landé interval rule: E(J) − E(J−1) ∝ J.
• Sodium D lines: 3²P₃/₂ → 3²S₁/₂ (589.0 nm), 3²P₁/₂ → 3²S₁/₂ (589.6 nm); doublet separation 0.6 nm (≈ 17 cm⁻¹).
• Alkali energy levels: Eₙ = −13.6/(n − δ_l)² eV, quantum defect δ_s > δ_p > δ_d.
• Term symbols for configurations:
• np²: ³P₀,₁,₂, ¹D₂, ¹S₀ (ground ³P₀ for C; ³P₂ for O (np⁴)).
• nd² etc. Pauli exclusion for equivalent electrons restricts terms; number of microstates np²: 15 (C(6,2)).
• 2p ground states: B ²P₁/₂, N ⁴S₃/₂, O ³P₂, F ²P₃/₂.
• Helium excited: singlets (para ¹S, ¹P…) and triplets (ortho ³S₁…); no 1³S (Pauli); 2³S₁ metastable.
• Selection rules (E1): ΔJ = 0, ±1 (not 0 ↔ 0); Δl = ±1; parity changes (Laporte); ΔM_J = 0, ±1; LS coupling: ΔL = 0, ±1; ΔS = 0 (intercombination lines weak).
• Zeeman effect (weak B, B ≪ internal field): ΔE = g_J μ_B B M_J, where Landé g_J = 1 + [J(J+1) + S(S+1) − L(L+1)] / [2J(J+1)]. g_J: ²S₁/₂ = 2; ²P₁/₂ = 2/3; ²P₃/₂ = 4/3; ¹L: 1; pure spin 2.
• Normal Zeeman (S = 0, g = 1): equally spaced triplet; ΔE = μ_BB; Δν = eB/4πm; Lorentz triplet (π: Δm = 0, σ: Δm = ±1). Splitting: μ_B/hc = 0.4669 cm⁻¹ per tesla.
• Anomalous (S ≠ 0): D₁: 4 components; D₂: 6 components (Na).
• Transitions: π (ΔM = 0, polarized ∥ B), σ± (ΔM = ±1, circularly polarized along B).
• Paschen–Back (strong B): L,S decouple; ΔE = μ_B B (m_l + 2m_s); pattern reverts to normal triplet. Stark effect: electric-field splitting, quadratic in general, linear in hydrogen.
• Hyperfine: ΔE = (A/2)[F(F+1) − I(I+1) − J(J+1)], hydrogen 21 cm line (1420 MHz); Na ground state 1772 MHz.
• Isotope shift (mass shift ∝ 1/M, field shift) and Lamb shift (QED).
• Width of lines: natural ΔE ≈ ℏ/τ; Doppler Δν/ν = √(8kT ln2/mc²); pressure.
• Electron spin resonance: hν = g μ_B B (g = 2.0023 free electron: 28 GHz/T); NMR: hν = γℏB, proton 42.58 MHz/T; chemical shift δ = (ν − ν_ref)/ν_ref × 10⁶ ppm.

C. High-Yield Points

1. Landé g-factor for pure spin (L = 0) = 2; pure orbital (S = 0) = 1; for ²P₁/₂ = 2/3, ²P₃/₂ = 4/3 — the key numbers.
2. Closed shells contribute L = S = 0; filled subshells ignored — equivalent electrons (np²) allow only terms with L + S even.
3. Fine-structure depends on j; hydrogen's 2s½ and 2p½ are degenerate in Dirac theory, the Lamb shift (QED) splits them.
4. Normal Zeeman triplet occurs only for singlet transitions (S = 0); Na D lines show anomalous pattern; D₁ splits into 4, D₂ into 6 lines.
5. Spin–orbit coupling ∝ Z⁴; thus fine structure grows rapidly with Z; LS → jj crossover for heavy atoms.
6. Hund's third rule: less-than-half-filled → smallest J; more-than-half → largest J. For d⁵ high-spin ⁶S₅/₂.
7. Zeeman splitting for J = 0 vanishes (g undefined, ΔE = 0). Intercombination lines ΔS ≠ 0 forbidden in pure LS coupling but observed in heavy atoms (Hg 253.7 nm 6³P₁ → 6¹S₀).

D. Comparison Table

DAY 18 — Molecular Spectra, Raman, Lasers

A. Core Definitions

• Born–Oppenheimer: nuclear motion much slower than electronic; ψ_total ≈ ψ_el(r;R)·χ_nuc(R); E = E_el + E_vib + E_rot with E_el ≫ E_vib ≫ E_rot (ratio ~ 1 : √(m/M) : m/M ~ eV : 0.1 eV : 10⁻³ eV).
• Franck–Condon principle: electronic transitions are vertical (nuclei fixed, R unchanged); intensity ∝ |⟨χ′|χ″⟩|² overlap.
• Raman effect: inelastic scattering; Stokes (ν₀ − ν_vib), anti-Stokes (ν₀ + ν_vib).
• Laser: light amplification by stimulated emission — requires population inversion and optical feedback.

B. Key Equations / Rules

• Rigid rotor: E_J = BJ(J+1) (B = ℏ²/2I = h/8π²Ic in cm⁻¹ units), I = μr²; degeneracy 2J+1. Spectrum (microwave): ν̃ = 2B(J+1) (J → J+1) — lines equally spaced by 2B; selection ΔJ = ±1, permanent dipole needed. B for CO ≈ 1.93 cm⁻¹. Isotope: B ∝ 1/μ. Centrifugal distortion E = BJ(J+1) − DJ²(J+1)².
• Relative population N_J ∝ (2J+1)e^{−BJ(J+1)hc/kT}; J_max = √(kT/2hcB) − ½.
• Vibrational (harmonic): E_v = (v + ½)hν₀; ν₀ = (1/2π)√(k/μ); Δv = ±1; anharmonic (Morse): E_v = (v+½)ω_e − (v+½)²ω_e x_e; overtones Δv = ±2,… weak; dissociation energy D_e ≈ ω_e/(4x_e) (in wavenumber units). Hot bands. Zero-point E₀ = ½ω.
• Vib–rot (rigid rotor, harmonic oscillator): R branch (ΔJ = +1) ν̃ = ν̃₀ + 2B(J+1), P branch (ΔJ = −1) ν̃ = ν̃₀ − 2BJ; line spacing 2B; no Q branch (ΔJ = 0) for ¹Σ molecules (HCl, CO), leaving a gap of 4B at the band centre ν̃₀; NO (²Π) shows a Q branch. Heavier isotopologue: lower ν̃₀ and smaller B.
• Combined E = (v+½)hν₀ + BJ(J+1) (rigid rotor-harmonic oscillator).
• IR selection rule: vibration must change the dipole moment (heteronuclear diatomics active; H₂, N₂, O₂ inactive).
• Raman selection: polarisability must change during vibration: Δv = ±1; ΔJ = 0, ±2 (O, Q, S branches); homonuclear diatomics Raman active. Pure rotational Raman: ΔJ = ±2; lines at ν̃₀ ± 2B(2J+3) → first line 6B from the Rayleigh line, spacing 4B (Stokes ν̃ = ν̃₀ − 2B(2J+3)).
• Mutual exclusion principle: in a molecule with centre of inversion, vibrations IR-active are Raman-inactive and vice versa (CO₂ symmetric stretch Raman-active only; antisymmetric stretch and bend IR-active).
• Stokes intensity > anti-Stokes (population of v = 0 ≫ v = 1); I_AS/I_S = ((ν₀+ν_v)/(ν₀−ν_v))⁴ e^{−hν_v/kT}.
• Raman shift independent of excitation wavelength; intensity ∝ ν⁴.
• Electronic spectra: Δv unrestricted (Franck–Condon); bands with vibrational progressions; ΔΛ = 0, ±1 (Λ: Σ, Π, Δ).
• Diatomic nuclear statistics: ortho/para H₂ (I = ½): ortho (I = 1, odd J), para (I = 0, even J), ratio 3:1 at high T.
• Lasers: Einstein coefficients: A₂₁ spontaneous, B₁₂ absorption, B₂₁ stimulated. Equilibrium: B₁₂ = B₂₁ (equal degeneracy; g₁B₁₂ = g₂B₂₁); A₂₁/B₂₁ = 8πhν³/c³ ; ratio of spontaneous to stimulated rate = e^{hν/kT} − 1 (at thermal equilibrium; spontaneous dominates for visible light at room T).
• Rate: dN₂/dt = −A₂₁N₂ − B₂₁ρ(ν)N₂ + B₁₂ρN₁. Lifetime τ = 1/A₂₁.
• Population inversion N₂ > N₁ (negative temperature); needs ≥ 3 levels (3-level: ruby 694.3 nm; 4-level: Nd:YAG 1064 nm; He–Ne 632.8 nm, CO₂ 10.6 µm). Threshold gain: R₁R₂e^{2(g−α)L} = 1.
• Resonator longitudinal modes: ν_q = qc/2nL, spacing Δν = c/2nL; coherence length L_c = c/Δν_laser; Gaussian beam divergence θ ≈ λ/πw₀; Fabry–Pérot cavity Q. Stability condition 0 ≤ (1 − L/R₁)(1 − L/R₂) ≤ 1.
• Gain: g = σ(N₂ − N₁); σ ~ λ²A₂₁/8πn²Δν. Doppler broadened linewidth for He–Ne ≈ 1.5 GHz.
• Q-switching (giant pulses), mode locking (fs pulses, Δν·Δt ≈ 1).

C. High-Yield Points

1. Rotational constants: B ∝ 1/I; heavier isotopologue ⇒ smaller B (closer lines); D₂ vs H₂ and ¹²CO vs ¹³CO standard problems.
2. Homonuclear diatomics (N₂, O₂, H₂) show no pure rotational IR/microwave and no IR vibrational spectrum but are Raman-active.
3. Pure rotational Raman spacing is 4B; microwave spacing 2B; first Raman Stokes line at 6B from the exciting line.
4. In vib–rot IR spectrum there's no Q branch (ΔJ = 0) for ¹Σ molecules (HCl, CO) — a gap in the band centre of width 4B; NO shows a Q branch.
5. Vibrational anharmonicity makes levels converge; spacing decreases with v; overtones at slightly less than integral multiples of ν₀.
6. At thermal equilibrium A₂₁/B₂₁ = 8πhν³/c³: spontaneous emission dominates at high ν (X-ray lasers are hard to build); stimulated emission is coherent (same phase, direction, polarisation, frequency).
7. Three-level laser needs > half of atoms excited (high pumping threshold); four-level laser lower threshold; population inversion is not possible in a two-level system (optical pumping limit N₂ = N₁).

D. Comparison Table

MODULE 4 — THERMODYNAMICS, STATISTICAL PHYSICS & CONDENSED MATTER

DAY 19 — Thermodynamic Potentials, Maxwell Relations, Phase Equilibria

A. Core Definitions & Laws

• Zeroth: thermal equilibrium is transitive ⇒ temperature. First: dU = δQ − δW (δW = PdV by system). Second: Clausius: dS ≥ δQ/T; Kelvin–Planck: no cyclic process converts heat wholly into work. Third (Nernst): S → 0 (or constant) as T → 0; C → 0; absolute zero unattainable in finite steps.
• Entropy: dS = δQ_rev/T; isolated system ΔS ≥ 0.
• Thermodynamic potentials: U(S,V), H = U + PV (S,P), F = U − TS (T,V), G = U − TS + PV = H − TS (T,P), Ω = F − μN (grand potential, T,V,μ).

B. Key Equations / Rules

• Differentials: dU = TdS − PdV (+μdN); dH = TdS + VdP; dF = −SdT − PdV; dG = −SdT + VdP + μdN.
• Equilibrium: isolated (U,V) max S; (T,V) min F; (T,P) min G; (S,P) min H; (S,V) min U.
• Maxwell relations: (∂T/∂V)_S = −(∂P/∂S)_V; (∂T/∂P)_S = (∂V/∂S)_P; (∂S/∂V)_T = (∂P/∂T)_V; (∂S/∂P)_T = −(∂V/∂T)_P.
• Gibbs–Helmholtz: U = −T²∂(F/T)/∂T|_V; H = −T²∂(G/T)/∂T|_P.
• Heat capacities: C_V = T(∂S/∂T)_V = (∂U/∂T)_V; C_P = T(∂S/∂T)_P = (∂H/∂T)_P; C_P − C_V = TVβ²/κ_T (β = volume expansion coefficient, κ_T isothermal compressibility) = R (ideal gas, per mole); γ = C_P/C_V = κ_T/κ_S.
• (∂U/∂V)_T = T(∂P/∂T)_V − P (energy equation); (∂H/∂P)_T = V − T(∂V/∂T)_P.
• Joule expansion (free): ΔT = 0 for ideal gas; Joule–Thomson μ_JT = (∂T/∂P)_H = [T(∂V/∂T)_P − V]/C_P; ideal gas 0; inversion where T(∂V/∂T)_P = V; van der Waals T_inv = 2a/Rb (max), T_c = 8a/27Rb, P_c = a/27b², V_c = 3b.
• Ideal gas: PV = nRT; adiabatic PV^γ = const; TV^{γ−1} = const; work W = nRT ln(V₂/V₁) isothermal; W = (P₁V₁ − P₂V₂)/(γ−1) adiabatic. Entropy S = nC_V ln T + nR ln V + const.
• Carnot efficiency η = 1 − T_c/T_h; COP refrigerator = T_c/(T_h − T_c); Otto η = 1 − r^{1−γ}; Diesel, Stirling, Brayton.
• Phase equilibria: μ₁ = μ₂ (coexistence); T, P equal. Gibbs phase rule: F = C − P + 2 (triple point F = 0 for one component).
• Clapeyron: dP/dT = L/(TΔV) = ΔS/ΔV. Clausius–Clapeyron (vapour; V_vapour ≫ V_liquid, ideal gas): d ln P/dT = L/RT² ⇒ P = P₀exp(−L/RT) (L constant). Ice melting: dP/dT < 0 (ΔV < 0) — pressure lowers melting point (slope −13.5 MPa/K); about 0.0075 K/atm.
• Van der Waals: (P + a/V²)(V − b) = RT; Boyle temperature a/Rb; critical compressibility Z_c = 3/8; law of corresponding states.
• Ehrenfest classification: first-order discontinuity in first derivatives of G (S, V) with latent heat; second-order discontinuity in C_P, κ, β (S, V continuous).
• Landau theory: F = a(T−T_c)m² + bm⁴ ⇒ m = ±√[a(T_c−T)/2b] (β = ½); χ ∝ 1/|T−T_c| (γ = 1). Mean-field exponents: α = 0, β = ½, γ = 1, δ = 3.
• Magnetic: dU = TdS + B·dM; Curie law χ = C/T; adiabatic demagnetisation cooling.
• Blackbody: u = aT⁴; P_rad = u/3; S = (4/3)aT³V; Stefan: j = σT⁴, σ = 5.67 × 10⁻⁸ W/m²K⁴. Wien: λ_maxT = 2.898 × 10⁻³ m·K.

C. High-Yield Points

1. C_P > C_V always (β² ≥ 0, κ_T > 0); equal only if β = 0 (water at 4 °C, where C_P = C_V).
2. Ideal gas: U and H depend on T alone ⇒ Joule coefficient and JT coefficient both zero.
3. Clausius–Clapeyron assumes ideal-gas vapour and constant L; for ice → water slope is negative; for most substances positive.
4. Triple point of water 273.16 K, 611.7 Pa; critical point 647 K, 22.06 MPa. At triple point F = 0 (invariant); at critical point latent heat vanishes.
5. Entropy change in irreversible process > integral δQ/T; entropy of universe never decreases; free expansion ΔS = nR ln(V₂/V₁) with ΔU = 0.
6. Maxwell relations follow from exactness of potentials; choose the potential with the right natural variables (G for T, P).
7. Carnot efficiency is the maximum for reversible engines between two reservoirs; independent of working substance; η = 1 impossible (T_c = 0 unreachable).

D. Comparison Table

DAY 20 — Ensembles, Partition Functions, Equipartition

A. Core Definitions & Postulates

• Postulate of equal a priori probability: in equilibrium, all accessible microstates of an isolated system are equally probable.
• Microcanonical (N,V,E): S = k_B ln Ω(E); 1/T = ∂S/∂E.
• Canonical (N,V,T): P_i = e^{−βE_i}/Z, β = 1/k_BT; Z = Σe^{−βE_i}; F = −k_BT ln Z.
• Grand canonical (μ,V,T): 𝒵 = Σe^{β(μN − E)}; Ω = −k_BT ln 𝒵 = −PV.

B. Key Equations / Rules

• ⟨E⟩ = −∂ln Z/∂β; σ_E² = k_BT²C_V = ∂²ln Z/∂β²; S = −∂F/∂T = k_B(ln Z + β⟨E⟩) = −k_BΣP ln P (Gibbs); P = −∂F/∂V = k_BT ∂ln Z/∂V; μ = −k_BT∂ln Z/∂N.
• Grand: ⟨N⟩ = ∂ln𝒵/∂(βμ) = z∂ln𝒵/∂z (z = e^{βμ} fugacity); σ_N² = k_BT(∂N/∂μ)_T; PV = k_BT ln 𝒵.
• Independent subsystems: Z = Z₁Z₂; N identical, distinguishable-corrected: Z_N = Z₁ᴺ/N! (Gibbs correction fixes the mixing paradox and makes S extensive).
• Ideal gas: Z₁ = V/λ_T³, λ_T = h/√(2πmk_BT) (thermal de Broglie); Z_N = (V/λ³)ᴺ/N!; F = −Nk_BT[ln(V/Nλ³) + 1]; PV = Nk_BT; U = (3/2)Nk_BT; Sackur–Tetrode: S = Nk_B[ln(V/Nλ³) + 5/2]. Maxwell speed distribution f(v) = 4π(m/2πk_BT)^{3/2}v²e^{−mv²/2k_BT}; v_mp = √(2kT/m), v̄ = √(8kT/πm), v_rms = √(3kT/m) — ratio 1 : 1.128 : 1.225.
• Equipartition theorem: each quadratic degree of freedom (p² or q² term) contributes ½k_BT to ⟨E⟩ (classical, T high enough). Monatomic C_V = 3R/2; diatomic rigid: 5R/2; with vibration 7R/2; solids Dulong–Petit 3R; γ = 1 + 2/f.
• Quantum-limited degrees of freedom (freeze-out): rotational Θ_rot = ℏ²/2Ik_B (H₂ 85 K), vibrational Θ_vib = ℏω/k_B (H₂ 6100 K, N₂ 3400 K, Cl₂ 800 K).
• Two-level system (energies ±ε): Z = 2cosh(βε); U = −ε tanh(βε); C = k(βε)² sech²(βε) (Schottky anomaly). Paramagnet (spin ½, magnetic moment μ): M = Nμ tanh(μB/kT) → Curie χ = Nμ²/kT at low field.
• Quantum harmonic oscillator: Z = e^{−βℏω/2}/(1 − e^{−βℏω}); ⟨E⟩ = ℏω[½ + 1/(e^{βℏω} − 1)]; C = k(Θ/T)²e^{Θ/T}/(e^{Θ/T} − 1)² → k_B at T ≫ Θ; ∝ e^{−Θ/T} at T ≪ Θ.
• Rotor (heteronuclear diatomic) Z = Σ(2J+1)e^{−J(J+1)Θ_r/T} ≈ T/Θ_r (high T); U = k_BT.
• Fluctuation: relative fluctuation ∼ 1/√N; ⟨(ΔE)²⟩ = k_BT²C_V.
• Gibbs paradox, density of states in classical phase space: Ω = (1/h^{3N}N!)∫d³Nq d³Np.
• Detailed balance: W_{ij}P_j = W_{ji}P_i at equilibrium; Boltzmann H-theorem dH/dt ≤ 0; Liouville.
• Boltzmann statistics of occupation: n_i = g_i/e^{(ε_i − μ)/kT}; ratio N₂/N₁ = (g₂/g₁)e^{−ΔE/kT}.
• Entropy of mixing: ΔS = −Nk_B Σxᵢ ln xᵢ (distinguishable gases); zero if identical.
• Random walk: ⟨x²⟩ = 2Dt (1D), D = k_BT/6πηa (Stokes–Einstein), ⟨r²⟩ = 6Dt (3D); Einstein relation D = μ_mobility k_BT; Brownian motion; diffusion equation ∂n/∂t = D∇²n; Gaussian solution n = (4πDt)^{−1/2}e^{−x²/4Dt}.

C. High-Yield Points

1. Equipartition fails at low T (quantum freeze-out): H₂ C_V drops from 5R/2 to 3R/2 below ~85 K; vibrational modes appear above Θ_vib.
2. Classical ideal gas needs the 1/N! factor for extensivity (Gibbs paradox); omit it and S isn't extensive.
3. Energy fluctuations in canonical ensemble related to heat capacity: σ_E² = k_BT²C_V; in microcanonical E fixed exactly.
4. Ratio v_mp : v̄ : v_rms = √2 : √(8/π) : √3. Average KE = 3k_BT/2 independent of mass.
5. Partition function of N distinguishable non-interacting subsystems multiplies; for indistinguishable divide by N! (classical limit). Quantum degeneracy sets in when nλ³ ≳ 1.
6. Ensembles are equivalent in thermodynamic limit; microcanonical for isolated, canonical for exchange of energy, grand canonical for exchange of energy and particles.
7. Schottky anomaly: two-level C peaks near kT ≈ 0.42ε — characteristic of finite-level systems (a bump, then decay).

D. Comparison Table

DAY 21 — Quantum Statistics: FD, BE, Condensation

A. Core Definitions

• Identical particles: wavefunction symmetric (bosons, integer spin) or antisymmetric (fermions, half-integer spin) under exchange; Pauli exclusion: no two fermions in the same state. Spin-statistics theorem.
• Distributions: Maxwell–Boltzmann: n̄ = e^{−(ε−μ)/kT}; Fermi–Dirac: n̄ = 1/(e^{(ε−μ)/kT} + 1); Bose–Einstein: n̄ = 1/(e^{(ε−μ)/kT} − 1). Photon/phonon: μ = 0.
• Fermi energy: ε_F = μ(T = 0) = (ℏ²/2m)(3π²n)^{2/3}.

B. Key Equations / Rules

• Classical limit when nλ_T³ ≪ 1 (e^{−(ε−μ)/kT} ≪ 1).
• Density of states g(ε) (per volume, 3D, spin degeneracy 2): g(ε) = (1/2π²)(2m/ℏ²)^{3/2}ε^{1/2}; N(ε) = (V/3π²)(2mε/ℏ²)^{3/2}.
• Ideal Fermi gas (T = 0): ε_F = (ℏ²/2m)(3π²n)^{2/3}; k_F = (3π²n)^{1/3}; T_F = ε_F/k_B; U₀ = (3/5)Nε_F; mean energy (3/5)ε_F; P₀ = (2/5)nε_F (degeneracy pressure) = (2/3)u; bulk modulus B = (5/3)P₀ = (2/3)nε_F. Copper: ε_F = 7.0 eV, T_F = 8.2 × 10⁴ K.
• Finite T (Sommerfeld, T ≪ T_F): μ = ε_F[1 − (π²/12)(kT/ε_F)²]; C_V = (π²/2)Nk_B(T/T_F) = γT (electronic specific heat; only fraction ~kT/ε_F of electrons excited); paramagnetic Pauli susceptibility χ_P = μ_B²g(ε_F) (independent of T).
• White dwarfs: electron degeneracy pressure supports; Chandrasekhar mass ≈ 1.4 M_☉; neutron stars supported by neutron degeneracy.
• Bose gas: μ ≤ 0 (ε₀ = 0); N = Σ1/(e^{(ε−μ)/kT} − 1). Number density integral ∝ ζ(3/2) = 2.612.
• BEC temperature (3D, nonrelativistic ideal gas): T_c = (2πℏ²/mk_B)(n/ζ(3/2))^{2/3} = (h²/2πmk_B)(n/2.612)^{2/3}; condition nλ_T³ = ζ(3/2) = 2.612 (phase-space density).
• For T < T_c: N₀/N = 1 − (T/T_c)^{3/2}; μ = 0; U = (3/2)Nk_BTζ(5/2)/ζ(3/2)^{3/2} = 0.770Nk_BT(T/T_c)^{3/2}; C_V = 1.925Nk_B(T/T_c)^{3/2} (peak at T_c ≈ 1.925Nk_B, cusp; then → 1.5Nk_B above); P = (ζ(5/2)/λ³)k_BT (independent of volume).
• ⁴He superfluid λ-point 2.17 K (ideal-gas T_c = 3.1 K); BEC in dilute alkali vapours (Rb-87, 170 nK, 1995).
• BEC absent in 2D (and 1D) uniform ideal gas; trapped gas allowed.
• Photon gas: n̄ = 1/(e^{ℏω/kT} − 1); Planck: u(ω)dω = (ℏω³/π²c³)dω/(e^{ℏω/kT} − 1); total u = (π²k⁴/15ℏ³c³)T⁴ = aT⁴; Stefan σ = π²k⁴/60ℏ³c². Wien displacement: wavelength form hc/λ_maxkT = 4.965 (from x = 5(1 − e^{−x})); frequency form hν_max/kT = 2.82. Rayleigh–Jeans u = ω²kT/π²c³ (ultraviolet catastrophe). Photon number density n = 0.244(kT/ℏc)³ ≈ 2.03 × 10⁷T³ m⁻³; mean energy per photon 2.70kT.
• Phonon gas (Debye): see Day 23.
• Fermi vs Bose at low T: Pauli pressure vs condensation. Relativistic ultra-degenerate: u ∝ n^{4/3}, P ∝ n^{4/3}.
• Occupation number fluctuation: ⟨(Δn)²⟩ = n̄(1 − n̄) (fermions); n̄(1 + n̄) (bosons); n̄ (classical).
• Equation of state of ideal quantum gas: PV = (2/3)U (nonrelativistic), PV = U/3 (ultrarelativistic or photon).
• Chemical potential for classical gas: μ = kT ln(nλ³) < 0.

C. High-Yield Points

1. FD at T = 0 is a step function; at T > 0 smearing width ~kT around ε_F; f(ε = μ) = ½ at all T.
2. Electronic specific heat C = γT is linear in T, much smaller than classical 3R/2; combined with lattice C = γT + AT³ (plot C/T vs T²: intercept γ, slope A).
3. BEC occurs for bosons with μ → 0⁻ and macroscopic occupation of ground state; no BEC for photons (number not conserved, μ = 0 always) — they have Planck spectrum instead.
4. Specific heat of ideal Bose gas peaks at T_c, with value 1.925Nk_B (cusp), higher than classical 3/2 Nk_B.
5. Pauli paramagnetism small and T-independent vs Curie paramagnetism ∝ 1/T; Landau diamagnetism = −⅓χ_P.
6. Fermi energy scales as n^{2/3}; metals ε_F few eV ≫ kT at room temperature (0.025 eV) ⇒ degenerate; in 2D, g(ε) constant.
7. Total number of photons ∝ T³; energy ∝ T⁴; each of ε-peak (Wien) λ_max ∝ 1/T.

D. Comparison Table

DAY 22 — Crystal Lattices, Miller Indices, Reciprocal Lattice, XRD

A. Core Definitions

• Bravais lattice: infinite array R = n₁a₁ + n₂a₂ + n₃a₃. 14 Bravais lattices in 3D (7 crystal systems), 5 in 2D. Primitive cell: one lattice point; Wigner–Seitz cell is a symmetric primitive cell; basis: atoms associated with each lattice point.
• Miller indices (hkl): reciprocals of intercepts (in lattice units) cleared to smallest integers; plane (hkl) ⟂ [hkl] direction in cubic.
• Reciprocal lattice: vectors G with e^{iG·R} = 1: bᵢ = 2π(aⱼ × aₖ)/[a₁·(a₂ × a₃)] (aᵢ·bⱼ = 2πδᵢⱼ). First Brillouin zone = Wigner–Seitz cell of reciprocal lattice, volume (2π)³/V_cell.

B. Key Equations / Rules

• Interplanar spacing (cubic): d_hkl = a/√(h² + k² + l²); orthorhombic 1/d² = h²/a² + k²/b² + l²/c²; hexagonal 1/d² = (4/3)(h² + hk + k²)/a² + l²/c². G_hkl = 2π/d_hkl.
• Cubic cells: SC (1 atom/cell; CN 6; packing 0.52), BCC (2; CN 8; 0.68; nn distance √3a/2), FCC (4; CN 12; 0.74; nn a/√2), diamond (8; CN 4; 0.34; nn √3a/4), HCP (c/a = √(8/3) = 1.633; CN 12; 0.74), NaCl (FCC with 2-atom basis; CN 6), CsCl (SC with 2-atom basis; CN 8), ZnS (zinc-blende).
• Reciprocal of FCC is BCC and vice versa; reciprocal of SC is SC (2π/a). Primitive vectors FCC: (a/2)(0,1,1),(1,0,1),(1,1,0); BCC: (a/2)(−1,1,1),(1,−1,1),(1,1,−1).
• Planar density, direction [uvw], family {hkl}, ⟨uvw⟩; angle between planes (cubic) cosθ = (h₁h₂ + k₁k₂ + l₁l₂)/(|n₁||n₂|).
• Bragg's law: 2d sinθ = nλ (nλ ≤ 2d); equivalent Laue condition k′ − k = G; Ewald sphere construction. Powder method: peaks at sin²θ ∝ h² + k² + l² (cubic); ratios sin²θ_i/sin²θ₁ identify lattice: SC 1:2:3:4:5:6:8; BCC 2:4:6:8:10:12:14 (h+k+l even) = 1:2:3:4:5:6:7; FCC 3:4:8:11:12:16 (unmixed indices) ; diamond 3:8:11:16:19.
• Structure factor: S_G = Σ_j f_j e^{iG·r_j} = Σ f_j e^{2πi(hu_j + kv_j + lw_j)}; intensity ∝ |S|². Atomic form factor f(0) = Z (electrons) for X-rays.
• Selection (extinction) rules:
• SC: all (hkl).
• BCC: S = f[1 + e^{iπ(h+k+l)}] ⇒ nonzero (= 2f) only if h + k + l even; (100) absent, (110) present.
• FCC: nonzero (=4f) only if h, k, l all odd or all even (unmixed); first reflections (111), (200), (220), (311).
• Diamond: FCC rule plus; if all even, need h + k + l = 4n (e.g., (222) forbidden, (400) allowed); (111) allowed with S = 4f(1 ∓ i).
• NaCl: all unmixed; all even: S = 4(f₊ + f₋); all odd: S = 4(f₊ − f₋) (weak lines — KCl's (111) nearly absent, as K⁺ and Cl⁻ isoelectronic).
• HCP: (hk.l) with h + 2k = 3n and l odd forbidden.
• Lattice constant from density: a³ = nM/(ρN_A); e.g., Cu a = 3.61 Å.
• Debye–Waller factor e^{−2W} reduces intensity with T; thermal diffuse scatter.
• Quasicrystals: five-fold symmetry, long-range orientational order but no translational periodicity (Shechtman 1982; Al-Mn); crystallographic restriction theorem: only 2, 3, 4, 6-fold rotations in periodic lattices.
• Defects: vacancy (Schottky n = N e^{−E_v/kT}), interstitial (Frenkel n = √(NN′)e^{−E_f/2kT}), edge/screw dislocations (Burgers vector), grain boundaries; liquid crystals: nematic (orientational order only), smectic (layered), cholesteric (twisted).
• Neutron & electron diffraction: λ = h/√(2mE); neutron 0.286/√E(eV) Å; electron 12.26/√V Å.

C. High-Yield Points

1. BCC: h + k + l even; FCC: unmixed (all odd or all even); SC: all allowed; diamond: unmixed and (all even ⇒ h+k+l = 4n).
2. Reciprocal lattice of FCC is BCC with cube side 4π/a; of BCC is FCC; first BZ of FCC is a truncated octahedron.
3. Packing fractions: SC 52%, BCC 68%, FCC/HCP 74%, diamond 34%; FCC and HCP differ in stacking (ABCABC vs ABAB).
4. Bragg reflection requires λ ≤ 2d; in a periodic crystal electrons with k at BZ boundary satisfy Bragg ⇒ energy gaps.
5. Miller indices of a plane parallel to an axis have 0 for that axis; negative intercepts denoted h̄; (hkl) vs [hkl] vs {hkl} vs ⟨hkl⟩ notation.
6. NaCl-type crystals are FCC lattices with two-atom basis; CsCl is simple cubic (not BCC) with a two-atom basis.
7. Powder diffraction ratios identify cubic lattice type: BCC lines 110, 200, 211… ; FCC lines 111, 200, 220, 311…; FCC peaks sin²θ ratio 3:4:8:11:12.

D. Comparison Table

DAY 23 — Phonons, Debye/Einstein, Free Electron Gas

A. Core Definitions

• Phonon: quantum of lattice vibration, energy ℏω, crystal momentum ℏq; bosons with μ = 0.
• Dispersion branches: monatomic chain 1 acoustic branch/ dimension; diatomic basis → acoustic + optical (3N total branches in 3D: 3 acoustic + 3(p−1) optical for p atoms/cell; LA, TA, LO, TO).
• Free electron model (Sommerfeld): non-interacting Fermi gas in a box; periodic BC k = 2πn/L.

B. Key Equations / Rules

• Monatomic chain (mass M, spring C, lattice constant a): ω = 2√(C/M)|sin(qa/2)|; ω_max = 2√(C/M) at q = π/a; low q: v_s = a√(C/M). Group velocity v_g = (a√(C/M))cos(qa/2) → 0 at BZ edge (standing wave).
• Diatomic chain (M₁ > M₂): ω²± = C(1/M₁ + 1/M₂) ± C√[(1/M₁ + 1/M₂)² − 4sin²(qa/2)/M₁M₂]; at q = 0: optical ω = √(2C(1/M₁ + 1/M₂)), acoustic 0; at q = π/a: acoustic √(2C/M₁), optical √(2C/M₂); gap between √(2C/M₁) and √(2C/M₂). Optical modes: IR-active in ionic crystals (opposite ion motion).
• Classical Dulong–Petit C_V = 3Nk_B = 24.9 J/mol·K.
• Einstein model: 3N identical oscillators, ω_E: U = 3Nℏω_E/(e^{Θ_E/T} − 1); C_V = 3Nk_B(Θ_E/T)²e^{Θ_E/T}/(e^{Θ_E/T} − 1)²; high T → 3Nk_B; low T ∝ e^{−Θ_E/T} (too fast vs experiment).
• Debye model: linear dispersion ω = v_sq up to cutoff ω_D; g(ω) = 9Nω²/ω_D³; ω_D = v_s(6π²N/V)^{1/3}; Θ_D = ℏω_D/k_B. C_V = 9Nk_B(T/Θ_D)³∫₀^{Θ_D/T}x⁴eˣ/(eˣ − 1)²dx. T ≫ Θ_D: 3Nk_B. T ≪ Θ_D: C_V = (12π⁴/5)Nk_B(T/Θ_D)³ = 234Nk_B(T/Θ_D)³ (T³ law). Debye temperatures: Pb 105 K, Cu 343 K, Al 428 K, diamond 2230 K, Na 158 K.
• Thermal conductivity κ = (1/3)C v λ_mfp (kinetic); phonon umklapp scattering; Wiedemann–Franz κ/σT = L = (π²/3)(k_B/e)² = 2.44 × 10⁻⁸ WΩ/K².
• Elastic waves: v_L = √((B + 4G/3)/ρ); v_T = √(G/ρ); sound in solid velocity vs. phonon.
• Phonon spectra by inelastic neutron scattering (conservation: ℏω = E − E′, ℏq = ℏ(k − k′) + ℏG).
• Free electron gas: Eₖ = ℏ²k²/2m; each k-state allowed volume (2π)³/V; spin degeneracy 2.
• 1D: n = 2k_F/π; ε_F = ℏ²π²n²/8m; g(ε) = (L/π)(2m/ℏ²)^{1/2}ε^{−1/2} (∝ ε^{−1/2}); N(ε) ∝ ε^{1/2}.
• 2D: n = k_F²/2π; ε_F = πℏ²n/m; g(ε) = mA/πℏ² = const (per area m/πℏ²); N(ε) ∝ ε.
• 3D: n = k_F³/3π²; ε_F = (ℏ²/2m)(3π²n)^{2/3}; g(ε) = (V/2π²)(2m/ℏ²)^{3/2}ε^{1/2} = 3N/2ε_F · (ε/ε_F)^{1/2}; g(ε_F) = 3N/2ε_F; N(ε) ∝ ε^{3/2}.
• General: g(ε) ∝ ε^{d/2 − 1}.
• Electronic specific heat C_el = (π²/2)Nk_B(T/T_F) = γT, γ = (π²/3)k_B²g(ε_F); γ_exp ≈ γ_free × m*/m.
• Free electron: v_F = ℏk_F/m (Cu 1.57 × 10⁶ m/s).
• Drude model: J = σE, σ = ne²τ/m (conductivity), mobility μ = eτ/m; resistivity ρ = m/ne²τ; relaxation time τ ~ 10⁻¹⁴ s; mean free path λ = v_Fτ ~ 10–100 Å. Matthiessen's rule ρ = ρ_impurity + ρ_phonon(T) ∝ T at high T, T⁵ (Bloch–Grüneisen) at low T. AC: σ(ω) = σ₀/(1 − iωτ); plasma frequency ω_p = √(ne²/ε₀m) (metal ~10¹⁶ rad/s; reflectance up to UV).
• Thermal expansion: Grüneisen γ = βV B/C_V.
• Thermoelectric: Seebeck S = −(π²k_B²T/3e)(d ln σ/dε)|_{ε_F} (Mott).

C. High-Yield Points

1. Debye T³ law applies only for T ≪ Θ_D (T < Θ_D/50 for precision); Einstein model misses the T³ behaviour (exponential).
2. Electronic C ∝ T dominates over phonon C ∝ T³ only at very low temperature (<~ 1–3 K for typical metals).
3. Density of states exponent: 1D ε^{−1/2}, 2D constant, 3D ε^{1/2}; singularities (van Hove) at critical points of dispersion.
4. Number of phonon modes = 3N (all branches) — total fixed, Debye cutoff ensures this; acoustic mode ω → 0 as q → 0; optical mode ω ≠ 0 at q = 0.
5. Phonon wavevector q is a crystal momentum — not true momentum; q and q + G equivalent, hence restricted to the first BZ; umklapp (k₁ + k₂ = k₃ + G) causes thermal resistance.
6. Drude model fails: specific heat (predicts 3R/2 electrons); thermal conductivity ratio Lorenz number L ≈ 1.1 × 10⁻⁸ vs actual 2.44 × 10⁻⁸ (accidental cancellation); Hall sign anomalies; temperature dependence of ρ.
7. Dulong–Petit fails at low T (quantum freeze-out); diamond at room T still far below 3R because Θ_D = 2230 K.

D. Comparison Table

DAY 24 — Kronig–Penney, Effective Mass, Hall Effect, Superconductivity

A. Core Definitions

• Bloch theorem: ψ_k(r) = e^{ik·r}u_k(r), u_k with lattice periodicity; E_n(k) periodic in k with period G; crystal momentum ℏk.
• Energy band/gap: bands of allowed states separated by forbidden gaps from Bragg reflection at BZ boundaries k = ±nπ/a.
• Effective mass: m* = ℏ²/(d²E/dk²).
• Hole: missing electron in nearly full band; charge +e, positive effective mass, wavevector −k_missing.
• Superconductor: ρ = 0 below T_c and perfect diamagnetism (Meissner–Ochsenfeld, B = 0 inside).

B. Key Equations / Rules

• Kronig–Penney (periodic square wells; barrier height V₀, width b, well width a; P = mV₀ba/ℏ² in delta limit): cos(ka) = P sin(αa)/(αa) + cos(αa), α = √(2mE)/ℏ. Allowed bands where |RHS| ≤ 1; gaps otherwise. As P → 0 free electrons; P → ∞ isolated well levels (αa = nπ); bands widen with increasing energy; gap at ka = nπ.
• Near band edge: E ≈ E₀ + ℏ²(k − k₀)²/2m; m > 0 near band bottom, < 0 near band top (hole), diverges at inflection point; heavier m for narrow bands (d-bands). Tight binding: E(k) = ε₀ − α − 2γ cos(ka) (1D), bandwidth 4γ; m = ℏ²/2γa² at bottom.
• Group velocity v_g = (1/ℏ)dE/dk; semiclassical: ℏdk/dt = −e(E + v × B). Bloch oscillations in static field.
• Metal/insulator/semiconductor: partially filled band → metal; filled valence band + gap E_g > ~4 eV → insulator; E_g < ~2–3 eV → semiconductor (Si 1.12 eV indirect; Ge 0.67 eV indirect; GaAs 1.42 eV direct). Divalent metal overlap bands. Direct gap: optical transition vertical (k conserved) – LEDs, lasers.
• Intrinsic carriers: n_i = √(N_cN_v)e^{−E_g/2kT} = 2(2πk_BT/h²)^{3/2}(m_em_h)^{3/4}e^{−E_g/2kT}; E_F near mid-gap (E_F = E_g/2 + (3/4)kT ln(m_h/m_e)); np = n_i² (mass action); doped: n-type n ≈ N_D; p-type p ≈ N_A; E_F moves toward band edge.
• Conductivity σ = e(nμ_e + pμ_h). Resistivity of semiconductor ρ ∝ e^{E_g/2kT} decreases with T; metals increase.
• Hall effect: current J_x, field B_z ⇒ transverse E_y = R_H J_xB_z; R_H = −1/ne (electrons), +1/pe (holes); two carriers: R_H = (pμ_h² − nμ_e²)/[e(pμ_h + nμ_e)²]. Hall voltage V_H = IB/(nte) (t = thickness); Hall mobility μ_H = |R_H|σ; Hall angle tanθ = ω_cτ. Cyclotron ω_c = eB/m*. Landau levels E = (n+½)ℏω_c; quantum Hall resistance R_H = h/νe² (von Klitzing 25.8128 kΩ).
• Magnetoresistance; de Haas–van Alphen oscillations periodic in 1/B give Fermi surface area.
• Superconductivity: critical temperature T_c (Hg 4.15 K, Nb 9.2 K, Pb 7.2 K; YBCO ~92 K; MgB₂ 39 K); Isotope effect T_c ∝ M^{−1/2} (BCS: phonon-mediated).
• Critical field B_c(T) = B_c(0)[1 − (T/T_c)²]. Type I: one critical field, complete Meissner state below B_c, abrupt transition (soft, pure elements). Type II: B_c1 (Meissner → mixed/vortex state with flux quantised Φ₀ = h/2e = 2.07 × 10⁻¹⁵ Wb) and B_c2 (normal); hard, high-field magnets (Nb₃Sn, NbTi). Ginzburg–Landau κ = λ/ξ: type I κ < 1/√2, type II κ > 1/√2.
• London equations: ∂(ΛJ_s)/∂t = E; ∇×(ΛJ_s) = −B with Λ = m/n_se²; ⇒ ∇²B = B/λ_L², λ_L = √(m/μ₀n_se²) (~ 50 nm): field decays exponentially inside, B(x) = B₀e^{−x/λ_L}. Coherence length ξ.
• BCS: Cooper pairs (k↑, −k↓), binding via phonon-mediated attraction; energy gap Δ(0) = 1.76k_BT_c (E_g = 2Δ = 3.52k_BT_c); specific heat jump at T_c, exponentially activated below.
• Josephson effect: DC: I = I_c sin(φ₂ − φ₁) (no voltage); AC: with V applied, f = 2eV/h = 483.6 MHz/µV; Shapiro steps V_n = nhf/2e; SQUID flux sensitivity ~ Φ₀ — magnetometry.
• Flux quantisation Φ = nh/2e (Cooper pair charge 2e).
• Persistent currents, critical current density J_c; thermodynamic: free-energy difference F_n − F_s = B_c²/2μ₀.
• High-T_c cuprates (d-wave, layered CuO₂), iron pnictides.
• Superfluidity: ⁴He below T_λ = 2.17 K: zero viscosity, two-fluid model (normal + superfluid), Landau critical velocity v_c = min(ε(p)/p), quantised vortices (circulation h/m), fountain effect; ³He superfluid at ~2.6 mK via p-wave Cooper pairing.

C. High-Yield Points

1. Effective mass is negative near top of a band; hole = absence of electron with positive mass and +e charge; R_H sign tells the majority carrier.
2. A superconductor is more than a perfect conductor: a perfect conductor would trap flux (ZFC vs FC), but a superconductor expels flux (Meissner) — this is the thermodynamic equilibrium state.
3. Type II superconductors have B_c2 ≫ B_c; vortices carry exactly h/2e; mixed state with zero resistance (pinned vortices).
4. T_c ∝ M^{−1/2} confirms phonon mediation; BCS gap 2Δ = 3.52 k_BT_c; flux quantum h/2e shows pairs of charge 2e.
5. In intrinsic semiconductor E_F lies at mid-gap (shifted slightly if m_h ≠ m_e); adding donors raises E_F towards E_c; n_ip = n_i² regardless of doping.
6. Hall coefficient of a metal is independent of B and τ; sign anomalies (Al, Be: positive R_H) show band-structure (hole-like Fermi surface) not accounted by Drude.
7. Josephson: ac effect f = 2eV/h ⇒ voltage standard; London penetration depth λ_L ∝ (n_s)^{−1/2} diverges as T → T_c.

D. Comparison Table

MODULE 5 — NUCLEAR & PARTICLE PHYSICS, ELECTRONICS & EXPERIMENTAL METHODS

DAY 25 — Nuclear Size, Binding Energy, Liquid Drop, SEMF

A. Core Definitions

• Nuclide ᴬ_ZX: Z protons, N = A − Z neutrons. Isotopes (same Z), isotones (same N), isobars (same A), isomers (same Z, A; different energy state).
• Mass defect Δm = Zm_p + Nm_n − M(A,Z); binding energy B = Δm c²; B/A ≈ 8 MeV (peak at ⁵⁶Fe/⁶²Ni ≈ 8.8 MeV).
• Nuclear radius: R = r₀A^{1/3}, r₀ ≈ 1.2 fm ⇒ density ρ ≈ 2.3 × 10¹⁷ kg/m³ (constant, independent of A); charge distribution Fermi/Woods–Saxon with surface thickness ~2.3 fm.
• Nuclear force: short range (~1–2 fm), strong, charge independent (nn ≈ pp ≈ np) and charge symmetric (nn ≈ pp), spin-dependent (deuteron triplet bound; singlet unbound), tensor component (deuteron quadrupole moment 0.286 e·fm²), saturation, repulsive core (~0.5 fm), meson-exchange (Yukawa: range ℏ/m_πc ≈ 1.4 fm; V ∝ e^{−r/R}/r).

B. Key Equations / Rules

• 1 u = 931.494 MeV/c²; m_p = 938.272 MeV, m_n = 939.565 MeV, m_e = 0.511 MeV. Q-value Q = (Σm_i − Σm_f)c² (>0 exothermic).
• Semi-empirical mass formula (Bethe–Weizsäcker): B(A,Z) = a_vA − a_sA^{2/3} − a_cZ(Z−1)/A^{1/3} − a_a(A − 2Z)²/A + δ(A,Z) a_v ≈ 15.8, a_s ≈ 17.8, a_c ≈ 0.71, a_a ≈ 23.7 MeV; δ = +a_pA^{−1/2} (even–even), 0 (odd A), −a_pA^{−1/2} (odd–odd), a_p ≈ 12 MeV.
• Volume: saturation, ∝ A. Surface: nucleons at the surface less bound, ∝ R² ∝ A^{2/3}. Coulomb: ∝ Z²/R. Asymmetry: Pauli principle (N = Z preferred), ∝ (N−Z)²/A. Pairing: spin-paired nucleons more stable.
• Most stable Z for fixed A: Z₀ = A/(2 + 0.0150A^{2/3}) (≈ A/2 for light nuclei; N/Z ≈ 1.5 for heavy). Isobar parabola M vs Z; odd A: single parabola ⇒ one stable isobar; even A: two parabolas (even–even lower).
• Mass of mirror nuclei difference gives a_c; Fission if Z²/A ≳ 49 (spontaneous); fission barrier ~6 MeV (²³⁵U + n → ²³⁶U* excitation 6.5 MeV > barrier).
• Separation energy S_n = B(A,Z) − B(A−1,Z); abrupt drop after magic numbers.
• Fission: ²³⁵U + n → fragments (A ~ 95, 140), Q ≈ 200 MeV (~0.9 MeV/nucleon), ν̄ ≈ 2.4 neutrons; chain reaction k = 1 critical; moderators (D₂O, graphite); thermal cross-section σ ≈ 580 b. Fusion: D + T → ⁴He + n, Q = 17.6 MeV; p–p chain 26.7 MeV per ⁴He; Coulomb barrier ~ 1 MeV, solar core T ≈ 1.5 × 10⁷ K via tunnelling; Lawson criterion nτ ≳ 10²⁰ s/m³.
• Deuteron: B = 2.224 MeV; only bound state (I = 1, ³S₁ + 4% ³D₁; J^P = 1⁺; μ_d = 0.857 μ_N; Q = 0.286 fm²); no excited bound state; spherical square-well estimate: V₀R² ≈ const (k₀R = π/2 threshold). Scattering lengths a_t = 5.4 fm, a_s = −23.7 fm.
• Nuclear magnetic moment unit μ_N = eℏ/2m_p = 3.152 × 10⁻⁸ eV/T = 5.05 × 10⁻²⁷ J/T; μ_p = 2.793μ_N; μ_n = −1.913μ_N (neutral particle has moment ⇒ substructure).
• Electromagnetic multipoles: parity & spin constraints — electric quadrupole nonzero only for I ≥ 1; no static electric dipole moment (parity/time-reversal).
• Rutherford, electron scattering (form factor F(q) ⇒ charge radius); nuclear radius from mirror nuclei, muonic atoms, α-scattering.

C. High-Yield Points

1. B/A rises steeply to ~8 MeV by A ≈ 20, peaks at Fe/Ni (~8.8 MeV), declines slowly to ~7.6 MeV at U: fusion releases energy for A < 56, fission for A > 56.
2. Nuclear density is nearly constant (saturation) ⇒ volume term ∝ A; surface term lowers B/A for small nuclei; Coulomb term for large nuclei.
3. Even–even nuclei most numerous (~ 160 stable); only four stable odd–odd nuclei: ²H, ⁶Li, ¹⁰B, ¹⁴N (a few more are very long-lived, e.g. ⁵⁰V) — pairing term.
4. Deuteron has no excited state; nn and pp (diproton) are unbound ⇒ the nuclear force is spin-dependent (singlet unbound; triplet bound).
5. Z₀ ≈ A/2 for light nuclei; neutron excess grows for heavy nuclei due to Coulomb term; n-excess line N/Z → 1.5 (²⁰⁸Pb).
6. Nuclear force saturates (B ∝ A, not A²), has short range (~2 fm), is attractive at ~1 fm & repulsive core below 0.5 fm; Coulomb ∝ 1/r long range.
7. Proton magnetic moment 2.79μ_N (not 1) and neutron −1.91μ_N reflect internal quark structure (SU(6) predicts μ_p/μ_n = −3/2).

D. Comparison Table

DAY 26 — Shell Model, Spin-Parity, Moments, α–β–γ Decay

A. Core Definitions

• Magic numbers: 2, 8, 20, 28, 50, 82, 126 (neutrons and protons; doubly magic: ⁴He, ¹⁶O, ⁴⁰Ca, ⁴⁸Ca, ⁵⁶Ni, ¹³²Sn, ²⁰⁸Pb).
• Shell model: nucleons move independently in a mean-field (harmonic oscillator/Woods–Saxon) potential plus strong spin–orbit coupling V_so ∝ −(l·s) (opposite sign to atomic; j = l + ½ lowers energy).
• Spin-parity J^P: parity = (−1)^l of the odd nucleon.

B. Key Equations / Rules

• Level ordering (with notation nl_j): 1s½ | 1p₃/₂ 1p₁/₂ (8) | 1d₅/₂ 2s₁/₂ 1d₃/₂ (20) | 1f₇/₂ (28) | 2p₃/₂ 1f₅/₂ 2p₁/₂ 1g₉/₂ (50) | 1g₇/₂ 2d₅/₂ 2d₃/₂ 3s₁/₂ 1h₁₁/₂ (82) | 1h₉/₂ 2f₇/₂ 1i₁₃/₂ 2f₅/₂ 3p₃/₂ 3p₁/₂ (126). Each j-level holds 2j + 1 nucleons.
• Spin-parity rules: even–even ground state 0⁺; odd-A: J^P of the unpaired nucleon (e.g., ¹⁷O: 1d₅/₂ → 5/2⁺; ¹⁷F 5/2⁺; ¹³C: 1p₁/₂ → ½⁻; ¹⁵N ½⁻; ³⁹K (hole in d₃/₂) 3/2⁺; ⁴¹Ca 7/2⁻; ²⁰⁹Bi 9/2⁻; ²⁰⁷Pb ½⁻ (hole in p₁/₂)); odd–odd: |j_p − j_n| ≤ J ≤ j_p + j_n (Nordheim rules).
• Schmidt (single-particle) magnetic moments (μ in μ_N): for j = l + ½: μ = g_l(j − ½) + ½g_s ; for j = l − ½: μ = [j/(j+1)][g_l(j + 3/2) − ½g_s]. Proton g_l = 1, g_s = 5.586; neutron g_l = 0, g_s = −3.826. Explicitly: proton j = l + ½: μ = j + 2.293; neutron j = l + ½: μ = −1.913; proton j = l − ½: μ = j/(j+1); neutron j = l − ½: μ = +1.913·j/(j+1). Experimental values lie between the two Schmidt lines (e.g., ¹⁷O: −1.89 vs −1.91; ¹⁵N: −0.28 vs −0.26).
• Quadrupole moment: spherical shell ⇒ 0; deformed nuclei large Q; rotational spectrum E_J = (ℏ²/2𝒥)J(J+1) for even–even deformed (J = 0, 2, 4…): E₄/E₂ = 3.33 (rotor) vs 2.0 (vibrator) vs 2.0–2.5 (spherical); harmonic vibration levels 0⁺, 2⁺, (0⁺,2⁺,4⁺)…
• Alpha decay: ᴬ_ZX → ᴬ⁻⁴_{Z−2}Y + ⁴He; Q_α = (M_X − M_Y − M_He)c² (typ. 4–9 MeV); E_α = Q·M_Y/(M_Y + M_α) ≈ Q(1 − 4/A). Geiger–Nuttall: log λ = a − b/√E_α (b ∝ Z). Gamow tunnelling probability P = e^{−2G}, G = (√(2μ)/ℏ)∫_R^b√(V − Q) dr; λ = f·P (f ~ 10²¹ s⁻¹). Selection: ΔJ = l, Δπ = (−1)^l; 0⁺ → 0⁺ l = 0 favoured. Long-lived/short-lived span 10⁻⁷ s to 10¹⁷ yr (T½ strongly dependent on E_α).
• Beta decay: β⁻: n → p + e⁻ + ν̄_e (A,Z → A,Z+1; Q = [M(Z) − M(Z+1)]c² using atomic masses); β⁺: p → n + e⁺ + ν_e (Q = [M(Z) − M(Z−1) − 2m_e]c², atomic masses); EC: p + e⁻ → n + ν_e (Q = [M(Z) − M(Z−1)]c²; competes with β⁺; the only mode if Q < 2m_ec² = 1.022 MeV). Continuous energy spectrum (Pauli's neutrino 1930); Fermi theory: rate ∝ |M_fi|²·f(Z,E₀) with Sargent's rule λ ∝ E₀⁵; Kurie plot linear; end-point E₀.
• Selection rules: Fermi (ΔJ = 0, no parity change; e & ν spins antiparallel S = 0), Gamow–Teller (ΔJ = 0, ±1, 0 ↛ 0; spins parallel S = 1). Allowed: Δπ = no, ΔJ = 0,±1; first forbidden: Δπ = yes (l = 1), ΔJ = 0, ±1, ±2; ft values: superallowed 10³–10⁴ s (log ft ≈ 3.5), allowed 10⁴–10⁶, first-forbidden 10⁶–10⁹.
• Neutrino rest mass < 1 eV (oscillations ⇒ nonzero); parity violation (Wu experiment ⁶⁰Co, 1957: electrons emitted opposite to nuclear spin; left-handed ν, right-handed ν̄).
• Free neutron: τ = 880 s (886 s), Q = 0.782 MeV.
• Double β decay (²νββ: e.g. ⁷⁶Ge → ⁷⁶Se; 0νββ Majorana test).
• Gamma decay: E_γ = ΔE − E_R (E_R = E_γ²/2Mc²); multipole radiation of order L: |J_i − J_f| ≤ L ≤ J_i + J_f (L ≥ 1; 0⁺ → 0⁺ only by internal conversion/pair production). Parity rule: EL: Δπ = (−1)^L; ML: Δπ = (−1)^{L+1}. E1: yes, M1: no, E2: no, M2: yes. Transition rates: λ(E1) ≫ λ(M1) ≫ λ(E2), Weisskopf: λ(E1) ≈ 1.0 × 10¹⁴A^{2/3}E³ s⁻¹ (E in MeV); each higher L reduced by (kR)² ~ 10⁻⁵–10⁻³. Internal conversion competes with γ emission: e⁻ energy = E_γ − B_e; coefficient α = N_e/N_γ; Mössbauer effect (⁵⁷Fe 14.4 keV recoil-free).
• Isomers: metastable state with long half-life (ΔJ large).
• Radioactive decay law N = N₀e^{−λt}; T½ = ln2/λ = 0.693τ; activity A = λN (1 Ci = 3.7 × 10¹⁰ Bq); series: secular equilibrium λ₁N₁ = λ₂N₂ (T₁ ≫ T₂); transient equilibrium; carbon dating T½(¹⁴C) = 5730 y; branching ratios partial λ = Σλᵢ.
• Mean life relation to natural width Γ = ℏ/τ.

C. High-Yield Points

1. Shell model's key ingredient is strong spin–orbit coupling (lowers j = l + ½ levels) — responsible for magic numbers 28, 50, 82, 126 (harmonic oscillator alone gives 2, 8, 20, 40, 70, 112).
2. Ground-state spin-parity is determined by the last unpaired nucleon (odd-A); even–even always 0⁺; magnetic moments between the Schmidt lines.
3. α decay requires tunnelling: strong T½ dependence on E_α (Geiger–Nuttall); α with l = 0 for 0⁺ → 0⁺; Q_α > 0 for A ≳ 150.
4. β decay spectrum continuous because of third particle (antineutrino); average e energy ~ ⅓E₀; EC shows monoenergetic neutrino; use atomic masses with correct m_e terms.
5. Gamma rays: 0 → 0 transition can't proceed via single γ; E1 changes parity, M1/E2 does not; multipole selection ΔJ = L (min).
6. Parity is violated in weak interaction (β decay, ⁶⁰Co), conserved in strong and EM; helicity: ν_e left-handed.
7. ⁴He (doubly magic) exceptional binding (B = 28.3 MeV): why α emitted instead of other clusters.

D. Comparison Table

DAY 27 — Elementary Particles, Quarks, Conservation Laws

A. Core Definitions

• Fundamental interactions: strong (gluon, range ~1 fm, α_s ~ 1), EM (photon, α = 1/137), weak (W^±, Z⁰; range ~10⁻³ fm; G_F = 1.17 × 10⁻⁵ GeV⁻²), gravity (graviton; ~10⁻³⁸).
• Leptons: e, μ, τ + ν_e, ν_μ, ν_τ (spin ½, no strong interaction). Hadrons: baryons (qqq, B = 1, fermions) and mesons (qq̄, B = 0, bosons).
• Quarks: charges u, c, t: +2/3; d, s, b: −1/3; baryon number 1/3, spin ½, three colours. Masses: u ~2 MeV, d ~5 MeV, s ~95 MeV, c ~1.27 GeV, b ~4.2 GeV, t ~173 GeV. Confinement; asymptotic freedom.

B. Key Equations / Rules

• Quantum numbers: charge Q; baryon number B; lepton numbers L_e, L_μ, L_τ; isospin I, I₃; strangeness S; charm C; bottomness B̃; topness T; hypercharge Y = B + S (+C+B̃+T).
• Gell-Mann–Nishijima: Q = I₃ + Y/2 = I₃ + (B + S)/2.
• Quark content: p = uud (I = ½, I₃ = +½); n = udd (I₃ = −½); Λ = uds (S = −1, I = 0); Σ⁺ = uus, Σ⁰ = uds, Σ⁻ = dds (S = −1, I = 1); Ξ⁰ = uss, Ξ⁻ = dss (S = −2); Ω⁻ = sss (S = −3; spin 3/2⁺; predicted by Gell-Mann, found 1964); Δ⁺⁺ = uuu (needs colour). π⁺ = ud̄, π⁻ = dū, π⁰ = (uū − dd̄)/√2; K⁺ = us̄ (S = +1), K⁻ = sū, K⁰ = ds̄, K̄⁰ = sd̄; J/ψ = cc̄ (3.097 GeV), Υ = bb̄ (9.46 GeV); D⁰ = cū, B⁰ = db̄.
• Meson multiplet: J^P = 0⁻ pseudoscalar octet (π, K, η) and 1⁻ vector nonet (ρ, K, ω, φ); baryon octet (½⁺) and decuplet (3/2⁺): SU(3)-flavour. Decuplet mass spacing equal (~150 MeV): Δ(1232), Σ(1385), Ξ*(1530), Ω⁻(1672).
• Conservation laws: see the conservation table in section D below.
• Parity: intrinsic: bosons vs. fermions: P(p) = P(n) = +1, P(π) = −1, P(γ) = −1; fermion–antifermion opposite; for particle–antiparticle boson pair same; parity of orbital (−1)^l. Pion spin from π⁺ + d ⇌ p + p. For meson qq̄ with orbital l: P = (−1)^{l+1}, C = (−1)^{l+s}. C-parity: π⁰ → γγ (C = +1), forbidden π⁰ → 3γ; γ has C = −1.
• Isospin: nucleon doublet; pion triplet I = 1; Δ(1232) I = 3/2; strong-reaction amplitudes factorise by total I via Clebsch–Gordan coefficients (π⁺p is a pure I = 3/2 state → Δ⁺⁺(1232) resonance). Mass formula Gell-Mann–Okubo.
• Reaction checks: e.g., p + p → p + p + π⁰ allowed; p + n → d + γ; K⁻ + p → Λ + π⁰ (strong, S: −1 + 0 → −1 + 0); π⁻ + p → K⁰ + Λ (associated production, S 0 → +1 − 1); Λ → p + π⁻ (weak, ΔS = 1, T ~ 2.6 × 10⁻¹⁰ s); π⁺ → μ⁺ + ν_μ; μ → e + ν̄_e + ν_μ; forbidden: p → e⁺ + π⁰ (B violated), μ → e + γ (lepton flavour), p + p → Σ⁺ + n (ΔS = −1: forbidden in strong interaction), n → p + e (L violated).
• Antiparticles: same mass & spin, opposite additive quantum numbers; discovery of positron (Anderson 1932), antiproton (1955); e⁺e⁻ → γγ.
• Lifetimes & interaction: strong ~10⁻²³ s (resonances, Δ, ρ), EM ~10⁻¹⁶ s (π⁰ 8.5 × 10⁻¹⁷ s), weak 10⁻¹⁰–10⁻⁸ s (K, Λ, π^± 2.6 × 10⁻⁸ s), muon 2.2 µs, neutron 880 s. Width Γτ = ℏ.
• Standard Model: gauge group SU(3)_C × SU(2)_L × U(1)_Y; W⁺, W⁻ (80.4 GeV), Z (91.2 GeV), Higgs (125 GeV, discovered 2012); three generations; electroweak unification (Glashow–Weinberg–Salam), sin²θ_W ≈ 0.23; CKM mixing matrix (Cabibbo angle ~13°); neutrino oscillations (PMNS).
• Feynman exchange: Yukawa range r = ℏ/mc. Cross-section; Rutherford; deep inelastic scattering ⇒ partons (quarks), R = σ(e⁺e⁻ → hadrons)/σ(e⁺e⁻ → μ⁺μ⁻) = N_cΣe_q² (=2 for u,d,s; 10/3 with c; 11/3 with b).
• Relativistic kinematics: threshold; invariant mass s = (E₁ + E₂)² − |p₁ + p₂|²; fixed-target s = m₁² + m₂² + 2E₁m₂.

C. High-Yield Points

1. Q = I₃ + (B + S)/2 holds for all hadrons (with Y = B + S for u,d,s only); charm etc. extend Y.
2. Strong & EM conserve strangeness; weak does not (ΔS = ±1 per decay); strange particles produced in pairs (associated production) yet decay slowly (weak).
3. Parity is violated in weak interactions (Lee–Yang 1956, Wu 1957); CP violation in neutral K (Cronin–Fitch 1964); CPT is conserved always.
4. Ω⁻ (sss) and Δ⁺⁺ (uuu) require a new quantum number (colour) for Pauli's principle; confinement ⇒ no free quark; hadrons are colour singlets.
5. Leptons: μ → e + γ forbidden (separate lepton family numbers); neutrino oscillations violate individual L_i but conserve total L.
6. Pion is a pseudoscalar (J^P = 0⁻); π⁰ → 2γ (EM), π⁺ → μ⁺ν (weak); mass difference m_π± − m_π⁰ = 4.6 MeV.
7. Baryon decuplet of equal mass-spacing predicted Ω⁻ mass; ratio R in e⁺e⁻ gives number of colours (N_c = 3).

D. Comparison Table

DAY 28 — Diodes, Zener Regulators, BJT, Biasing

A. Core Definitions

• p–n junction: diffusion of carriers creates depletion region with built-in potential V_bi = (kT/e)ln(N_AN_D/n_i²) (Si ≈ 0.6–0.8 V). Forward bias reduces barrier; reverse increases it.
• Zener diode: heavily doped p–n junction operating in reverse breakdown (Zener tunnelling V_Z < ~5 V, negative temperature coefficient; avalanche > ~7 V, positive T.C.).
• BJT: three-layer device (npn/pnp); emitter heavily doped, base thin & lightly doped, collector large.

B. Key Equations / Rules

• Shockley diode: I = I_s(e^{V/ηV_T} − 1), V_T = kT/e = 25.9 mV at 300 K (≈ 26 mV), η = 1–2. Forward turn-on Si 0.7 V, Ge 0.3 V, LED 1.8–3.3 V (GaAs 1.4 V, GaN blue ~3 V), Schottky 0.2–0.4 V. I_s doubles every ~10 °C; V decreases ~2 mV/°C at constant I. Dynamic resistance r_d = ηV_T/I (26 Ω at 1 mA).
• Depletion width W = √[2ε(V_bi − V)(N_A + N_D)/eN_AN_D]; junction capacitance C_j = C₀/(1 − V/V_bi)^{1/2} (varactor); diffusion capacitance forward.
• Rectifiers: half-wave V_dc = V_m/π, V_rms = V_m/2, ripple factor 1.21, efficiency 40.6%, PIV = V_m; full-wave (centre-tap/bridge) V_dc = 2V_m/π, V_rms = V_m/√2, ripple 0.48, efficiency 81.2%, PIV = 2V_m (centre-tap), V_m (bridge). Capacitor filter ripple V_r = I_L/(fC) (full-wave: f → 2f_line). Clamper, clipper, voltage doubler.
• Zener shunt regulator: series R_S; I_S = (V_in − V_Z)/R_S = I_Z + I_L; load regulation: I_Z,min ≤ I_Z; R_S limits: R_S,max = (V_in,min − V_Z)/(I_Z,min + I_L,max), R_S,min = (V_in,max − V_Z)/(I_Z,max + I_L,min). Power P_Z = V_ZI_Z ≤ P_max. Output resistance ≈ r_Z.
• BJT currents: I_E = I_B + I_C; I_C = αI_E + I_CBO; α = I_C/I_E (0.95–0.99); β = I_C/I_B = α/(1 − α); I_C = βI_B + (1+β)I_CBO.
• Active region: BE forward, BC reverse; saturation: both forward (V_CE,sat ≈ 0.2 V); cut-off: both reverse; reverse active: BE reverse, BC forward.
• Common-emitter characteristics: input I_B–V_BE (diode), output I_C–V_CE family; Early effect (base-width modulation) slope; CB: input I_E–V_EB, output I_C–V_CB. Configurations: CB (A_i ≈ 1, low R_in, high R_out, no phase shift), CE (high A_v & A_i, 180° phase shift, medium R_in), CC/emitter follower (A_v ≈ 1, high R_in, low R_out, buffer).
• Transconductance g_m = I_C/V_T; r_π = β/g_m; r_e = V_T/I_E; CE small-signal A_v = −g_mR_C (unbypassed R_E: −R_C/(r_e + R_E)); h-parameters.
• Biasing: fixed bias: I_B = (V_CC − V_BE)/R_B; Q: I_C = βI_B, V_CE = V_CC − I_CR_C; highly β-dependent (stability factor S = 1 + β). Collector-to-base (feedback) bias: S lower than fixed bias. Voltage-divider (self) bias: V_B = V_CCR₂/(R₁+R₂) (Thevenin R_B = R₁‖R₂), I_E = (V_B − V_BE)/(R_E + R_B/β) ≈ (V_B − 0.7)/R_E, S = (1 + β)/(1 + βR_E/(R_E + R_B)) ≈ 1 + R_B/R_E (best stability when R_B ≪ βR_E). DC load line: V_CE = V_CC − I_C(R_C + R_E); Q-point at centre for maximum symmetrical swing; thermal runaway (I_CBO doubles per 10 °C).
• Transistor as switch: cut-off (OFF), saturation (ON): I_B > I_C(sat)/β_min.
• Classes: A (360° conduction, η ≤ 25% (resistive) / 50% (transformer-coupled)), B (180°, η ≤ 78.5%, crossover distortion), AB, C (< 180°, tuned RF, η > 78.5%).
• FETs: JFET (n-channel, depletion type): I_D = I_DSS(1 − V_GS/V_P)², g_m = (2I_DSS/|V_P|)(1 − V_GS/V_P); pinch-off V_DS ≥ V_GS − V_P. MOSFET enhancement NMOS: I_D = (k/2)(V_GS − V_T)² (saturation), triode I_D = k[(V_GS − V_T)V_DS − V_DS²/2]; high input impedance; CMOS.
• Opto-electronic devices: Solar cell I = I_L − I_s(e^{eV/kT} − 1); V_oc = (kT/e)ln(1 + I_L/I_s); short-circuit I_sc = I_L; fill factor FF = P_max/(V_ocI_sc) (~0.7–0.85); efficiency η = P_max/P_in; Shockley–Queisser limit ≈ 33.7% (E_g ≈ 1.3 eV). Photodiode reverse biased photocurrent ∝ intensity (PIN, APD gain); LED forward biased radiative recombination, λ(µm) = 1.24/E_g(eV), direct gap material (GaAs, GaN, GaAsP); laser diode needs inversion + feedback.
• Tunnel diode (negative resistance, heavily doped), Schottky (majority carrier, fast), varactor, PIN, Gunn.

C. High-Yield Points

1. β = α/(1−α); α < 1 always; CE gives both voltage and current gain and 180° phase reversal; CC is a voltage buffer with A_v ≈ 1.
2. Voltage-divider bias is the most stable: stability depends on R_B/R_E; fixed bias is very β-sensitive; Q-point should lie at the middle of the load line.
3. Zener: breakdown mechanism: Zener (tunnelling, < 5 V, −ve T.C.) vs avalanche (> 7 V, +ve T.C.); ~5.6 V zero temperature coefficient. Regulation requires I_Z ≥ I_Z,min even at maximum load and minimum supply.
4. Diode V_F decreases ~2.1 mV/°C; reverse saturation current rises with temperature (doubles ~10 °C for Ge/Si).
5. Full-wave bridge PIV = V_m; centre-tapped FWR PIV = 2V_m; HWR efficiency 40.6%, FWR 81.2%.
6. Early effect: output conductance in the active region (I_C rises slightly with V_CE).
7. Solar cell operates in fourth quadrant of I–V curve; V_oc ∝ ln(I_L); under-bandgap photons not absorbed, excess lost as heat.

D. Comparison Table

DAY 29 — Operational Amplifiers

A. Core Definitions

• Ideal op-amp: infinite open-loop gain A_OL, infinite input impedance, zero output impedance, infinite bandwidth, infinite CMRR, zero offset, infinite slew rate.
• Golden rules (negative feedback): (1) no current into inputs; (2) V₊ = V₋ (virtual short; virtual ground when non-inverting input grounded).
• CMRR = |A_d/A_cm| (dB: 20log; 741: 90 dB). Slew rate SR = dV_o/dt|max (741: 0.5 V/µs); full-power bandwidth f_max = SR/2πV_peak. Gain–bandwidth product GBW = A_v × f{−3dB} = f_unity (741: 1 MHz).

B. Key Equations / Rules

• Inverting amplifier: A_v = −R_f/R_in; R_in(input) = R_in(resistor); output = −(R_f/R₁)V_in.
• Non-inverting amplifier: A_v = 1 + R_f/R₁; input impedance very high. Voltage follower: A_v = 1 (buffer).
• Summing (inverting): V_o = −R_f(V₁/R₁ + V₂/R₂ + V₃/R₃); with equal R's V_o = −(V₁ + V₂ + V₃). Non-inverting summer averaging; difference amplifier: V_o = (R₂/R₁)(V₂ − V₁) with matched ratios (R₁/R₂ = R₃/R₄).
• Instrumentation amplifier (three op-amp): A_v = (1 + 2R/R_G)(R₃/R₂); high CMRR, high Z_in.
• Integrator: V_o = −(1/RC)∫V_in dt (square → triangular, sine → −cosine); practical: shunt R_f across C to limit DC gain. Differentiator: V_o = −RC dV_in/dt (sine → −cosine ⇒ noisy; high-frequency gain, add series R).
• Log amplifier V_o = −V_T ln(V_in/RI_s); antilog; precision rectifier; comparator (open loop; Schmitt trigger with hysteresis ΔV = 2R₁V_sat/(R₁ + R₂)); current-to-voltage (transimpedance) V_o = −I R_f; V-to-I converter.
• Closed-loop gain from finite A_OL: A_CL = A_OL/(1 + A_OLβ) ≈ 1/β; feedback factor β = R₁/(R₁ + R_f); bandwidth f_CL = GBW/A_CL.
• Input offset voltage, bias currents (I_B = (I_B1 + I_B2)/2), offset current; compensate R = R₁‖R_f.
• Active filters (first order): low-pass: A(jω) = A₀/(1 + jω/ω_c), f_c = 1/2πRC, roll-off −20 dB/decade; high-pass f_c = 1/2πRC, +20 dB/dec rising; second-order (Sallen–Key) −40 dB/decade, Butterworth maximally flat Q = 1/√2 = 0.707 (A₀ = 1.586 for equal R,C); Chebyshev ripple in pass band steeper cut-off; Bessel linear phase. Sallen–Key low-pass: f_c = 1/(2π√(R₁R₂C₁C₂)). Band-pass: Q = f₀/BW; band-reject (notch) f₀ = 1/2πRC; all-pass shifts phase. Order n ⇒ −20n dB/decade.
• Oscillators: Barkhausen: loop gain |Aβ| = 1, phase 0° (2πn). Wien-bridge f = 1/2πRC (A_v ≥ 3 non-inverting); RC phase-shift f = 1/(2πRC√6), gain ≥ 29 (inverting, 3 sections); Colpitts f = 1/[2π√(L C₁C₂/(C₁ + C₂))]; Hartley f = 1/[2π√((L₁ + L₂ + 2M)C)]; crystal oscillators (series/parallel resonance, very high Q, stability 10⁻⁶–10⁻⁹).
• 555 timer: astable f = 1.44/[(R_A + 2R_B)C], duty D = (R_A + R_B)/(R_A + 2R_B); monostable t = 1.1RC.
• Feedback in amplifiers: negative feedback ⇒ gain stability, bandwidth ↑ (×(1+Aβ)), distortion & noise ↓, Z_in ↑ (series mixing) or ↓ (shunt mixing), Z_out ↓ (voltage sampling) or ↑ (current sampling).
• Impedance matching: max power transfer R_L = R_s (50% efficiency); signal conditioning, shielding/grounding; lock-in (phase-sensitive detection) & boxcar integrator (S/N ∝ √N); noise: thermal (Johnson) V_n² = 4kTRΔf, shot noise I_n² = 2eIΔf, 1/f noise.

C. High-Yield Points

1. Inverting configuration: gain −R_f/R₁, input impedance = R₁ (not infinite); non-inverting: gain 1 + R_f/R₁ ≥ 1 (can't give less than 1), input impedance very high; follower used for buffering.
2. GBW is constant for voltage-feedback op-amps: gain 100 ⇒ BW = f_unity/100.
3. Integrator output for a constant input is a ramp (saturates); for square input → triangle; differentiator of triangle → square; the integrator with no DC feedback drifts due to offsets.
4. Comparator: no negative feedback ⇒ output saturates at ±V_sat; virtual short does not apply; Schmitt trigger has positive feedback (hysteresis) to reject noise.
5. Wien-bridge requires gain ≥ 3 at f = 1/2πRC (phase 0°); RC phase-shift oscillator 3 sections each 60° ⇒ gain ≥ 29.
6. Slew-rate limiting causes distortion of large-amplitude high-frequency sines: f_max = SR/(2πV_p). 741: SR = 0.5 V/µs; for 10 V amplitude f_max ≈ 8 kHz.
7. Differential amplifier rejects common-mode signals; CMRR high for tail-current source; Butterworth filter pass-band flat; −3 dB at cut-off f_c = 1/2πRC.

D. Comparison Table

DAY 30 — Digital Logic, Error Analysis, Curve Fitting

A. Core Definitions

• Boolean algebra: AND (·), OR (+), NOT (′). Universal gates: NAND, NOR. Combinational circuit: output depends only on current inputs; sequential: depends on inputs and stored state (flip-flops, clocked).
• Minterm (product with all variables, 1), maxterm (sum with all variables, 0). SOP (sum of minterms) and POS. K-map: gray-code ordered grid for minimisation; groups of 1, 2, 4, 8 (power of 2) adjacent cells (wraps around edges); don't-cares (X) used to enlarge groups.
• Accuracy (closeness to true value; systematic error) vs precision (reproducibility; random error).

B. Key Equations / Rules

• Boolean identities: A + AB = A; A + A′B = A + B; AB + A′C + BC = AB + A′C (consensus); De Morgan: (AB)′ = A′ + B′; (A + B)′ = A′B′. NAND = AND-OR inverter; XOR A ⊕ B = A′B + AB′ (odd parity), XNOR equality; A ⊕ A = 0; A ⊕ 1 = A′.
• NAND-only: NOT: A NAND A; AND: NAND followed by NAND-as-NOT; OR: (A′ NAND B′).
• Half-adder: S = A ⊕ B, C = AB. Full-adder: S = A ⊕ B ⊕ C_in; C_out = AB + C_in(A ⊕ B) (two half-adders + OR). Half-subtractor: D = A ⊕ B, Borrow = A′B. Ripple-carry adder delay ∝ n; carry-lookahead faster.
• K-map procedure: group adjacent 1-cells (wrap-around allowed); each group eliminates variables that change. Number of variables eliminated = log₂(group size). 4-variable map: 16 cells; pairs eliminate 1 variable, quads 2, octets 3.
• Multiplexer (2ⁿ:1): Y = Σ(I_k · m_k(select)); implement any function of n+1 variables with 2ⁿ:1 MUX; demultiplexer/decoder n → 2ⁿ lines (each output a minterm); encoder 2ⁿ → n; priority encoder; comparator; parity generator; BCD, Gray code (Gray: G = B ⊕ (B >> 1)), excess-3.
• Flip-flops: SR: Q⁺ = S + R′Q (SR = 0 forbidden); JK: Q⁺ = JQ′ + K′Q (J = K = 1 toggles; no invalid state; race-around cured by master-slave/edge-triggering); D: Q⁺ = D; T: Q⁺ = T ⊕ Q (T = 1 toggles; JK with J = K = T). Latch (level-sensitive) vs flip-flop (edge-triggered); setup/hold times.
• Counters: n flip-flops → mod-2ⁿ counter, divides frequency by 2ⁿ; asynchronous (ripple) counter: propagation delay accumulates, max clock f_max = 1/(n·t_pd); synchronous counter all FFs clocked simultaneously; mod-N counter (reset at N via NAND on outputs; need n = ⌈log₂N⌉ FFs; e.g., decade/BCD counter 7490 mod-10 needs 4 FFs); up/down; ring counter (n states with n FFs; self-starting needs correction; initial 1000…); Johnson (twisted-ring) counter 2n states with n FFs (e.g., 4 FFs → 8 states).
• Registers: shift registers SISO, SIPO, PISO, PIPO (n bits, n clock pulses to serialise); universal shift register; left shift multiplies by 2, right shift divides by 2 (unsigned).
• Number systems: binary ↔ hex conversion by 4-bit groups; 2's complement: −x = ~x + 1; range −2^{n−1} to 2^{n−1} − 1; 1's complement; sign-magnitude. ADC resolution Δ = V_FS/2ⁿ; DAC: V_out = V_ref(Σbᵢ2^{i})/2ⁿ; binary-weighted resistor vs R-2R ladder; flash ADC (2ⁿ − 1 comparators, fastest), successive-approximation (n clock cycles), dual-slope (accurate, slow), counter/ramp; Nyquist f_s ≥ 2f_max (aliasing); quantisation error ±½LSB; dynamic range ≈ 6.02n + 1.76 dB. Memory: 2^n × m bits; ROM/PROM/EPROM, SRAM/DRAM. Logic families: TTL (V_CC 5 V, V_IH min 2 V, V_IL max 0.8 V), CMOS (low power, wide noise margin); noise margin; fan-out. Microprocessor basics: ALU, registers, program counter, buses (address 2ⁿ locations, 8085: 16-bit address = 64 KB, 8-bit data), microcontroller adds on-chip ROM/RAM/peripherals (8051).
• Error analysis: mean x̄ = Σxᵢ/N; sample standard deviation s = √[Σ(xᵢ − x̄)²/(N − 1)]; standard error of mean σ_m = s/√N; Gaussian ±1σ = 68.3%, ±2σ = 95.4%, ±3σ = 99.7%; Poisson counting: σ = √N, relative error 1/√N.
• Propagation (independent errors, quadrature): z = x ± y: σ_z² = σ_x² + σ_y²; z = xy or x/y: (σ_z/z)² = (σ_x/x)² + (σ_y/y)²; z = xⁿ: σ_z/z = |n|σ_x/x; z = ln x: σ_z = σ_x/x; z = eˣ: σ_z/z = σ_x; general σ_z² = Σ(∂z/∂xᵢ)²σᵢ². Correlated: add covariance term; worst-case: linear sum. Weighted mean x̄_w = Σ(xᵢ/σᵢ²)/Σ(1/σᵢ²), σ_w = (Σ1/σᵢ²)^{−1/2}. Significant figures; systematic errors (calibration, zero offset) do not reduce with N.
• Least-count error (±LC/2 or LC); vernier, screw gauge.
• Least-squares fit y = mx + c: m = [NΣxy − ΣxΣy]/[NΣx² − (Σx)²]; c = [Σy − mΣx]/N (= ȳ − mx̄); σ_m² = Nσ_y²/Δ, σ_c² = σ_y²Σx²/Δ, Δ = NΣx² − (Σx)²; σ_y² = Σ(yᵢ − mxᵢ − c)²/(N − 2); correlation r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)²Σ(y − ȳ)²], |r| ≤ 1; r² = fraction of variance explained. Fit through origin: m = Σxy/Σx². Weighted fit: weights 1/σᵢ².
• Linearisation: y = aeᵇˣ → ln y vs x; y = axᵇ → log y vs log x; y = a/x → y vs 1/x.
• χ² test: χ² = Σ[(yᵢ − f(xᵢ))/σᵢ]²; degrees of freedom ν = N − (number of fitted parameters); reduced χ²_ν = χ²/ν ≈ 1 for good fit (≪ 1 overestimated errors / overfitting; ≫ 1 poor model or underestimated errors); mean of χ² distribution = ν, variance 2ν. Nonlinear fit: Levenberg–Marquardt, Gauss–Newton; Gaussian/binomial/Poisson distributions: Poisson mean = variance; binomial mean Np, variance Npq; Gaussian approximates binomial for large N.
• Interpolation (Lagrange/Newton), numerical integration: trapezoid error ∝ h², Simpson ∝ h⁴ (needs even number of intervals); Newton–Raphson x_{n+1} = x_n − f/f′ (quadratic convergence); bisection (linear); Runge–Kutta 4th order (error ∝ h⁵ per step), Euler (h²).
• Transducers: thermocouple (Seebeck), RTD (Pt100: R = R₀(1 + αT), α = 0.00385/K), thermistor (NTC: R = R₀exp[B(1/T − 1/T₀)]); strain gauge (gauge factor G = (ΔR/R)/ε ≈ 2); LVDT; Pirani (thermal conductivity) and ionisation gauges (high vacuum); Hall probe, fluxgate/SQUID; photodiode/PMT; Geiger–Müller, scintillation, semiconductor detectors.
• Lock-in: multiplies signal by reference, low-pass filters ⇒ narrow-band detection (noise reduction ∝ √bandwidth); S/N improved by signal averaging ∝ √N.

C. High-Yield Points

1. NAND and NOR are universal; XOR is not. De Morgan converts AND-OR to NAND-NAND. A K-map group size must be a power of 2; larger groups ⇒ fewer literals.
2. Johnson counter: n FFs → 2n states; ring counter: n states; binary: 2ⁿ states; mod-N needs ⌈log₂N⌉ flip-flops.
3. JK with J = K = 1 toggles; SR with S = R = 1 forbidden/indeterminate; D latch is a transparent delay; T FF divides frequency by 2.
4. Full-adder equations: Sum = A⊕B⊕C; Carry = AB + BC + CA; in a ripple counter the total delay accumulates (limits speed).
5. Standard error of the mean decreases as 1/√N (random error only); systematic errors don't average out. Error in a difference of nearly equal numbers is relatively large; for a power xⁿ relative error multiplies by n.
6. Reduced χ² ≈ 1 indicates good fit; χ²_ν ≫ 1 poor fit or underestimated errors; χ²_ν ≪ 1 overestimated errors. Degrees of freedom = N − p.
7. Flash ADC fastest (2ⁿ − 1 comparators); dual-slope best noise rejection; 8-bit ADC has 256 levels, resolution V_FS/255 (or /256); Nyquist rate = 2f_max; uncertainty from least count = ± half the smallest division.

D. Comparison Table

Appendix — Constants & Quick Reference