GATE PH cheat sheet

Cauchy–Riemann: for f(z)=u+iv analytic, u_x=v_y and u_y=−v_x; then u and v satisfy Laplace's equation.

Residue theorem: ∮C f(z)dz=2πi Σ Res(f,z_k) for poles inside C; simple-pole residue = lim{z→z0}(z−z0)f(z).

Fourier transform pair with F(k)=∫f(x)e^{−ikx}dx and f(x)=(1/2π)∫F(k)e^{ikx}dk; Parseval: ∫|f|²dx=(1/2π)∫|F|²dk.

Hermitian matrices have real eigenvalues and orthogonal eigenvectors; Pauli matrices σ_x, σ_y, σ_z are Hermitian, traceless, and satisfy σ_iσ_j=δ_ij I+iε_ijkσ_k.

Legendre equation (1−x²)y″−2xy′+l(l+1)y=0 gives P_l(x); Bessel equation x²y″+xy′+(x²−n²)y=0 gives J_n(x).

Gamma function: Γ(z+1)=zΓ(z), Γ(n+1)=n! for integer n≥0, Γ(1/2)=√π; beta function B(p,q)=Γ(p)Γ(q)/Γ(p+q).

Separation of Laplace's equation in spherical coordinates gives spherical harmonics Y_lm(θ,φ), with l=0,1,2,... and m=−l,...,l.

Jordan's lemma: for f(z)→0 uniformly as |z|→∞, ∫ along a large semicircle of f(z)e^{iaz}dz→0 for a>0, enabling real integral evaluation.

Lagrangian L=T−V; Euler–Lagrange: d/dt(∂L/∂q̇_i)−∂L/∂q_i=0.

Hamiltonian H=Σp_iq̇_i−L; Hamilton equations q̇_i=∂H/∂p_i and ṗ_i=−∂H/∂q_i.

Noether theorem: every continuous symmetry of the action has a conserved charge; time translation→energy, spatial translation→momentum, rotation→angular momentum.

Central force: U_eff(r)=U(r)+L²/(2mr²); circular orbit satisfies dU_eff/dr=0, and stability requires d²U_eff/dr²>0.

Small oscillations: expand V and T to second order; normal-mode frequencies ω satisfy det(V_ij−ω²T_ij)=0.

Rigid-body Euler equations: I_1ω̇_1+(I_3−I_2)ω_2ω_3=τ_1, with cyclic permutations; principal axes diagonalize the inertia tensor.

Poisson brackets: {q_i,p_j}=δ_ij and {q_i,q_j}={p_i,p_j}=0; dA/dt=∂A/∂t+{A,H}.

Special relativity: invariant interval s²=c²t²−r²; four-momentum p^μ=(E/c,p) with E²=p²c²+m²c⁴.

Maxwell differential forms: ∇·E=ρ/ε0, ∇·B=0, ∇×E=−∂B/∂t, ∇×B=μ0J+μ0ε0∂E/∂t.

Vacuum wave speed c=1/√(μ0ε0); plane wave has E⊥B⊥k, B=E/c, and phase velocity ω/k.

Poynting vector S=E×H; energy density u=(1/2)(ε0E²+B²/μ0); continuity ∂u/∂t+∇·S=−J·E.

Boundary conditions: tangential E continuous; normal B continuous; normal D jump = σ_free; tangential H jump = K_free×n̂.

Lorenz gauge ∇·A+(1/c²)∂φ/∂t=0 gives □²A=−μ0J and □²φ=−ρ/ε0; gauge invariance leaves E,B unchanged.

Larmor formula for nonrelativistic accelerated charge: P=q²a²/(6πε0c³); electric dipole radiation scales as P∝p̈².

Lorentz force F=q(E+v×B); magnetic force does no work, while electric field changes kinetic energy.

Good-conductor skin depth δ=√(2/(ωμσ)); fields decay as e^{−z/δ}, and waveguide cutoff requires ω>ω_c for propagation.

Time-dependent Schrödinger: iℏ∂ψ/∂t=Ĥψ; probability density ρ=|ψ|² and current j=(ℏ/2mi)(ψ*∇ψ−ψ∇ψ*).

Canonical commutator [x,p]=iℏ; uncertainty ΔxΔp≥ℏ/2; incompatible observables cannot have simultaneous eigenstates.

Infinite well E_n=n²π²ℏ²/(2mL²), ψ_n=√(2/L)sin(nπx/L), n=1,2,...; zero-point energy is nonzero.

Harmonic oscillator E_n=(n+½)ℏω; a|n⟩=√n|n−1⟩ and a†|n⟩=√(n+1)|n+1⟩.

Angular momentum: L²|l,m⟩=l(l+1)ℏ²|l,m⟩, L_z|l,m⟩=mℏ|l,m⟩, with m=−l,...,l.

Hydrogen: E_n=−13.6 eV/n²; degeneracy n² including spin; radial probability peaks near Bohr radius a0.

First-order nondegenerate perturbation: E_n¹=⟨n|V|n⟩; degenerate case diagonalizes V in the degenerate subspace.

Spin-1/2: S_i=(ℏ/2)σ_i; measurement of S_z gives ±ℏ/2, and spin addition uses Clebsch–Gordan coefficients.

First law dU=đQ−đW; for reversible P dV work, đW=P dV; U is a state function and ΔU=0 over a cycle.

Second law: dS≥đQ/T; isolated-system entropy never decreases; reversible processes have equality.

Thermodynamic potentials: H=U+PV, F=U−TS, G=H−TS; Maxwell relations follow from equality of mixed second derivatives.

Ideal gas PV=nRT; U=C_VT for constant C_V; C_P−C_V=nR; adiabatic reversible PV^γ=constant with γ=C_P/C_V.

Canonical ensemble: Z=Σe^{−βE_i}, F=−kT lnZ, ⟨E⟩=−∂lnZ/∂β, β=1/kT.

Fermi–Dirac and Bose–Einstein occupancies: 1/(e^{β(E−μ)}±1); classical Maxwell–Boltzmann limit when e^{β(E−μ)}≫1.

Planck law u(ν)=8πhν³/c³ · 1/(e^{hν/kT}−1); Stefan–Boltzmann P=σAT⁴; Debye heat capacity ∝T³ at low T.

Clausius–Clapeyron dP/dT=L/(TΔv); phase boundary slope depends on latent heat and volume change.

Bohr model: E_n=−13.6Z²/n² eV; radius r_n=n²a0/Z; angular momentum L=nℏ.

Rydberg formula 1/λ=RZ²(1/n_f²−1/n_i²); Lyman, Balmer, Paschen series correspond to n_f=1,2,3.

Selection rules for electric dipole: Δl=±1, Δm=0,±1, Δs=0; parity must change.

Fine structure from spin–orbit coupling, relativistic correction, and Darwin term; total J=L+S and term symbol ^{2S+1}L_J.

Landé g_J=1+[J(J+1)+S(S+1)−L(L+1)]/[2J(J+1)]; Zeeman shift ΔE=g_Jμ_BBm_J.

Rigid rotor E_J=BJ(J+1), B=ℏ²/(2I); infrared ΔJ=±1; Raman ΔJ=0,±2.

Harmonic vibration E_v=ℏω(v+½); infrared active if dipole moment changes; Raman active if polarizability changes.

X-ray Moseley law √ν∝(Z−σ); Auger electron emission competes with X-ray fluorescence after inner-shell ionization.

Bravais lattice: r=R+n1a1+n2a2+n3a3; reciprocal lattice vectors satisfy a_i·b_j=2πδ_ij.

Bragg law 2d sinθ=nλ; Laue condition Δk=G; structure factor determines absent reflections.

Free electron gas: E_F=ℏ²(3π²n)^{2/3}/(2m); density of states g(E)∝√E in 3D.

Bloch theorem ψ_k(r)=e^{ik·r}u_k(r); band gaps open at Brillouin-zone boundaries; effective mass m*=ℏ²/(d²E/dk²).

Semiconductor carriers: n_i²=N_CN_V exp(−E_g/kT); n-type Fermi level near conduction band, p-type near valence band.

Drude conductivity σ=ne²τ/m; Hall coefficient R_H=1/(ne) for single-carrier free-electron model.

Phonons: Debye model gives C_V∝T³ at T≪Θ_D; Einstein model gives exponentially small C_V at low T.

Superconductors: Meissner effect expels B; type-II has vortices; BCS gap Δ≈1.76kT_c and Cooper pairs.

Binding energy B(Z,A)=[Zm_p+(A−Z)m_n−M(A,Z)]c²; binding-energy curve peaks near A≈56–62.

Semi-empirical mass formula: volume − surface − Coulomb − asymmetry ± pairing; explains fission of heavy nuclei and fusion of light nuclei.

Radioactive decay N=N0e^{−λt}; t_1/2=ln2/λ; activity A=λN; mean lifetime τ=1/λ.

Alpha decay Q=(m_parent−m_daughter−m_alpha)c²; beta decay conserves charge and lepton number; gamma emission conserves A and Z.

Nuclear reaction conservation: A, Z, energy–momentum, angular momentum, and baryon number; Q=(m_reactants−m_products)c².

Standard model fermions: six quarks (u,d,c,s,t,b) and six leptons; gauge bosons are photon, W±, Z, gluons; Higgs gives mass.

Conservation in particle reactions: baryon number, lepton number, electric charge, energy; strangeness is conserved in strong/EM but not weak interactions.

QED vertex: electron–photon coupling strength √α; Feynman rules multiply vertices by coupling and conserve four-momentum at each vertex.

Diode: I=I_S(e^{V/(nV_T)}−1); V_T=kT/q≈26 mV at 300 K; forward bias lowers barrier, reverse bias widens depletion.

BJT active region: emitter–base forward, collector–base reverse; I_C=βI_B, I_E=(β+1)I_B.

MOSFET saturation: I_D=(1/2)μ_nC_ox(W/L)(V_GS−V_th)²; triode region for small V_DS.

Ideal op-amp with negative feedback: V+=V− and input currents zero; inverting gain −R_f/R_in, non-inverting gain 1+R_f/R_in.

Barkhausen criterion: loop gain magnitude 1 and phase shift 2πn at oscillation frequency; positive feedback sustains oscillation.

Boolean algebra: De Morgan (AB)′=A′+B′, (A+B)′=A′B′; NAND and NOR are universal gates.

Flip-flops are edge-triggered bistable elements; counters use cascaded flip-flops, registers store binary words.

Negative feedback: closed-loop gain A_f=A/(1+Aβ); improves bandwidth and linearity, and can stabilize operating point.

Percentage change: a price increased by x% then decreased by x% gives net change −x²/100 %, not zero.

AM–GM–HM: for positive a,b, AM=(a+b)/2, GM=√(ab), HM=2ab/(a+b); AM≥GM≥HM with equality when a=b.

Probability: P(A∪B)=P(A)+P(B)−P(A∩B); independent events satisfy P(A∩B)=P(A)P(B).

Permutations nPr=n!/(n−r)!; combinations nCr=n!/[r!(n−r)!]; circular arrangements of n distinct objects: (n−1)!.

Ratio: if a:b=c:d then ad=bc; compound ratio of a:b and c:d is ac:bd; duplicate ratio of a:b is a²:b².

Syllogism: a conclusion is valid only if it follows in every possible Venn diagram consistent with the premises; particular conclusions need existential support.

Subject–verb agreement: singular subject takes singular verb; with neither...nor, verb agrees with nearer subject; collective nouns usually singular when acting as unit.

Data interpretation: average=sum/count; weighted average uses weights; median is middle ordered value; mode is most frequent.