GATE PH flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Cauchy-Riemann equations

For f(z)=u+iv analytic, u_x=v_y and u_y=−v_x; equivalently ∂f/∂z̄=0, and u,v are harmonic.

Residue theorem

For f meromorphic inside a simple closed contour, ∮ f dz = 2πi times sum of residues at enclosed poles; simple pole residue is lim_{z→z0}(z−z0)f(z).

Gamma function recurrence

Γ(z+1)=zΓ(z); for integer n≥0 Γ(n+1)=n!, and Γ(1/2)=√π; used to evaluate beta and Gaussian integrals.

Fourier transform Parseval

If F(k)=∫f(x)e^{−ikx}dx, then ∫|f(x)|²dx=(1/2π)∫|F(k)|²dk with this convention; energy is conserved between domains.

Euler-Lagrange equation

For L(q,qdot,t), stationary action gives d/dt(∂L/∂qdot)−∂L/∂q=0; equivalent to Newton's equations for conservative systems.

Effective potential central force

U_eff(r)=U(r)+L²/(2mr²); radial motion is one-dimensional with this potential, and circular orbits occur at its minimum.

Poisson bracket

{f,g}=Σ(∂f/∂q_i ∂g/∂p_i−∂f/∂p_i ∂g/∂q_i); {q_i,p_j}=δ_ij, and dA/dt=∂A/∂t+{A,H}.

Poynting vector

S=E×H gives electromagnetic power per unit area; its divergence gives energy flux, and for a plane wave the time average is (1/2)E0H0.

Lorentz force

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

Boundary conditions Maxwell

Tangential E and normal B are continuous; normal D jumps by free surface charge σ_f; tangential H jumps by free surface current K_f.

Commutator x p

[x,p]=iℏ; this gives ΔxΔp≥ℏ/2 and means x and p cannot share a simultaneous eigenstate.

Infinite square well energies

E_n=n²π²ℏ²/(2mL²), n=1,2,...; wavefunctions are sin(nπx/L), with nodes at walls and zero-point energy.

Harmonic oscillator spectrum

E_n=(n+½)ℏω for a one-dimensional harmonic oscillator; ladder operators a and a† raise and lower n, with ground-state Gaussian.

Hydrogen energy levels

E_n=−13.6 eV/n² for hydrogen; degeneracy is n² including spin, and transitions obey Δl=±1 with photon energy equal to level difference.

First law thermodynamics

dU=đQ−đW; for reversible pressure work đW=P dV, and U is a state function, so cyclic processes have ΔU=0.

Canonical partition function

Z=Σ_i e^{−E_i/kT}; free energy F=−kT ln Z, average energy ⟨E⟩=−∂lnZ/∂β, and β=1/kT.

Clausius-Clapeyron

dP/dT=L/(TΔv); it gives the slope of a phase boundary, with L latent heat and Δv volume change per particle or mole.

Rydberg formula

1/λ=R(1/n_f²−1/n_i²) for hydrogen-like transitions; R≈1.097×10⁷ m⁻¹, and n_f<n_i gives emission.

Landé g factor

g_J=1+[J(J+1)+S(S+1)−L(L+1)]/[2J(J+1)]; magnetic moment μ=−g_J μ_B J/ℏ and Zeeman splitting ΔE=g_J μ_B B m_J.

Rotational energy rigid rotor

E_J=B J(J+1) with B=ℏ²/(2I); infrared transitions obey ΔJ=±1, giving equally spaced lines separated by 2B.

Bragg law

2d sinθ=nλ; constructive interference from lattice planes with spacing d occurs at angle θ, used in X-ray and neutron diffraction.

Bloch theorem

In a periodic potential, ψ_k(r)=e^{ik·r}u_k(r) with u_k periodic; k is crystal momentum and energy bands E(k) are periodic in reciprocal lattice.

Fermi energy

At T=0, electrons fill states up to E_F; E_F=ℏ²(3π²n)^{2/3}/(2m) for a three-dimensional free electron gas.

Semi-empirical mass formula

Binding energy is volume minus surface, Coulomb, asymmetry, and pairing terms; it explains nuclear stability and the binding-energy peak near A≈56–62.

Radioactive decay law

N(t)=N0e^{−λt}; half-life t_1/2=ln2/λ, activity A=λN, and mean lifetime τ=1/λ.

Quark content proton neutron

Proton is uud with charge +1; neutron is udd with charge 0; quarks carry fractional charges +2/3 and −1/3 and interact by gluons.

Ideal op-amp rules

With negative feedback, input currents are zero and V+=V− (virtual short); these rules set inverting and non-inverting gains.

Barkhausen criterion

For sustained oscillation, loop gain magnitude must equal one and loop phase shift must be an integer multiple of 2π at the oscillation frequency.

BJT active region

Emitter-base forward biased and collector-base reverse biased; I_C=βI_B, and the transistor acts as a current-controlled current source/amplifier.

Probability union rule

P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events the intersection term is zero, and for independent events P(A∩B)=P(A)P(B).

AM GM HM inequality

For positive a and b, AM=(a+b)/2, GM=√(ab), HM=2ab/(a+b), and AM≥GM≥HM with equality only when a=b; useful for optimization.

Syllogism validity rule

A categorical conclusion follows only if it is true in every Venn diagram satisfying the premises; conclusions about some objects require an existential premise.