Quantum Mechanics
Quantum Mechanics in GATE PH tests the postulates, operator algebra, commutators, angular momentum, exactly solvable systems, perturbation theory, variational methods, WKB approximation, scattering, and identical particles. The exam often rewards recognising a symmetry or a standard result before doing a long integral.
Core rules and concepts
- Physical observables are Hermitian operators; eigenfunctions belonging to distinct eigenvalues are orthogonal. The uncertainty relation follows from [A,B] ≠ 0.
- For the infinite square well of width L, ψ_n = √(2/L) sin(nπx/L) and E_n = n²π²ħ²/(2mL²).
- Angular momentum eigenstates satisfy L²|l,m⟩ = l(l+1)ħ²|l,m⟩ and L_z|l,m⟩ = mħ|l,m⟩, with m = -l,...,l.
- First-order nondegenerate perturbation: E_n¹ = ⟨n⁰|V'|n⁰⟩. Degenerate perturbation theory requires diagonalising V' in the degenerate subspace.
- Identical fermions require antisymmetric wavefunctions; identical bosons require symmetric ones.
Worked example
A particle in the infinite well 0 < x < L is perturbed by V' = λx. The first-order correction to the ground-state energy is E₀¹ = λ⟨0|x|0⟩. The ground-state probability density is symmetric about x = L/2, so ⟨x⟩ = L/2. Hence E₀¹ = λL/2. No integral is needed once the symmetry is recognised.
Traps that cost marks
- Forgetting the boundary conditions when solving the Schrödinger equation.
- Applying nondegenerate perturbation theory to a degenerate level.
- Sign errors in commutators such as [x,p] = iħ.
- Ignoring the antisymmetrisation requirement for identical fermions.