Quantum Mechanics
Quantum Mechanics accounts for roughly 12% of the GRE Physics Subject Test. It tests the postulates, the Schrödinger equation in solvable potentials, hydrogen, angular momentum and spin, and approximation methods - usually one short calculation plus a conceptual check.
Core rules
- Wavefunctions must be normalized, ∫|ψ|²dx = 1; |ψ|² is a probability density and observables are expectation values ⟨A⟩ = ∫ψ*Aψ dx.
- Stationary states satisfy Ĥψ = Eψ. Boundary conditions (ψ continuous, ψ′ continuous for finite potentials) quantize the spectrum.
- Infinite square well: E_n = n²h²/(8mL²), with n = 1, 2, 3… and no n = 0 state. Harmonic oscillator: E_n = (n + ½)ħω, so the ground state has zero-point energy.
- Tunneling falls exponentially, roughly e^{−2κL} with κ = √(2m(V − E))/ħ. It is significant when the barrier is thin and low.
- Uncertainty principle: ΔxΔp ≥ ħ/2. A state with definite energy in a well still has spread in momentum.
- Angular momentum: L² has eigenvalues l(l+1)ħ² and L_z has mħ. Spin-½ admits two states, and the Pauli principle permits one fermion per state.
- Hydrogen: E_n = −13.6 eV/n²; degeneracy n² (2n² with spin); radial nodes = n − l − 1.
- Commutators: [x, p] = iħ. Two observables share eigenstates only if their operators commute.
Worked example
An electron is confined to a 1.0 nm infinite well. Then E₁ = h²/(8mL²) = (6.63×10⁻³⁴)²/(8 × 9.11×10⁻³¹ × 10⁻¹⁸) ≈ 6.0×10⁻²⁰ J ≈ 0.38 eV. The n = 2 level is four times higher, so a 2→1 transition emits a photon of 3 × 0.38 = 1.13 eV, giving λ = 1240/1.13 ≈ 1100 nm.
Traps
Omitting the oscillator zero-point energy; treating n = 0 as the well's ground state; using E = p²/2m with a de Broglie wavelength that was never adjusted to the well; confusing radial and total quantum numbers in hydrogen; assuming degeneracies survive once a perturbation is switched on.