Quantum Mechanics – Unit II: Perturbation & Approximation Methods

These flashcards review key ideas, formulas, and definitions from the Unit II lecture notes on perturbation theory, variational methods, WKB approximation, Stark effect, and time-dependent transitions.

What is the basic goal of time-independent perturbation theory?

To calculate successive corrections to the energy levels and wave-functions of a system whose Hamiltonian can be written as an exactly solvable part H₀ plus a small perturbation H′.

When is perturbation theory applicable to a Hamiltonian?

When the total Hamiltonian can be expressed as H = H₀ + λ H′, with H′ sufficiently small compared to H₀ so that a power-series expansion in λ converges.

Name the three main approximation methods discussed and their typical use.

(1) Perturbation theory – small corrections about a solvable H₀; (2) Variational method – estimating ground-state properties when exact solutions are unknown; (3) WKB (Wentzel-Kramers-Brillouin) – semi-classical approximation for 1-D, slowly varying potentials.

For what purpose is the variational method primarily used?

To obtain an upper bound to the ground-state energy and an approximate ground-state wave-function of complex systems.

What is the principal drawback of the variational method?

Its accuracy depends entirely on the choice of the trial wave-function; a poor guess yields a poor estimate.

What does WKB stand for, and what does it approximate?

Wentzel–Kramers–Brillouin; it provides approximate solutions to the one-dimensional, time-independent Schrödinger equation in regions where the potential varies slowly compared with the particle’s de Broglie wavelength.

How is the perturbed Hamiltonian written in Rayleigh–Schrödinger theory?

H = H₀ + λ H′, where λ is treated as a bookkeeping parameter set to 1 at the end.

Define a non-degenerate energy level.

An energy eigenvalue that corresponds to a single, unique eigenfunction; no other independent state shares the same energy.

Define a degenerate energy level.

An energy eigenvalue shared by two or more linearly independent eigenfunctions (different quantum states with identical energy).

What is the energy of the n-th level of a one-dimensional harmonic oscillator?

Eₙ = (n + ½) ℏω, with n = 0,1,2,…

What extra term appears in the Hamiltonian of an anharmonic oscillator?

An anharmonic potential term, commonly λ x⁴/4, added to the harmonic potential ½ mω²x².

Qualitatively, what does the first-order energy shift of an anharmonic oscillator represent?

The expectation value ⟨n| H′ |n⟩, giving the leading correction ΔEₙ(1) to the harmonic energy due to the x⁴ term.

Write the perturbation Hamiltonian for the Stark effect of a hydrogen atom in a uniform electric field E along z.

H′ = − e E z (or −𝐌·𝐄, with dipole moment 𝐌 = −e 𝐫).

Why is there no first-order Stark shift for the hydrogen ground state (n = 1)?

Because the ground-state wave-function is spherically symmetric; ⟨1s| z |1s⟩ = 0, so ⟨H′⟩ vanishes.

What is the degeneracy of the first excited state (n = 2) of hydrogen before an electric field is applied?

Four-fold: the states |200⟩ and |21m⟩ with m = −1,0,1.

After applying a weak electric field, how are the n = 2 hydrogen levels split?

Degeneracy is lifted into energies E₂ ± 3 a₀ eE (two states) and two unchanged levels, corresponding to induced dipole moments parallel or antiparallel to the field.

State the variational principle for the expectation value W of H with a trial function ψ_trial.

W = ⟨ψtrial| H |ψtrial⟩ / ⟨ψtrial|ψtrial⟩ ≥ E₀, where E₀ is the true ground-state energy.

What is the ‘upper-bound’ result of the variational method?

Any properly normalized trial wave-function yields an energy expectation W that is never lower than the exact ground-state energy.

Why is an effective nuclear charge Z′ introduced in the helium variational calculation?

To account for the shielding of the nucleus by one electron felt by the other; Z′ is treated as a variational parameter that minimizes the energy.

What effective charge gives the best simple variational estimate for helium?

Z′ ≈ 1.67 (instead of the bare Z = 2).

What ground-state energy does the simple shielding variational calculation predict for helium?

≈ −77.5 eV, within about 2 % of the experimental −78.975 eV.

In the WKB method, what is k(x) in a classically allowed region (E > U(x))?

k(x) = √[2m(E − U(x))]/ℏ (real).

What is k(x) in a classically forbidden region (E < U(x))?

k(x) = i κ(x) with κ(x)= √[2m(U(x) − E)]/ℏ, making the wave-function exponential.

Give the zeroth-order WKB wave-function form in an allowed region.

ψ(x) ≈ C /√k(x) · exp[ ± i ∫ k(x) dx ].

What does the WKB connection formula relate?

It matches the oscillatory solution on one side of a turning point to the exponentially growing/decaying solution on the other side, ensuring a continuous wave-function.

Write the WKB transmission coefficient for tunnelling through a barrier of width L at constant U > E.

T ≈ exp[ − 2 ∫₀^L κ(x) dx ] = exp[ − 2L √{2m(U − E)}/ℏ ] for a rectangular barrier.

State the time-dependent Schrödinger equation for a Hamiltonian H = H₀ + H′(t).

iℏ ∂Ψ/∂t = [H₀ + H′(t)] Ψ(x,t).

What quantity gives the probability of transition from initial state |i⟩ to |f⟩?

|af(t)|², where af(t) is the time-dependent transition amplitude obtained from perturbation theory.

Give the first-order transition amplitude in time-dependent perturbation theory.

af^{(1)}(t) = (1/iℏ) ∫₀^t dt′ H′{fi}(t′) e^{iω{fi} t′}, with ω{fi} = (Ef − Ei)/ℏ.

What is the selection rule for first-order transitions?

A transition i → f is allowed only if the matrix element H′_{fi} ≠ 0; otherwise it is forbidden to first order.

State Fermi’s Golden Rule for the transition rate.

W{i→f} = (2π/ℏ) |H′{fi}|² ρ(Ef), where ρ(Ef) is the density of final states at energy E_f.

Under a harmonic perturbation of frequency ω, what resonance condition gives absorption?

ω{fi} = +ω, i.e., Ef = E_i + ℏω (the system absorbs a quantum ℏω).

Under the same perturbation, what condition gives stimulated emission?

ω{fi} = −ω, i.e., Ef = E_i − ℏω (the system emits a quantum ℏω).

What is meant by an absorptive transition?

A transition in which the system moves to a higher energy state by absorbing energy from the perturbing field.

What is meant by an emissive (downward) transition?

A transition in which the system moves to a lower energy state and releases energy to the perturbing field.

Give the upward (absorption) transition rate induced by a harmonic perturbation.

W↑ = (2π/ℏ) |H{mn}(ω{mn})|² ρ(ω{mn}), for Em > En.

Give the downward (emission) transition rate for the same perturbation.

W↓ = (2π/ℏ) |H{nm}(ω{mn})|² ρ(ω{mn}), with ω{mn} = (Em − E_n)/ℏ.

What equations yield zeroth, first and second-order energies in non-degenerate perturbation theory?

(H₀ − En^{(0)})|ψn^{(0)}⟩ = 0; (H₀ − En^{(0)})|ψn^{(1)}⟩ = (En^{(1)} − H′)|ψn^{(0)}⟩; similarly for second order with (H₀ − En^{(0)})|ψn^{(2)}⟩ = (En^{(1)} − H′)|ψn^{(1)}⟩ + En^{(2)} |ψn^{(0)}⟩.

What is a classical turning point in WKB analysis?

A position x = a where E = U(a); k(x) changes from real to imaginary, marking the boundary between classically allowed and forbidden regions.

State the basic validity condition for WKB.

|dλ/dx| ≪ 1, i.e., the potential varies slowly over a distance comparable to the local de Broglie wavelength λ = 2π/k(x).