11. The physical picture behind approximation methods

The physical picture behind approximation methods

Work out what perturbation theory, the variational method and WKB each actually change, and where each one breaks.

Draft: the scene works, but the text and the observation guide are still being polished.

What you will see

Levels and wavefunctions before and after a perturbation, and how corrections accumulate order by order
A trial wavefunction being pushed down towards the true ground state
The WKB phase integral, and why it fails at a turning point

Assumed background

Bound states and energy levels (module 03)
Eigenvalue problems in linear algebra

Almost no real quantum system can be solved exactly. Hydrogen is a lucky exception, and adding a single electron (helium) already puts closed-form solutions out of reach.

So a large part of practical quantum mechanics is about computing accurately when you cannot solve. Three workhorses:

Perturbation theory: write the hard problem as easy plus a bit

\hat H = \hat H_0 + \lambda \hat V,\qquad E_n = E_n^{(0)} + \lambda\langle n|\hat V|n\rangle + \lambda^2\sum_{m\ne n}\frac{|\langle m|\hat V|n\rangle|^2}{E_n^{(0)}-E_m^{(0)}} + \cdots

The first-order correction is just the expectation value of the perturbation in the unperturbed state — no new wavefunction needed. The second-order correction always pushes the ground state down (the denominator is negative for the ground state), which is where “levels repel” comes from.

Where it fails: when a denominator goes to zero, that is, for degenerate or nearly degenerate levels. Then you must first diagonalise within the degenerate subspace (degenerate perturbation theory), or the expansion diverges.

The variational method: guess, then optimise

For any normalised trial wavefunction,

E[\psi_{\text{trial}}] = \frac{\langle\psi|\hat H|\psi\rangle}{\langle\psi|\psi\rangle} \ \ge\ E_0 .

The inequality holds unconditionally, so pushing the expectation value down is guaranteed to approach the true ground-state energy.

The advantage: no small parameter is needed, so it works at strong coupling too. The price: it gives only an upper bound and no error estimate; and because the energy is stationary with respect to the trial function (errors enter at second order), a very accurate energy can come from a rather poor wavefunction.

A single “effective nuclear charge” trial function gets the helium ground-state energy within 2% of experiment.

WKB: the semiclassical limit of a slowly varying wavelength

\psi(x) \approx \frac{C}{\sqrt{p(x)}}\exp\!\left[\pm\frac{\ii}{\hbar}\int^x p(x')\,\dd x'\right],\qquad p=\sqrt{2m(E-V)} .

It is valid when the de Broglie wavelength changes little over one wavelength: $\left|\dd\lambda/\dd x\right|\ll 1$ .

At a classical turning point ( $E=V$ , $p\to0$ ) that condition necessarily fails, and the solutions must be connected through Airy functions. The connection formulas produce the quantisation condition

\oint p\,\dd x = \left(n+\tfrac12\right)h,

the modern form of Bohr–Sommerfeld quantisation, where the $\tfrac12$ comes from a $\pi/4$ phase at each turning point.

WKB also gives the general tunnelling formula behind module 04:

T \approx \exp\!\left[-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V-E)}\,\dd x\right],

valid for any barrier shape. The Gamow factor for α decay is this integral.

Think it through

The second-order correction to the ground state is always negative. What does that imply about the ground state relative to that of $H_0$ ? Does the same hold for excited states?
Can the variational method reach excited states? (Hint: add an orthogonality constraint.)
WKB is excellent at large $n$ and often poor for the ground state. Is that consistent with the correspondence principle, or in tension with it?