Quantum Mechanics · Chapter 13

The variational method and WKB

Almost nothing can be solved exactly, so this chapter is about getting good answers anyway. The variational principle turns any guess into an upper bound on the ground state energy; the WKB approximation reads a slowly varying potential as a wave of slowly varying wavelength, with turning points, Bohr–Sommerfeld quantisation and tunnelling rates falling out of it.

E [ψ] \geq E_{0}

E [ψ] = \frac{⟨ ψ ∣ H ^ ∣ ψ ⟩}{⟨ ψ ∣ ψ ⟩}

p (x) = 2 m (E - V (x))

ψ_{WKB} \propto \frac{e ^{\frac{i}{ℏ} \int p d x}}{p}

Lessons

Build the foundation

Near a classical turning point the WKB formulas blow up, but there the potential is nearly a straight line, and the Schrödinger equation in a linear potential is solved exactly by the Airy function. Far from the turning point on either side, the Airy function turns into a WKB wave: an oscillation on the allowed side and a decay on the forbidden side, joined with a definite phase of $\pi/4$. These connection formulas let WKB solutions be carried across turning points, and so give energy levels and tunnelling rates.

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05Bohr–Sommerfeld quantisationInside a well, WKB gives an oscillating solution starting from each turning point. The two must be the same function, which is possible only when the phase accumulated across the well is a half-integer multiple of $\pi$. Written as the area of the classical orbit in the plane of position and momentum, the condition says: each state occupies an area $2\pi\hbar$ of phase space, the first one half of it. For the harmonic oscillator this is exact; for other wells it is best for high levels.Read lesson →

06Tunnelling in the WKB approximationUnder a barrier the WKB solution decays, and across the forbidden region its amplitude falls by the exponential of the integral of $|p|/\hbar$. The transmission is the square of that: an exponential of twice the integral, the Gamow factor. For a rectangular barrier it gives back the $e^{-2\kappa a}$ of the chapter on one dimension; for a real barrier of changing height it gives what that chapter could not — the rates of alpha decay, of field emission from a metal tip, of nuclear fusion in the Sun.Read lesson →

07Chapter practiceBring the chapter together: the variational principle and its upper bounds, trial states and the Rayleigh–Ritz matrix, the WKB approximation, turning points and connection formulas, Bohr–Sommerfeld quantisation and tunnelling through smooth barriers. No worked example sits above the answers.Read lesson →

Mapping the lesson.

The variational method and WKB

Build the foundation