Introduction to Quantum Mechanics · Problem 9.16

The WKB Approximation — David J. Griffiths

Physics — Quantum Mechanics · wkb-hydrogen-bound-state-energies

Problem

Use the WKB approximation in the form of Equation 9.52, ∫r1r2p(r) dr=(n′−1/2)πℏ\int_{r_1}^{r_2} p(r)\,dr = \left(n' - 1/2\right)\pi\hbar to estimate the bound state energies for hydrogen. Don't forget the centrifugal term in the effective potential (Equation 4.38). The following integral may help: ∫ab1x(x−a)(b−x) dx=π2(b−a)2.\int_a^b \frac{1}{x}\sqrt{(x-a)(b-x)}\,dx = \frac{\pi}{2}\left(\sqrt{b}-\sqrt{a}\right)^2. *Answer:* En′ℓ≈−13.6 eV[n′−(1/2)+ℓ(ℓ+1)]2.E_{n'\ell} \approx \frac{-13.6\text{ eV}}{\left[n' - (1/2) + \sqrt{\ell(\ell+1)}\right]^2}. I put a prime on n′n', because there is no reason to suppose it corresponds to the nn in the Bohr formula. Rather, it orders the states for a given ℓ\ell, counting the number of nodes in the radial wave function. In the notation of Chapter 4, n′=N=n−ℓn' = N = n - \ell (Equation 4.67). Put this in, expand the square root (1+ϵ=1+12ϵ−18ϵ2+⋯ )\left(\sqrt{1+\epsilon} = 1 + \frac{1}{2}\epsilon - \frac{1}{8}\epsilon^2 + \cdots\right), and compare your result to the Bohr formula.
Intuitive explanation

The WKB quantization condition captures how many wavelengths fit between the classical turning points of the effective potential. For hydrogen, the effective potential includes both the Coulomb attraction and the centrifugal barrier; you'll need to find where the particle's kinetic energy vanishes, then integrate the momentum between those turning points. The provided integral is tailored to this problem—recognize its form and use the substitution that matches your effective potential to evaluate it cleanly.

Intuitive explanation

The WKB quantization condition captures how many wavelengths fit between the classical turning points of the effective potential. For hydrogen, the effective potential includes both the Coulomb attraction and the centrifugal barrier; you'll need to find where the particle's kinetic energy vanishes, then integrate the momentum between those turning points. The provided integral is tailored to this problem—recognize its form and use the substitution that matches your effective potential to evaluate it cleanly.

Related theory

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