The WKB approximation and the Bohr-Sommerfeld quantization condition worked perfectly for calculating the energy eigenvalues of the infinite square well and the hydrogen atom. Does it work for the other system we solved exactly, the quantum harmonic oscillator?

(a) Apply the Bohr-Sommerfeld quantization condition to the quantum harmonic oscillator. What does it predict for the energy eigenvalues? Is this correct?

(b) A difference between the infinite square well and the harmonic oscillator is that a particle can tunnel a short distance outside of the classically allowed region of the harmonic oscillator, while it cannot in the infinite square well. In the classically allowed region, the momentum is real-valued; that is, \(E-V(x)>0\). On the other hand, outside the harmonic oscillator well, the momentum becomes imaginary, because \(E-V(x)<0\) and we take its square root. Thus, when thinking about momentum as a general complex number, its phase or argument changes by \(\pi / 2\) in going from real to imaginary valued, exactly how \(e^{0}=1\) is real while \(e^{i \pi / 2}=i\) is imaginary. This phase change at the boundary, beyond the phase of the wavefunction accumulated over the classically allowed region, suggests that a more general quantization condition is

\[\begin{equation*}\frac{1}{\hbar} \int_{x_{\min }}^{x_{\max }} d x^{\prime} \sqrt{2 m\left(E-V\left(x^{\prime}\right)\right)}=\left(n+\frac{1}{2}\right) \pi \tag{10.129}\end{equation*}\]

This additional \(\pi / 2\) phase is called the Maslov correction. \({ }^{8}\) Apply this quantization condition to the harmonic oscillator. Are the energy eigenvalues correctly predicted now?

\footnotetext{

8 V. P. Maslov, Theory of Perturbations and Asymptotic Methods (in Russian), Izv. MGU Moscow (1965). [Translation into French, 1972.]

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(c) In principle, we should have included a Maslov correction to the calculation of the hydrogen atom energy eigenvalues because the electron could have tunneled a small distance beyond the upper turning point at \(r_{0}\), from Example 10.4. Why was no Maslov correction needed there?

(d) Now, use the Bohr-Sommerfeld quantization condition with the Maslov correction to estimate the energy eigenvalues for a potential of the form \(V(x)=k|x|^{\alpha}\), for \(\alpha>0\) and \(k\) some constant with units of energy/distance \({ }^{\alpha}\). What do you find in the limit that \(\alpha \rightarrow \infty\) ?
For the next problems, we'll use the power method to study the hydrogen atom and extensions. Unlike the power method introduced in this chapter, there's no need to invert the hydrogen atom's Hamiltonian to use the power method to focus on the ground state. The ground-state energy of hydrogen has the largestin-magnitude value of any bound-energy eigenstate of hydrogen, so we can simply apply increasing powers of the Hamiltonian itself on a state to return a better estimate of the ground state.