# Bohr-Sommerfeld quantization for different potentials

Let's have Bohr-Sommerfeld quantization for one-dimensional case: $$\int \limits_{a}^{b} p(x)dx ~=~ \pi \hbar (n + \nu ).$$ Here $p(x) = \sqrt{2m(E - U)}$, $a, b$ are turning points, and the area which isn't located at interval $(a, b)$ is classically forbidden. I know that when we don't have infinite wells at $a, b$, i.e. the wave-function isn't equal to zero out of the range $(a, b)$, we have $\nu = \frac{1}{2}$, when we have infinite well in, for example, point $a$, $\nu = \frac{3}{4}$, and finally in case when there are two wells, we have $\nu = 1$.

I know that in the first case $\nu = |\varphi_{1} - \varphi_{2}|$ is combined as result of matching of function out of $(a, b)$, $\frac{1}{\sqrt{p}}e^{-|\int pdx|}$ and solution in $(a, b)$, $$\frac{C_{1}}{\sqrt{p}}e^{i \int pdx + i\varphi_{1}} + \frac{C_{2}}{\sqrt{p}}e^{-i \int pdx + i\varphi_{2}},$$ after bypassing of turning points. But what to do in the second and third cases, when function is equal to zero outside at least on one side of the rangle?

## 1 Answer

OP is considering the Bohr-Sommerfeld quantization rule

$$\tag{1} \oint k(x) \mathrm{d}x ~=~2\pi (n + \frac{1}{4}\sum_i\mu_i) , \qquad n\in\mathbb{N}_0,$$ where the sum $\sum_i$ is over turning points $i$ with positions $x_i$ and where $\mu_i\in\mathbb{Z}$ is the metaplectic correction/Maslov index of the $i$th turning point. See also this Phys.SE post for references.

I) A turning point of generic type has Maslov index $\mu_i=1$. This is seen from the semiclassical connection formulas. The connection formulas with the well-known $\exp(\pm i\frac{\pi}{4})$ phase shift [which induces a $2\times \frac{\pi}{4}=\frac{\pi}{2}$ phase shift between the left and the right mover, and corresponds to $\mu_i=1$] are derived under the assumption that there exists an overlapping region where the following two conditions are both satisfied:

1. The quasi-classical condition: $|\lambda^{\prime}(x)| \ll 1$. This typically holds away from the turning point.

2. A linearization of the potential $V(x)$ is valid. This typically holds only in a small neighborhood around the turning point.

II) An infinite square well potential (say located at $x=x_i$) typically satisfies condition 1 for $x\neq x_i$ but fails condition 2. The connection formula is then replaced by a boundary condition (BC)

$$\tag{2} \psi(x_i)~=~0,$$

cf. e.g. this Phys.SE post. The BC (2) implies a $\pi$ phase shift between the left and the right mover, which corresponds to a Maslov index $\mu_i=2$.