The energy $E$ of the partical within the well must be invariant with respect to coordinate system chosen by the observer. Therefore, its value must be derived in a fashion independent of the position of the well. The Bohr-Sommerfeld rule for the quantizaation of momentum allows for this derivation of the value of $k$ in a frame-independent fashion.¹

It can be shown that for a constant potential $V(x)$ in a region with turning points $x_a$ and $x_b$, the ground state energy $E_1$ can be calculated from

$$\begin{array}{}& {\int }_{x_a}^{x_b}{k}_{1}dx=n\pi \end{array}$$

with

$$\begin{array}{}& k_1=\sqrt{\frac{2m{E}_1}{{\hslash }^2}}\end{array}$$

as a generalization of the Bohr-Sommerfeld quantizatuion rule.²

The next set of energy levels is given by

$$\begin{array}{}& {\int }_{x_0}^{x_0+L}{k}_{n}dx=n\pi n=2,3\cdots \end{array}$$

which gives

$$\begin{array}{}\text{(1)}& k_n=\frac{n\pi }{L}n=\pm 1,\pm 2,\pm 3,\dots \end{array}$$

Keeping in mind that

$$\begin{array}{}\text{(2)}& k_n=\sqrt{\frac{2m{E}_n}{{\hslash }^2}}\end{array}$$

Substituting Eq.$\text{(2)}$ into Eq.$\text{(1)}$ and solving for $E$ results in

$$\begin{array}{}& E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}\end{array}$$

The quantization of energy is the same as that noted in the traditional approach. However, obtaining the result from the Wilson-Sommerfeld rule relies on the length $L$ of the well which is invariant.