The traditional solutions involve the boundary conditions $\mathrm{\Psi }(x_0)=\mathrm{\Psi }(x_0+L)=0$ with the expectation value of the momentum being $⟨p⟩=0$. However, neither ${\psi }_1(x)=A{e}^{ıkx}$ nor ${\psi }_2(x)=B{e}^{-ıkx}$ can ever equal zero. Furthermore, the expectation value of the momentum for the latter two solutions is $⟨p⟩=\pm \hslash k$. In this section, I show how the traditional values for the solutions at the boundary and for the momentum can be obtained from a combination of the states represented by ${\psi }_1(x)$ and ${\psi }_2(x)$ by using the principle of quantum superposition.

Quantum Superposition

The principle of superposition states that any linear combination of solutions is itself a valid solution. As a result, a new state ${\psi }_3(x)$ can be formed as follows:

$${\psi }_3(x)=c_1{\psi }_1(x)+c_2{\psi }_2(x)c_1,c_2\in \mathbb{C}$$

where $|c_1{|}^2$ and $|c_2{|}^2$ represent the probability of finding the particle in each respective state.

Therefore, in order for the boundary condition to be ${\psi }_3(x)=0$, ${\psi }_1(x)$ and ${\psi }_2(x)$ can be combined in a fashion that gives the desired result and which still is a viable solution. Keeping in mind that $|A|=|B|=\sqrt{1/L}$ and $k=n\pi /L$, combining the two wave equations in the following manner

$$\begin{array}{}& {\psi }_3(x)=\frac{1}{2}{\psi }_1(x)–\frac{1}{2}{\psi }_2(x)\end{array}$$

$$\begin{array}{}& =\frac{1}{2}\frac{1}{\sqrt{L}}{e}^{ıkx}–\frac{1}{2}\frac{1}{\sqrt{L}}{e}^{-ıkx}\end{array}$$

$$=-ı\frac{\sqrt{2}}{\sqrt{L}}\sin (kx)$$

gives a solution ${\psi }_3(x)$ which has the desired results. If the well has its left-hand edge at $x=0$ and the right-hand edge at $x=L$, ${\psi }_3(0)=0$ and ${\psi }_3(L)=0$.

Furthermore for the traditional solutions for the asymmtric and symmetric well, it is clearly seen that the reason that $⟨p⟩=0$ is that indeed these states consist of equal parts of a particle, with the same energy $E$, traveling to the right  – as given by ${\psi }_1(x)$ – and of a particle traveling to the left – as given by ${\psi }_2(x)$.

Dirac on Superposition

This fits in perfectly with what Dirac states in his original book:

“The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$, such that there exists an observation which, when made on the system in state $\mathbf{\text{A}}$, is certain to lead to one particular result, $\mathit{\text{a}}$ say, and when made on the sytem in state $\mathbf{\text{B}}$ is certain to lead to some different result, $\mathit{\text{b}}$ say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes $\mathit{\text{a}}$ and sometimes $\mathit{\text{b}}$, according to a probability law depending on the relative weights of $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ in the superposition process. It will never be different from both $\mathit{\text{a}}$ and $\mathit{\text{b}}$. The intermediate character of the state formed by superposition expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.” [italics in the original]¹