The traditional solutions involve the boundary conditions $\Psi(x_{0}) = \Psi(x_{0}+L) = 0$ with the expectation value of the momentum being $\langle p \rangle = 0$. However, neither $\psi_{1}(x) = Ae^{\imath kx}$ nor $\psi_{2}(x) = Be^{-\imath kx}$ can ever equal zero. Furthermore, the expectation value of the momentum for the latter two solutions is $\langle p \rangle = \pm \hbar k$. 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:

\[ \psi_{3}(x) = c_{1} \psi_{1}(x) + c_{2} \psi_{2}(x) \quad c_{1}, c_{2} \in \mathbb{C} \]

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

\[ \psi_{3}(x)  = \frac{1}{2}\psi_{1}(x) – \frac{1}{2} \psi_{2}(x) \notag \]

\[ = \frac{1}{2}\frac{1}{\sqrt{L}} e^{\imath k x} – \frac{1}{2}\frac{1}{\sqrt{L}}e^{-\imath kx} \notag \]

\[= -\imath \frac{\sqrt{2}}{\sqrt{L}} \sin (kx) \label{Eq:EqualParts} \]

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 $\langle p \rangle = 0$ 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, $ \textbf{A}$ and $\textbf{B}$, such that there exists an observation which, when made on the system in state $\textbf{A}$, is certain to lead to one particular result, $\textit{a}$ say, and when made on the sytem in state $\textbf{B}$ is certain to lead to some different result, $\textit{b}$ 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 $\textit{a}$ and sometimes $\textit{b}$, according to a probability law depending on the relative weights of $\textbf{A}$ and $\textbf{B}$ in the superposition process. It will never be different from both $\textit{a}$ and $\textit{b}$. 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