Particle In A 3D Box

Suppose a particle of mass $m \neq 0$ is confined to a 3D square box of edge length = a.

The Time Independent Schrodinger Equation reads

\begin{align} -\frac{\hbar^2}{2m}\nabla^{2}\psi(x,y,z) + V(x,y,z)\psi(x,y,z) = E\psi(x,y,z) \end{align}

For a free particle, $V(x,y,z) = 0$ and so

\begin{align} -\frac{\hbar^2}{2m}\nabla^{2}\psi(x,y,z) = E\psi(x,y,z) \end{align}

which can be readily solved by separation of variables, taking

\begin{align} \psi(x,y,z) = \psi_{x}(x)\psi_{y}(y)\psi_{z}(z) \end{align}

Substituting this into the wave equation and dividing throughout by $\psi(x,y,z) = \psi_{x}(x)\psi_{y}(y)\psi_{z}$ we get

\begin{align} \frac{1}{\psi_{x}(x)}\frac{d^{2}\psi_{x}}{dx^2} + \frac{1}{\psi_{y}(y)}\frac{d^{2}\psi_{y}}{dy^2} + \frac{1}{\psi_{z}(z)}\frac{d^{2}\psi_{z}}{dz^2} = -\frac{2mE}{\hbar^2} \end{align}

The left hand side is of the form

\begin{equation} f_{1}(x) + f_{2}(y) + f_{3}(z) = const \end{equation}

The only way such an equation can be satisfied is for each of the $f_{i}$s to be constant. Let

\begin{align} \frac{1}{\psi_{x}(x)}\frac{d^{2}\psi_{x}}{dx^2} = -\frac{2mE_{x}}{\hbar^2} \end{align}
\begin{align} \frac{1}{\psi_{y}(y)}\frac{d^{2}\psi_{y}}{dy^2} = -\frac{2mE_{y}}{\hbar^2} \end{align}
\begin{align} \frac{1}{\psi_{z}(z)}\frac{d^{2}\psi_{z}}{dz^2} = -\frac{2mE_{z}}{\hbar^2} \end{align}


\begin{equation} E = E_{x} + E_{y} + E_{z} \end{equation}

The general solution is thus

\begin{align} \psi(x,y,z) = \left(\frac{2}{a}\right)^{3/2}\sin\frac{n_{x}\pi}{a}}\sin\frac{n_{y}\pi}{a}}\sin\frac{n_{z}\pi}{a}} \end{align}


\begin{align} E_{i} = \frac{n_{i}^{2}h^{2}}{8ma^{2}} \end{align}

and hence

\begin{align} E = \frac{h^{2}}{8ma^{2}}\left(n_{x}^2 + n_{y}^2 + n_{z}^2\right) \end{align}
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