The Particle in a Box

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The Particle in a Box

Consider a particle constrained to move in a single dimension, under the influence of a potential V(x) which is zero for $0 \leq x \leq a$ and infinite elsewhere. Since the wavefunction is not allowed to become infinite, it must have a value of zero where V(x) is infinite, so $\psi(x)$ is nonzero only within [0,a]. The Schrödinger equation is thus

$\begin{displaymath}- \frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E \psi(x) \hspace{0.5cm} 0 \leq x \leq a \end{displaymath}$

(115)

It is easy to show that the eigenvectors and eigenvalues of this problem are

$\begin{displaymath}\psi_n(x) = \sqrt{\frac{2}{a}} sin \left( \frac{n \pi x}{a} \... ...\hspace{1.0cm} 0 \leq x \leq a \hspace{1.0cm} n = 1,2,3,\ldots \end{displaymath}$

(116)

$\begin{displaymath}E_n = \frac{h^2 n^2}{8 m a^2} \hspace{1.0cm} n=1,2,\ldots \end{displaymath}$

(117)

Extending the problem to three dimensions is rather straightforward; see McQuarrie [1], section 6.1.