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

$$- \frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E \psi(x) \hspace{0.5cm} 0 \leq x \leq a$$ (115)

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

$$\psi_n(x) = \sqrt{\frac{2}{a}} sin \left( \frac{n \pi x}{a} ... ...\hspace{1.0cm} 0 \leq x \leq a \hspace{1.0cm} n = 1,2,3,\ldots$$ (116)

$$E_n = \frac{h^2 n^2}{8 m a^2} \hspace{1.0cm} n=1,2,\ldots$$ (117)

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