Infinite potential well: quantized energy and wave function

A particle in an infinite potential well of width $L$ is free inside the well ($V = 0$ for $0 < x < L$) but cannot leave it ($V = \infty$ outside). Applying the Schrödinger equation with the boundary conditions shows that only certain discrete energies are allowed.

 V = inf |                    | V = inf
         |                    |
         |      V = 0         |
         |   (particle here)  |
         +--------------------+
         0                    L    x

Solving the Schrödinger equation

Outside the well $\psi = 0$. Inside, with $V = 0$:

$$\frac{d^2\psi}{dx^2} + k^2\psi = 0, \qquad k^2 = \frac{2mE}{\hbar^2}$$

General solution:

$$\psi(x) = A\sin kx + B\cos kx$$

Applying boundary conditions

• At $x = 0$: $\psi(0) = 0 \Rightarrow B = 0$, so $\psi = A\sin kx$.
• At $x = L$: $\psi(L) = A\sin kL = 0$. Since $A \neq 0$, $\sin kL = 0$:

$$kL = n\pi \quad\Rightarrow\quad k_n = \frac{n\pi}{L}, \qquad n = 1, 2, 3, \ldots$$

($n = 0$ gives $\psi = 0$ everywhere, i.e. no particle, so it is excluded.)

Quantized energy

$$\begin{aligned} E_n &= \frac{\hbar^2 k_n^2}{2m} = \frac{\hbar^2}{2m}\cdot\frac{n^2\pi^2}{L^2} \\ &= \frac{n^2 h^2}{8 m L^2} \end{aligned}$$

Since $n$ can only be a whole number, the energy can take only the values $E_1, 4E_1, 9E_1, \ldots$ where $E_1 = h^2/(8mL^2)$. The energy is quantized. The lowest energy $E_1$ is not zero (zero-point energy).

Normalized wave function

The particle must be somewhere in the well:

$$\begin{aligned} \int_0^L |\psi|^2 dx &= A^2\int_0^L \sin^2\frac{n\pi x}{L}\,dx = A^2\cdot\frac{L}{2} = 1 \\ A &= \sqrt{\frac{2}{L}} \end{aligned}$$ $$\psi_n(x) = \sqrt{\frac{2}{L}}\,\sin\frac{n\pi x}{L}, \qquad 0 \le x \le L$$

Graphs of $\psi_n$ and $|\psi_n|^2$

        psi_n(x)                 |psi_n(x)|^2
 n=3 |  _         _  |      |  _    _    _  |  E3=9E1
     | / \       / \ |      | / \  / \  / \ |
     |/   \     /   \|      |/   \/   \/   \|
     0     \_ _/     L      0               L

 n=2 |   _           |      |   _       _   |  E2=4E1
     |  / \          |      |  / \     / \  |
     |_/   \       _/|      |_/   \___/   \_|
     0      \_____/  L      0               L

 n=1 |    .----.     |      |     .--.      |  E1
     |  /        \   |      |   /      \    |
     |/            \ |      |__/        \___|
     0               L      0               L

• $\psi_n$ has $(n - 1)$ nodes inside the well; it fits exactly $n$ half-wavelengths, $L = n\lambda/2$.
• $|\psi_n|^2$ gives the probability density. For $n = 1$ the particle is most likely at the centre; for $n = 2$ it is never found at $x = L/2$, which is quite unlike a classical particle.