Statement#

A particle of mass $m$ is confined to a one-dimensional box (infinite potential well) of length $L$:

$V(x) = \begin{cases} 0 & \text{if } 0 < x < L \\ \infty & \text{otherwise} \end{cases}$

1. Solve the time-independent Schrödinger equation inside the box
2. Find the allowed energy levels $E_n$
3. Find the normalized wave functions $\psi_n(x)$

Required Topics#

• Time-independent Schrödinger equation
• Boundary conditions
• Wave function normalization
• Energy quantization
• Second-order differential equations

The Schrödinger Equation#

Inside the box where $V = 0$: $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$

Boundary Conditions#

• $\psi(0) = 0$ (wave function vanishes at walls)
• $\psi(L) = 0$ (wave function vanishes at walls)
• $\int_0^L |\psi(x)|^2 dx = 1$ (normalization)

What to Find#

1. Allowed energy eigenvalues: $E_n$ (quantized energies)
2. Corresponding eigenfunctions: $\psi_n(x)$ for $n = 1, 2, 3, \ldots$
3. Show that energies are proportional to $n^2$

Solution#

Solution coming soon.