The Schrödinger Equation for the Particle in a Box

The Mathematical Setup

We imagine a particle confined to a one-dimensional line segment of length L. The walls are impenetrable, represented by an infinite potential energy barrier.

V(x) = 0 for 0 < x < L
V(x) = ∞ for x ≤ 0 and x ≥ L

Inside the box, where the potential energy V(x) is zero, the time-independent Schrödinger equation simplifies to the following form.

The Governing Equation

-ħ²/2m * d²ψ(x)/dx² = Eψ(x)

In this equation:

ħ (h-bar) is the reduced Planck's constant.

m is the mass of the particle.

ψ(x) is the wavefunction, which contains all the information about the quantum state of the particle.

E is the total energy of the particle, which is the value we are solving for.

The Solution: Quantized Energy Levels

Solving this differential equation with the boundary condition that the wavefunction ψ(x) must be zero at x=0 and x=L leads to the crucial result of energy quantization.

The allowed energy levels are given by:

E_n = (n² h²) / (8 m L²)

Here, n is the quantum number, which can only be a positive integer (n = 1, 2, 3, ...). This restriction of n to integer values is what forces the energy to be quantized. The particle can only exist in states with these specific, discrete energies.

The Solution: The Wavefunctions

For each allowed energy level E_n, there is a corresponding wavefunction. These wavefunctions are standing waves, similar to the vibrations on a guitar string.

ψ_n(x) = √(2/L) * sin( nπx / L )

The physical significance of the wavefunction is that its square, |ψ_n(x)|², gives the probability density of finding the particle at a specific point x inside the box. This probabilistic interpretation is a fundamental departure from classical physics.