Particle in a Three‑Dimensional Box

1. Introduction

Consider a particle of mass \(m\) confined inside a rectangular box of side lengths \(a, b, c\) along the \(x, y, z\) axes. The potential is zero inside the box and infinite outside:

2. Time‑Independent Schrödinger Equation

The Hamiltonian operator in three dimensions is:

The time‑independent Schrödinger equation \(\hat{H}\psi = E\psi\) gives:

Rearranging:

3. Solution Outside the Box (\(V=\infty\))

Outside the box, \(V=\infty\). Since \(E\) is finite, equation (4) becomes:

The term \(\frac{2m}{\hbar^{2}}(E-\infty)\to -\infty\). For the equation to hold, the wavefunction must be zero:

Thus the particle cannot exist outside. The wavefunction vanishes at the boundaries and beyond.

4. Solution Inside the Box (\(V=0\))

Inside the box, \(V=0\). Equation (4) reduces to:

Define the constant:

Then equation (7) becomes:

We solve this using separation of variables. Assume:

Substituting (10) into (9) and dividing by \(XYZ\):

Since each term depends on a different variable, each must equal a constant. Write:

with

Each of the equations in (12) is a one‑dimensional harmonic oscillator equation. For \(X(x)\):

The general solution is \(X(x) = A\sin(k_{x}x) + B\cos(k_{x}x)\). The boundary conditions require \(\psi=0\) at \(x=0\) and \(x=a\) (since the wavefunction must vanish at the infinite walls). At \(x=0\):

At \(x=a\):

Thus

The normalized solution for \(X(x)\) is obtained by requiring \(\int_{0}^{a} |X|^{2}dx = 1\):

Hence

Analogously, for the \(y\) and \(z\) directions:

with \(n_{y}, n_{z}=1,2,3,\dots\).

5. Total Wavefunction and Energy Quantization

Combining (10), (19) and (20), the complete normalized wavefunction is:

From (13), (17) and the analogous expressions:

Using (8) \(k^{2}=2mE/\hbar^{2}\), we get:

Solving for \(E\):

Since \(\hbar = h/(2\pi)\), \(\hbar^{2}\pi^{2} = \frac{h^{2}}{4}\). Substituting gives the familiar form:

Equation (25) is the energy quantization condition. The quantum numbers \(n_{x}, n_{y}, n_{z}\) are positive integers. The ground state is \((n_{x},n_{y},n_{z})=(1,1,1)\).

6. Probability Density

The probability density is the square of the wavefunction:

This is always non‑negative and integrates to 1 over the volume of the box. The maxima occur where all sine functions equal ±1.

7. Discussion of Degeneracy

When the box is not cubic (\(a\neq b\neq c\)), the energy levels are generally non‑degenerate because the sum \(\frac{n_{x}^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}+\frac{n_{z}^{2}}{c^{2}}\) takes different values for different triples. However, if two sides are equal (e.g., \(a=b\)), degeneracy appears. In a fully cubic box, degeneracy is high and is essential for understanding the electronic structure of atoms and quantum dots.

The complete set of wavefunctions (21) forms an orthonormal basis for the Hilbert space of square‑integrable functions on the box.

8. Video Lecture (Urdu/Hindi)

9. Summary of Key Results

• Wavefunction: \(\displaystyle \psi_{n_{x}n_{y}n_{z}}(x,y,z) = \sqrt{\frac{8}{abc}}\sin\left(\frac{n_{x}\pi x}{a}\right)\sin\left(\frac{n_{y}\pi y}{b}\right)\sin\left(\frac{n_{z}\pi z}{c}\right)\)
• Energy: \(\displaystyle E_{n_{x}n_{y}n_{z}} = \frac{h^{2}}{8m}\left(\frac{n_{x}^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}+\frac{n_{z}^{2}}{c^{2}}\right)\)
• Boundary conditions force the wavefunction to vanish at the walls, leading to three quantum numbers \(n_{x},n_{y},n_{z}\in\mathbb{N}\).
• The ground state energy is non‑zero (zero‑point energy).
• Degeneracy arises when the box has symmetric dimensions.

These results form the basis for many advanced quantum systems, including quantum wells, wires, and dots.