AcademyAtomic Quantum Structure
Academy
Particle in a Three-Dimensional Box
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
A particle confined by hard walls in three dimensions has standing waves in each direction. Each direction contributes a quantum number and a kinetic-energy term.
Different quantum-number triples can have the same energy, producing degeneracy.
Notation
\(L_x,L_y,L_z\)
box side lengths
\(\mathrm{m}\)
\(n_x,n_y,n_z\)
positive integer quantum numbers
1
\(\psi\)
spatial wave function
m^{-3/2}
\(E\)
energy eigenvalue
\(\mathrm{J}\)
\(m\)
particle mass
\(\mathrm{kg}\)
\(h\)
Planck constant
\(\mathrm{J\,s}\)
Method
Derivation 1: Apply hard-wall boundary conditions
The wave function must vanish at every wall. The solution separates into standing waves along each axis.
Separated wave function
\[\psi(x,y,z)=X(x)Y(y)Z(z)\]
Allowed wavelengths
\[\lambda_i=\frac{2L_i}{n_i}\]
Derivation 2: Add kinetic energies
Each direction contributes a squared momentum component.
Momentum components
\[p_i=\frac{n_ih}{2L_i}\]
Energy
\[E=\frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)\]
Derivation 3: Recognize degeneracy
For a cube, permuting the same set of quantum numbers gives the same energy.
Cubic box energy
\[E=\frac{h^2}{8mL^2}(n_x^2+n_y^2+n_z^2)\]
Ground state
\[(n_x,n_y,n_z)=(1,1,1)\]
Rules
3D box energy
\[E=\frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)\]
Cubic box energy
\[E=\frac{h^2}{8mL^2}(n_x^2+n_y^2+n_z^2)\]
Cubic eigenfunction
\[\psi=\left(\frac{2}{L}\right)^{3/2}\sin\frac{n_x\pi x}{L}\sin\frac{n_y\pi y}{L}\sin\frac{n_z\pi z}{L}\]
Examples
Question
For a cubic box, compare \(E_{111}\) and \(E_{211}\).
Answer
\[E_{111}=3E_0\]
and \[E_{211}=6E_0\]
where \[E_0=h^2/(8mL^2)\]
Checks
- Each quantum number starts at 1.
- The ground state is \((1,1,1)\), not \((0,0,0)\).
- Energies add by squared components.
- Degeneracy depends on symmetry; a rectangular box usually breaks cubic degeneracies.