A Particle in a 2D Box
A system with Degeneracy
The particle-in-a-box model can be extended to two dimensions to describe a particle confined to a rectangular region. In this model, the particle is free to move in the \(x\) and \(y\) directions but is confined by infinitely high potential walls.
The particle is restricted to the region \(0 \le x \le L_x\) and \(0 \le y \le L_y\), and the wavefunction must vanish at all boundaries.
1. Hamiltonian
Inside the box, the potential energy is zero, so the Hamiltonian consists only of kinetic energy contributions from motion in the \(x\) and \(y\) directions:
\[ \hat{H} = -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right). \]
This Hamiltonian is the sum of two independent one-dimensional kinetic energy operators.
2. Schrödinger equation
The time-independent Schrödinger equation is
\[ \hat{H}\psi(x,y) = E\,\psi(x,y). \]
Because the Hamiltonian separates into independent \(x\) and \(y\) parts, the equation is separable. We can write the wavefunction as a product:
\[ \psi(x,y) = X(x)\,Y(y). \]
This allows us to separate the Schrödinger equation into two separate equations. This happens as follows:
\[-\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right)X(x)\,Y(y) = E\,X(x)\,Y(y)\]
Because the partial derivitives only operate on one of the variables, the other part of the function can be treated as a constant, and will passs through the partial derivitive operator. This will also turn the partial derivitives into total derivitives.
\[-\frac{Y(y)\hbar^2}{2m}\frac{d^2}{d x^2} X(x) - \frac{X(x)\hbar^2}{2m}\frac{d^2}{d y^2} Y(y) = E\,X(x)\,Y(y)\]
Dividing both sides by \[X(x)\,Y(y)\] and \[-\frac{\hbar^2}{2m}\] yields the following:
\[\frac{1}{X(x)}\,\frac{d^2}{d x^2} X(x) + \frac{1}{Y(y)}\,\frac{d^2}{d y^2} Y(y) = -\frac{2 m E}{\hbar^2}\]
Since \[x\] and \[y\] are completely independent variables, the only way this equation can hold is if each of the pieces on the left-hand side are constant. We will call these constants \[-k_x^2\] and \[-k_x^2\], and place on them the following constraint:
\[-k_x^2 - k_y^2 = -k^2\]
This generates the two (possibly familiar looking) equations:
\[\frac{d^2}{d x^2}\,X(x) = -k_x^2\,X(x)\] \[\frac{d^2}{d y^2}\,Y(y) = -k_y^2\,Y(y)\]
These equations can be solved to determine energies in both x and y directions:
\[E_x = \frac{n_x^2 h^2}{8 m L_x^2}\] \[E_y = \frac{n_y^2 h^2}{8 m L_y^2}\]
And
\[E_{tot} = E_x + E_y\]
3. Energy levels
Solving the separated equations yields quantized energies associated with motion in each direction. The total energy is the sum of the \(x\) and \(y\) contributions:
\[ E_{tot} = \frac{h^2}{8m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} \right), \qquad n_x,n_y = 1,2,3,\dots \]
Each allowed energy is specified by a pair of quantum numbers, \( (n_x,n_y) \), reflecting the independent quantization of motion in the two directions.
4. Wavefunctions
The corresponding normalized wavefunctions are products of the one-dimensional particle-in-a-box solutions:
\[ \psi_{n_x,n_y}(x,y) = \sqrt{\frac{2}{L_x}}\sin\!\left(\frac{n_x\pi x}{L_x}\right) \sqrt{\frac{2}{L_y}}\sin\!\left(\frac{n_y\pi y}{L_y}\right). \]
These wavefunctions have nodes wherever either sine factor is zero, producing characteristic two-dimensional standing-wave patterns inside the box.
Big idea: in two dimensions, confinement in each direction leads to independent quantization, and the total energy is the sum of the energies associated with each coordinate.
A particularly important special case of the two-dimensional particle in a box occurs when the box is square, so that \( L_x = L_y = L \). In this case, the energy levels simplify and new physical behavior appears.
For a square box, the allowed energies become
\[ E_{n_x,n_y} = \frac{h^2}{8mL^2} \left( n_x^2 + n_y^2 \right), \qquad n_x,n_y = 1,2,3,\dots \]
Because the energy depends only on the sum of the squares \( n_x^2 + n_y^2 \), different pairs of quantum numbers can correspond to the same energy. When two or more distinct quantum states have the same energy, the energy level is said to be degenerate.
Example of degeneracy
Consider the states \( (n_x,n_y)=(1,2) \) and \( (n_x,n_y)=(2,1) \). For both states,
\[ n_x^2 + n_y^2 = 1^2 + 2^2 = 2^2 + 1^2 = 5, \]
so the two states have the same energy and are therefore twofold degenerate.
In contrast, the ground state \( (n_x,n_y)=(1,1) \) is not degenerate, since no other pair of positive integers gives the same value of \( n_x^2 + n_y^2 \).
| level | nx | ny | nx2 + nx2 | degeneracy |
|---|---|---|---|---|
| 1 | 1 | 1 | 2 | 1 |
| 2 | 1 | 2 | 5 | 2 |
| 2 | 1 | |||
| 3 | 2 | 2 | 8 | 1 |
| 4 | 3 | 1 | 10 | 2 |
| 1 | 3 |
Physical meaning of degeneracy
Degeneracy arises from the symmetry of the square box. Exchanging the \(x\) and \(y\) directions leaves the system unchanged, so states that differ only by swapping \( n_x \) and \( n_y \) must have the same energy.
If the box is not square (\( L_x \neq L_y \)), this symmetry is broken and the degeneracy is lifted: states such as \( (1,2) \) and \( (2,1) \) no longer have the same energy.
Big idea: degeneracy reflects the symmetry of a quantum system, and changing that symmetry changes the energy-level structure.