Quantum Mechanics in Three Dimensions: Particle in a Box

Quantum Mechanics in Three Dimensions

Kinetic Energy and the Schrödinger Equation

In quantum mechanics, the behavior of a particle in three dimensions is governed by the three-dimensional Schrödinger equation. The kinetic energy operator in three dimensions is expressed as the sum of the kinetic energies in each spatial direction.

• Kinetic Energy Operator: The kinetic energy for a particle of mass m is given by: $K = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{p_z^2}{2m}$

• Time-Dependent Schrödinger Equation: $i\hbar \frac{\partial \Psi(x, y, z, t)}{\partial t} = \left[ -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) + U(x, y, z) \right] \Psi(x, y, z, t)$

• Normalization Condition: The wave function must satisfy: $\int_{-\infty}^{+\infty} |\Psi(x, y, z, t)|^2 dV = 1$

• Stationary States: For a state of definite energy, the wave function can be separated as: $\Psi(x, y, z, t) = \psi(x, y, z) e^{-iEt/\hbar}$

• Time-Independent Schrödinger Equation: $-\frac{\hbar^2}{2m} \nabla^2 \psi(x, y, z) + U(x, y, z) \psi(x, y, z) = E \psi(x, y, z)$

Particle in a Three-Dimensional Box

Potential and Boundary Conditions

The three-dimensional box (also called an infinite potential well) is a fundamental quantum system where a particle is confined within a cubical region of side L with infinitely high potential walls.

• Potential Energy Function:

  • $U(x) = 0$ for $0 \leq x \leq L$, otherwise $U(x) = \infty$

  • $U(y) = 0$ for $0 \leq y \leq L$, otherwise $U(y) = \infty$

  • $U(z) = 0$ for $0 \leq z \leq L$, otherwise $U(z) = \infty$

• Boundary Conditions: The wave function must vanish at the walls: $\psi = 0$ at $x=0, x=L, y=0, y=L, z=0, z=L$.

Separation of Variables and Solution

The time-independent Schrödinger equation inside the box can be solved using separation of variables, assuming a solution of the form $\psi(x, y, z) = X(x)Y(y)Z(z)$.

• Separation of Variables: Leads to three independent equations for $X(x)$, $Y(y)$, and $Z(z)$, each resembling the one-dimensional particle in a box problem.

• Allowed Solutions:

  • $X_n(x) = C_X \sin\left(\frac{n_X \pi x}{L}\right)$, $n_X = 1, 2, 3, ...$

  • $Y_n(y) = C_Y \sin\left(\frac{n_Y \pi y}{L}\right)$, $n_Y = 1, 2, 3, ...$

  • $Z_n(z) = C_Z \sin\left(\frac{n_Z \pi z}{L}\right)$, $n_Z = 1, 2, 3, ...$

• General Solution: $\psi_{n_X, n_Y, n_Z}(x, y, z) = C \sin\left(\frac{n_X \pi x}{L}\right) \sin\left(\frac{n_Y \pi y}{L}\right) \sin\left(\frac{n_Z \pi z}{L}\right)$

• Normalization: The constant $C$ is determined by the normalization condition.

Energy Levels and Quantum Numbers

The energy levels of a particle in a three-dimensional box depend on three quantum numbers: $n_X$, $n_Y$, and $n_Z$.

• Energy Eigenvalues: $E_{n_X, n_Y, n_Z} = \frac{\pi^2 \hbar^2}{2mL^2} (n_X^2 + n_Y^2 + n_Z^2)$

• Quantum Numbers: $n_X, n_Y, n_Z = 1, 2, 3, ...$

• Degeneracy: Different combinations of quantum numbers can yield the same energy (degenerate states).

Probability Distributions and Nodal Planes

The probability density $|\psi_{n_X, n_Y, n_Z}(x, y, z)|^2$ describes the likelihood of finding the particle at a given location. Nodal planes are regions where the probability density is zero.

• Nodal Planes: For each quantum number greater than one, there is a nodal plane perpendicular to the corresponding axis.

• Examples:

  • For $\psi_{2,1,1}$, there is a nodal plane at $x = L/2$ where the probability density is zero.

  • For $\psi_{1,2,1}$, the nodal plane is at $y = L/2$.

  • For $\psi_{1,1,2}$, the nodal plane is at $z = L/2$.

Energy Level Diagram and Degeneracy

The energy levels for a particle in a cubical box can be represented graphically, showing the degeneracy (number of states with the same energy) for each level. Degeneracy arises from the symmetry of the box; if the box is not symmetric (i.e., $L_X \neq L_Y \neq L_Z$), degeneracy is lifted.

• Energy Level Structure: The lowest energy state is $(n_X, n_Y, n_Z) = (1,1,1)$. Higher energy levels correspond to higher quantum numbers and may be degenerate.

• Degeneracy Table: The number of distinct quantum states for each energy level is shown in the diagram below.

Summary Table: Energy Levels and Degeneracy

Quantum Numbers (n_X, n_Y, n_Z)	Energy (in units of $E_{1,1,1}$)	Degeneracy
(1,1,1)	$E_{1,1,1}$	1
(2,1,1), (1,2,1), (1,1,2)	2$E_{1,1,1}$	3
(2,2,1), (2,1,2), (1,2,2)	3$E_{1,1,1}$	3
(2,2,2)	4$E_{1,1,1}$	1
(3,1,1), (1,3,1), (1,1,3)	$\frac{11}{3}E_{1,1,1}$	3
(3,2,1), (3,1,2), (2,3,1), (2,1,3), (1,3,2), (1,2,3)	$\frac{14}{3}E_{1,1,1}$	6

Additional info: Degeneracy is a direct consequence of the symmetry of the box. If the box dimensions are not equal, the degeneracy is removed, and each energy level becomes unique to its quantum numbers.