Problem 3 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

A particle of mass \(m\) is confined by an tnfinite potental barrier to rematn within a box \(0 \leq x, y, z \leq a\), so that the wave function vanishes on the boundary of the box. Shew that the energy levels are $$ E=\frac{1}{2 m} \frac{\pi^{2} \hbar^{2}}{a^{2}}\left(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\right) $$ where \(n_{1}, n_{2}, n_{3}\) are positive intcecrs, and calculate the stationary wave functions \(\psi_{E}(\mathrm{x})\). Verify that the lowest energy state is non-degenerate, but the next highest is triply degenerate.

Step-by-Step Solution

Answer

The energy levels of a particle in the three-dimensional box are given by \(E=\frac{1}{2m}\frac{\pi^2\hbar^2}{a^2}(n_x^2+n_y^2+n_z^2)\) where \(n_x\), \(n_y\), \(n_z\) are the positive integers representing the states in \(x\), \(y\), and \(z\) direction. The stationary wave functions are obtained as \(\psi_E(x) = \sqrt{\frac{8}{a^3}}\sin(k_x x)\sin(k_y y)\sin(k_z z)\), where \(k_x=\frac{n_x\pi}{a}\), \(k_y=\frac{n_y\pi}{a}\), \(k_z=\frac{n_z\pi}{a}\). The lowest energy state is non-degenerate, while the next highest state is triply degenerate.

Step 1: Deriving the Energy Function

The energy levels of a particle in a 3D box are given by solving the Schrödinger equation for the particle in a box scenario. The time independent Schrödinger equation in three dimensions is \(-\frac{\hbar^2}{2m} (\psi_{xx} + \psi_{yy} + \psi_{zz}) + V\psi = E\psi\). In a 3D box, the potential \(V\) is set to a finite value inside the box, i.e., \(0 \leq x,y,z \leq a\), and infinite outside it. Here, \(\psi_{xx}\) denotes the second derivative with respect to \(x\), and same applies for \(y\) and \(z\). Since the wave function \(\psi\) must vanish at the walls of the box (i.e., \(\psi(0,0,0)=\psi(a,a,a) = 0\), the solution to the Schrödinger equation inside the box is given as \(\psi(x,y,z) = A\sin(k_x x)\sin(k_y y)\sin(k_z z)\), where \(A\) is the normalization constant and \(k_x=\frac{n_x\pi}{a}\), \(k_y=\frac{n_y\pi}{a}\), \(k_z=\frac{n_z\pi}{a}\) with \(n_x\), \(n_y\), \(n_z\) being positive integers. Now, substitute \(\psi(x,y,z)\) into the Schrödinger equation to get the energy levels: \(E=\frac{1}{2m}\frac{\pi^2\hbar^2}{a^2}(n_x^2+n_y^2+n_z^2)\).

Step 2: Calculating the Stationary Wave Functions

The stationary wave functions of this system, denoted as \(\psi(x, y, z)\), can be found by solving the time-independent Schrödinger equation and applying the boundary conditions (\(\psi(0,0,0)=\psi(a,a,a) = 0\)). Consequently, the stationary wave functions \(\psi_E(x)\) can be expressed as a product of wave functions in the \(x\), \(y\), and \(z\) directions: \(\psi_E(x) = A\sin(k_x x)\sin(k_y y)\sin(k_z z)\), where \(A\) is the normalization constant that can be determined by solving the normalization condition, \(\int|\psi|^2 dx dy dz = 1\). This gives \(A=\sqrt{\frac{8}{a^3}}\). This makes \(\psi_E(x) = \sqrt{\frac{8}{a^3}}\sin(k_x x)\sin(k_y y)\sin(k_z z)\). Therefore, the stationary wave functions \(\psi_E(x)\) are derived.

Step 3: Determining the Energy Degeneracy

Energy degeneracy refers to the situation when two or more energy states have the same energy. This needs to be checked through the expression for \(E\) derived in step 1. When \(n_x=n_y=n_z=1\), we have the lowest energy state, \(E_1\). Increasing any one of the \(n\) values to 2 while keeping the others at 1 would provide the next energy states. There are three ways to do this (incrementing 'n' for either x, y, or z), hence these states are triply degenerate due to the three ways of obtaining the same energy level

Key Concepts

Infinite Potential Well

Imagine an infinite potential well as a deep and narrow trap from which a particle cannot escape. This is a theoretical model in quantum mechanics where potential energy is infinite outside a certain region and zero inside. It's an idealized concept used to simplify problems. Within this well, the particle behaves like it's in a perfect box, moving freely inside but bouncing back from the walls.

The potential energy inside the box is zero, while outside it's infinite, making it impossible for the particle to exist there. This forces the wave function of the particle to be zero at the boundaries, constraining its motion strictly within the box.
The infinite potential well is often used to model situations like an electron in a metal or in molecules, where the particle is confined to a space but free within that region. It's a foundational model for understanding quantum mechanics.

Schrödinger Equation

The Schrödinger Equation is the centerpiece of quantum mechanics, governing how quantum systems evolve over time. For an infinite potential well, we use the time-independent Schrödinger equation to find stationary states and energy levels. This form of the equation is useful when a system is not explicitly time-dependent.

In a three-dimensional box scenario, the Schrödinger equation is written as:

• \[-\frac{\hbar^2}{2m} (\psi_{xx} + \psi_{yy} + \psi_{zz}) + V\psi = E\psi\]

Here, \(V\) is the potential energy and becomes infinite outside the box, enforcing a zero wave function at boundaries. Solving this equation gives us the allowed wave functions and discrete energy levels. This equation is fundamental in predicting quantum behavior in confined systems.

Wave Function

The wave function, \(\psi(x, y, z)\), is a crucial concept that describes the quantum state of a particle. For a particle in an infinite potential well, this wave function reflects the probability of locating the particle at specific coordinates within the box.

The general form of the wave function within the box, using the boundary conditions imposed by the potential well, is:

• \[\psi(x, y, z) = A\sin(k_x x)\sin(k_y y)\sin(k_z z)\]

This expression is a product of sine functions, ensuring that the wave function vanishes at the box boundaries. The values \(k_x, k_y, \) and \(k_z\) are determined by:

• \[k_x = \frac{n_x\pi}{a}, \, k_y = \frac{n_y\pi}{a}, \, k_z = \frac{n_z\pi}{a}\]

where \(n_x, n_y,\) and \(n_z\) are positive integers representing quantum numbers in their respective directions. The normalization constant \(A\) ensures that the total probability of finding the particle within the box is 1.

Energy Levels

The energy levels of a particle in an infinite potential well are distinct, quantized values that emerge due to the conditions imposed by the well. Solving the Schrödinger equation reveals these energy levels, depicted by the formula:

• \[E=\frac{1}{2 m} \frac{\pi^{2}\hbar^{2}}{a^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)\]

where \(n_x, n_y, \) and \(n_z\) are the quantum numbers associated with different modes of vibration inside the well.

These values are integers, and each unique set corresponds to a specific energy state. The system's lowest energy, called the ground state, occurs when \(n_x = n_y = n_z = 1\), resulting in the non-degenerate lowest energy level.
Higher energy levels can be degenerate, meaning multiple quantum states have the same energy. For example, if any one of \(n_x, n_y,\) or \(n_z\) equals 2 while the others are 1, the energy level is triply degenerate. Degeneracy is a key feature affecting the system's symmetry and physical properties.