Problem 7 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Show that the eigenvalues of the three-dimensional harmonic oscillater have the form \(\left(n+\frac{3}{2}\right) \hbar \omega\) where \(n\) is a non-negative integer. Show that the dcpeneracy of the \(n\)th eigenvalue is \(\frac{1}{2}\left(n^{2}+3 n+2\right)\). Find the corresponding eigenfunctions.

Step-by-Step Solution

Answer

The eigenvalues of the 3D Quantum Harmonic Oscillator will take the form \((n + 3/2) ħω\). The degeneracy of any eigenvalue \(n\) of this system is given by the formula \(\frac{1}{2} (n^2 + 3n + 2)\). The eigenfunctions are a product of 1D QHO eigenfunctions for each dimension.

Step 1: Understand the 3D Quantum Harmonic Oscillator

The eigenstates of a 3D Quantum Harmonic Oscillator are product of one dimensional harmonic oscillator eigenstates for each dimension. The quantum numbers of a 3D QHO are given by three integers (n_x, n_y, n_z), each corresponding to one dimension. The eigenstates are signified by |n_x, n_y, n_z>. The energy of each state is: E = (n_x + n_y + n_z + 3/2) ħω

Step 2: Determine the eigenvalues

Substituting \(n=n_x + n_y + n_z\) in the energy equation we get: \(E= (n + 3/2) ħω\). This indicates that eigenvalues of a 3D QHO take the form \((n + 3/2) ħω\) where \(n=n_x + n_y + n_z\) is a non-negative integer.

Step 3: Calculate the degeneracy

Degeneracy is the number of different states corresponding to a particular energy level. For a 3D QHO, degeneracy is given by the number of different combinations of \(n_x, n_y, n_z\) that sum up to \(n\). This is given by the formula: \(\frac{1}{2} (n^2+3n+2)\). This can be shown mathematically through combinatorial principles.

Step 4: Finding the eigenfunctions

The 3D QHO eigenfunctions are a product of the 1D eigenfunctions for each dimension. The 1D QHO eigenfunctions are given by: \(\psi_n(x) = ((mω/(πħ))^(1/4) / √2^n n!) H_n((mω/ħ)^1/2 x) e^(−mωx^2 / 2ħ)\) where H_n is the Hermite polynomial of order n.

Key Concepts

Eigenvalues

In quantum mechanics, eigenvalues play a crucial role as they correspond to measurable quantities in a system, such as energy levels. For a three-dimensional quantum harmonic oscillator, determining the eigenvalues helps us understand the energy levels associated with different quantum states.
In this case, the energy eigenvalues can be expressed as

• \(E = \(n + \frac{3}{2}\) \hbar \omega\)

where \(n\) is a non-negative integer. This formula arises from the sum of quantum numbers \(n_x, n_y,\) and \(n_z\) along the three dimensions of space. Each dimension acts like a one-dimensional harmonic oscillator, contributing its own component to the overall energy. Understanding this formulation gives insight into how energy states are quantized in a multi-dimensional system.

Eigenfunctions

Eigenfunctions describe the positional probabilities of a quantum particle in a system. For the 3D Quantum Harmonic Oscillator, eigenfunctions are derived from the product of one-dimensional harmonic oscillator eigenfunctions for each dimension. These are represented as

• \(|n_x, n_y, n_z\>\)

Each component of the eigenfunction corresponds to the Hermite polynomial \(H_n\), which dictates the shape of the wavefunction along a particular axis.
The general form for one-dimensional eigenfunctions is given by:

• \(\psi_n(x) = \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n\left(\frac{m\omega}{\hbar}^{1/2} x\right) e^{-m\omega x^2 / 2\hbar}\)

Understanding these eigenfunctions helps visualize how particles behave within the potential wells of each dimension.

Degeneracy

In quantum systems, degeneracy refers to the number of quantum states that share the same energy level. For the 3D Quantum Harmonic Oscillator, multiple combinations of quantum numbers \(n_x, n_y,\) and \(n_z\) can lead to the same total \(n\). Degeneracy for this system is calculated using the formula:

• \(\frac{1}{2} (n^2 + 3n + 2)\)

This expression emerges from combinatorial methods, considering all the ways the integers \(n_x, n_y,\) and \(n_z\) can sum to provide the same \(n\). By understanding degeneracy, one can predict the multiplicity of an energy level, which is important in statistical physics and helps explain phenomena in spectroscopy and quantum statistics.

Quantum Mechanics

Quantum mechanics provides the framework to understand the behavior of systems at atomic and subatomic scales. It introduces concepts like wave-particle duality, quantization, and probability. At the heart of quantum mechanics is the Schrödinger equation, which describes how the quantum state of a system evolves over time.
In the context of the 3D Quantum Harmonic Oscillator, quantum mechanics allows us to predict energy eigenvalues and eigenfunctions by solving the Schrödinger equation for a potential well. This reveals quantized energy levels and probabilistic behaviors of particles in potentials. Understanding quantum mechanics is crucial for interpreting experimental data and predicting system behaviors across physics and chemistry. This understanding underpins modern technology, impacting fields from material science to quantum computing.