Harmonic Oscillator Wave Function Plotter

Last Updated on September 16, 2024 by Max

Theory of the Quantum Harmonic Oscillator

The wave functions generated in the above code correspond to the quantum harmonic oscillator, a fundamental model in quantum mechanics. This model describes the behavior of a particle subjected to a quadratic potential well, and it is essential for understanding various physical systems, including molecules, crystals, and fields in quantum field theory.

Description of the System

Consider a particle of mass \( m \) subject to a potential that is quadratic in position, such as:

\[V(x) = \frac{1}{2} m \omega^2 x^2,\]

where \( \omega \) is the angular frequency of the oscillator. This potential represents a restoring force proportional to the displacement \( x \), similar to a mass attached to a spring obeying Hooke’s law.

Schrödinger’s Equation

The time-independent Schrödinger equation for a particle in this potential is:

\[-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi(x) = E \psi(x),\]

where \( \hbar \) is the reduced Planck’s constant, \( \psi(x, t) \) is the wave function, and \( E \) is the corresponding energy eigenvalue.

Rearranging, we get:

\[\frac{d^2 \psi(x)}{dx^2} = \left( \frac{m^2 \omega^2}{\hbar^2} x^2 – \frac{2mE}{\hbar^2} \right) \psi(x).\]

Defining:

\[\xi = \sqrt{\frac{m \omega}{\hbar}} x, \quad \lambda = \frac{2E}{\hbar \omega},\]

the equation simplifies to:

\[\frac{d^2 \psi(\xi)}{d\xi^2} = (\xi^2 – \lambda) \psi(\xi).\]

Solution of the Schrödinger Equation

The general solution of this differential equation can be expressed as:

\[\psi_n(x) = N_n e^{-\xi^2 / 2} H_n(\xi),\]

where \( H_n(\xi) \) are the Hermite polynomials of order \( n \), and \( N_n \) is the normalization constant.

Normalization of Wave Functions

The normalization constant \( N_n \) is given by:

\[N_n = \left( \frac{1}{2^n n!} \sqrt{\frac{m \omega}{\pi \hbar}} \right)^{1/2}.\]

Thus, the normalized wave functions for the quantum harmonic oscillator are:

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

Quantized Energy Levels

The energy levels for the quantum harmonic oscillator are quantized and given by:

\[E_n = \left( n + \frac{1}{2} \right) \hbar \omega, \quad n = 0, 1, 2, 3, \ldots\]

These discrete energy levels reflect the quantum nature of the oscillator and the fact that the system cannot have arbitrary energy values.

Physical Interpretation

Nodes and Antinodes: Each wave function \( \psi_n(x) \) has \( n \) nodes (points where the wave function passes through zero), which indicates the number of times the probability amplitude changes sign. The higher the energy state, the more nodes the wave function has.

Wave Function \( (\psi_n(x)) \): Describes the probability amplitude of finding the particle at position \( x \). The square of the wave function, \( |\psi_n(x)|^2 \), represents the probability density, showing the likelihood of finding the particle in a particular region.

Quantization: The discrete energy levels result from the boundary conditions imposed by the potential and the requirement that the wave function remains finite everywhere. Unlike classical mechanics, where energy can vary continuously, quantum mechanics restricts the allowed energy values to discrete levels due to the wave nature of particles.

Harmonic Oscillator in Quantum Mechanics

The quantum harmonic oscillator is a model that appears in numerous physical systems, ranging from molecular vibrations to quantum fields. It demonstrates the principles of quantization, wave function behavior, and the uncertainty principle in a simple yet profound way. The solutions, involving Hermite polynomials, provide insights into the behavior of particles confined by a quadratic potential, a fundamental concept in both physics and chemistry.

Examples of quantum systems where the harmonic oscillator concept is used include:

1. Molecular Vibrations: The harmonic oscillator approximates the vibrational motion of atoms in a diatomic molecule around their equilibrium positions, providing a basis for understanding infrared spectroscopy.
2. Phonons in Crystals: The quantized vibrational modes of atoms in a crystalline solid, known as phonons, can be modeled as harmonic oscillators. This approach is fundamental to solid-state physics and helps describe thermal properties and lattice dynamics.
3. Quantum Field Theory: In quantum field theory, each mode of a free scalar field is treated as a quantum harmonic oscillator. This forms the basis for understanding particle creation and annihilation in fields.
4. Quantum Optics: The electromagnetic field in a cavity or a mode of light in free space can be quantized as a harmonic oscillator, which is essential for describing coherent states and the quantum nature of light.
5. Trapped Ions and Atoms: Ions or neutral atoms confined in electromagnetic traps experience a potential well that closely approximates a harmonic oscillator, allowing precise control of their quantum states for quantum computing and simulation.
6. Superconducting Circuits: The oscillatory modes of superconducting resonators or LC circuits, fundamental components in quantum circuits, are often described as quantum harmonic oscillators, which are used in superconducting qubits.
7. Quantum Dots: In quantum dots, the motion of electrons can be approximated by a harmonic potential, useful for understanding electronic states and optical properties in nanoscale materials.

These examples illustrate the importance of the quantum harmonic oscillator in modeling diverse phenomena across different areas of physics.

Conclusion

The harmonic oscillator wave function plotter visualizes wave functions and probability distributions for different quantum states \( n \). As the quantum number \( n \) increases, the wave functions display more nodes. These nodes indicate higher energy levels in the system. The plotter helps illustrate energy quantization, where each state has a unique waveform and probability density. It demonstrates fundamental principles of quantum mechanics, like the relationship between energy levels and wave function behavior. This tool offers an intuitive way to explore the properties of the quantum harmonic oscillator across various physical systems.