The eigenfunctions of a one-dimensional quantum harmonic oscillator are generally written as:

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

where:

• $n$ is the quantum number (non-negative integer $n = 0, 1, 2, \ldots$),
• $m$ is the mass of the particle,
• $\omega$ is the angular frequency of the oscillator,
• $\hbar$ is the reduced Planck's constant,
• $H_n$ is the Hermite polynomial of order $n$.

Explanation of Terms:

1. Normalization Factor: $\left( \frac{1}{\sqrt{2^n n!}} \right) \left( \frac{m \omega}{\pi \hbar} \right)^{1/4}$ ensures that the wave function is normalized.
2. Exponential Term: $e^{-\frac{m \omega x^2}{2 \hbar}}$ represents the Gaussian envelope of the wave function, which decays rapidly for large $|x|$.
3. Hermite Polynomial: $H_n \left( \sqrt{\frac{m \omega}{\hbar}} x \right)$ describes the oscillatory part of the wave function, with $n$ nodes corresponding to the energy level.

These eigenfunctions correspond to the energy eigenvalues:

$E_n = \left( n + \frac{1}{2} \right) \hbar \omega$

for $n = 0, 1, 2, \ldots$.

Would you like more details or have any questions?

Relative Questions:

1. How are Hermite polynomials defined and what are their properties?
2. What is the significance of the quantum number $n$ in the context of a harmonic oscillator?
3. How do we derive the eigenfunctions and eigenvalues of a quantum harmonic oscillator using the Schrödinger equation?
4. What is the role of the normalization constant in the wave function?
5. How does the potential energy of a quantum harmonic oscillator influence the form of its wave functions?

Tip:

For higher quantum numbers $n$, the wave functions exhibit more nodes, corresponding to higher energy levels, and the probability density spreads out more, reflecting the particle's increased energy.