eigenfunction for a one-dimensional quantum harmonic oscillator is generally written as:

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.

Explore the eigenfunctions of a one-dimensional quantum harmonic oscillator, covering key quantum mechanics concepts such as wave functions, Hermite polynomials, and energy levels. This detailed explanation is suitable for undergraduate physics students.

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