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The “mass on a spring” is a common example of simple harmonic motion. The quantum harmonic oscillator is a quantum mechanical analog of this scenario, in which a particle is placed in the following potential:

$V(x) = \frac{1}{2} m \omega^2 x^2$

The corresponding solutions to the Schrödinger equation have the following form:

$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} e^{-\frac{m \omega x^2}{2\hbar}} H_n\left( \sqrt{\frac{m \omega}{\hbar}} x \right), \quad n = 0, 1, 2, 3, \dots$

where $H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right)$ are the physicists' Hermite polynomials.

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Problem:

Use the Schrödinger equation to determine expressions for the first three energy levels of the quantum harmonic oscillator in terms of $\omega$ and $\hbar$. Then, use these to determine a general expression for the energy of any state $n$.

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This is the transcription of the provided text. Would you like a detailed breakdown or explanation of the quantum harmonic oscillator's energy levels?

Here are 5 related questions to expand on this topic:

1. What is the Schrödinger equation for a one-dimensional system?
2. How do Hermite polynomials relate to the solutions of the quantum harmonic oscillator?
3. What is the classical analogy for the quantum harmonic oscillator?
4. How is the potential energy term $V(x) = \frac{1}{2} m \omega^2 x^2$ derived in classical mechanics?
5. Can you explain the significance of the quantum number $n$ in the context of quantum harmonic oscillators?

Tip: The general energy of the $n$-th state of a quantum harmonic oscillator is $E_n = \left(n + \frac{1}{2}\right)\hbar \omega$.