Write the potential energy of a one-dimensional harmonic oscillator in terms of \(m\), \(\omega\), and \(x\).

Write the allowed energy levels of the quantum harmonic oscillator.

What is the ground-state energy of a quantum harmonic oscillator?

Find the energy spacing \(E_{n+1}-E_n\).

A harmonic oscillator has adjacent energy spacing \(0.20\,\mathrm{eV}\). Find \(\omega\) in terms of this spacing.

What photon energy is emitted in a transition from \(n=5\) to \(n=2\) for a harmonic oscillator?

Write the ground-state length scale \(x_0\) for a quantum harmonic oscillator.

How does the ground-state width change if \(\omega\) is quadrupled while \(m\) is fixed?

Find the classical turning-point magnitude for an oscillator state of energy \(E_n\).

How many nodes does the \(n=4\) harmonic oscillator eigenstate have?

State the parity pattern of one-dimensional harmonic oscillator eigenstates.

Find \(\langle x\rangle\) for any harmonic oscillator energy eigenstate centered at \(x=0\).

For the oscillator ground state, use \(\Delta x=\sqrt{\hbar/(2m\omega)}\) and \(\Delta p=\sqrt{\hbar m\omega/2}\) to find \(\Delta x\Delta p\).

Using the virial theorem for a harmonic oscillator, find \(\langle K\rangle\) and \(\langle V\rangle\) in the state \(n\).

Find \(\langle x^2\rangle\) for the oscillator ground state using the virial theorem.

Why is the oscillator ground-state energy nonzero?

Contrast the energy spacing of a harmonic oscillator with that of a particle in a box.

A perturbation can only cause transitions with \(\Delta n=\pm1\) when it is proportional to \(x\). Explain this selection rule qualitatively using parity.

Use the ladder-operator result \(\hat H=\hbar\omega(\hat a^\dagger\hat a+1/2)\) to justify the oscillator spectrum.

Estimate the oscillator zero-point energy by minimizing \(E\approx (\Delta p)^2/(2m)+m\omega^2(\Delta x)^2/2\) with \(\Delta p\approx\hbar/(2\Delta x)\).