Quantum Harmonic Oscillator: Energy Levels and Wave Functions

All flashcards

What potential describes a quantum harmonic oscillator?

The quadratic potential $U(x)=\frac{1}{2}m\omega^2x^2$, where $m$ is mass and $\omega$ is angular frequency.

What are the allowed energies of a quantum harmonic oscillator?

$E_n=(n+\frac{1}{2})\hbar\omega$, for nonnegative integer quantum numbers $n$.

What is the form of a harmonic-oscillator stationary-state wave function?

A normalization constant times a Gaussian factor and a Hermite polynomial: $\psi_n(x)=N_n e^{-\beta^2x^2/2}H_n(\beta x)$.

What is zero-point energy?

The nonzero ground-state energy of a quantum harmonic oscillator, $E_0=\frac{1}{2}\hbar\omega$.

What does Bohr’s correspondence principle predict for highly excited oscillators?

At high quantum numbers, quantum predictions approach the corresponding classical behavior.

Quiz questions

1. What is the energy difference between neighboring levels of a quantum harmonic oscillator?Answer: $\hbar\omega$
2. What characterizes the probability density of the quantum harmonic oscillator’s ground state?Answer: It is greatest at the center of the potential well.
3. What does it mean for an oscillator to be in a stationary state?Answer: It has definite energy, and its probability density is time-independent.
4. How does the parity of oscillator wave functions depend on quantum number $n$?Answer: Even $n$ gives an even wave function; odd $n$ gives an odd one.
5. What can be true of a quantum oscillator beyond its classical turning points?Answer: Its probability density can be nonzero in those classically forbidden regions.