Quantum mechanics

The quantum harmonic oscillator

The quantum oscillator has equally spaced levels $E_n=\hbar\omega(n+1/2)$ and a nonzero zero-point energy.

The quantum spring is an exactly solvable model for vibrations near equilibrium in molecules, lattices, and many field systems. Unlike a classical oscillator resting at the potential minimum, its ground state still fluctuates quantum mechanically.

V(x)=\frac12m\omega^2x^2,\qquad E_n=\hbar\omega\left(n+\frac12\right),\quad n=0,1,2,...

Definition: Zero-point energy

The lowest level is $E_0=\hbar\omega/2$ , not zero. The ground-state wavefunction is Gaussian, with $\Delta x=\sqrt{\hbar/(2m\omega)}$ and $\Delta p=\sqrt{m\hbar\omega/2}$ ; their product reaches $\hbar/2$ .

Change

n

to inspect

|\psi_n|^2

and equally spaced energies; zero-point motion leaves finite density even at the potential minimum.

Excited states

Eigenfunctions are Hermite polynomials times a Gaussian. The $n$ th state has $n$ nodes, with probability distributed on both sides of the nodes. Its evenly spaced spectrum makes the oscillator a useful basis for expanding nearly parabolic potentials.

Example: Level spacing

An oscillator has frequency $\nu=5.0\times10^{13}$ Hz. Find the adjacent-level spacing $E_{n+1}-E_n$ .

Solution

$\Delta E=\hbar\omega=h\nu=6.626\times10^{-34}\times5.0\times10^{13}=3.31\times10^{-20}$ J $=0.207$ eV.

Ladder operators $a$ and $a^\dagger$ construct the spectrum without directly solving a differential equation. They obey $[a,a^\dagger]=1$ , and $H=\hbar\omega(a^\dagger a+1/2)$ . The ground state is annihilated by $a$ , so its energy remains $\hbar\omega/2$ ; excited states follow by repeatedly applying $a^\dagger$ . The ground-state probability density is Gaussian, while higher states have nodes. The model approximates small oscillations near a potential minimum and provides the basis for phonons in crystals.

Quick check

What is the quantum oscillator's ground-state energy?

How does the spacing between adjacent levels depend on $n$ ?

References

David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics