The quantum harmonic oscillator is one of the most important exactly solvable systems in physics. It models a particle
subject to a restoring force proportional to displacement, which corresponds to the potential
Harmonic-oscillator potential.
\[
V(x)=\frac12 kx^2=\frac12 m\omega^2x^2.
\]
Here, \(k\) is the spring constant, \(m\) is the particle mass, and \(\omega\) is the angular frequency. These are
related by
\[
\omega=\sqrt{\frac{k}{m}}.
\]
In classical mechanics, the oscillator can have any energy. In quantum mechanics, the allowed energies are discrete:
Energy levels.
\[
E_n=\hbar\omega\left(n+\frac12\right),
\qquad n=0,1,2,\dots
\]
This formula shows two striking features. First, the levels are equally spaced, because the difference between
neighboring levels is always
\[
E_{n+1}-E_n=\hbar\omega.
\]
Second, the ground-state energy is not zero. When \(n=0\),
\[
E_0=\frac12\hbar\omega.
\]
This is called the zero-point energy. It means the oscillator still possesses energy even in its lowest possible state.
The reason is quantum uncertainty: a particle confined near the minimum of the potential cannot simultaneously have
exactly zero position spread and exactly zero momentum.
Wave functions and Hermite polynomials
The stationary-state wave functions are built from a Gaussian factor multiplied by Hermite polynomials. It is common to
introduce the reduced coordinate
\[
\xi=\sqrt{\frac{m\omega}{\hbar}}\,x.
\]
In terms of \(\xi\), the normalized eigenfunctions are
Reduced-coordinate eigenfunctions.
\[
\psi_n(\xi)=\frac{1}{\sqrt{2^n n!\sqrt{\pi}}}\,H_n(\xi)e^{-\xi^2/2}.
\]
Here, \(H_n(\xi)\) is the Hermite polynomial of order \(n\). The Gaussian factor keeps the wave function localized,
while the Hermite polynomial determines the number of oscillations. The \(n\)-th state has \(n\) nodes, so the
wave-function shape becomes more oscillatory as the energy increases.
Sample natural-units interpretation
If one works in scaled or natural oscillator units and sets \(\omega=1\), then the energy formula becomes especially
simple:
\[
E_n=\hbar\left(n+\frac12\right).
\]
Therefore,
\[
E_0=0.5\,\hbar,
\qquad
E_1=1.5\,\hbar,
\qquad
E_2=2.5\,\hbar.
\]
This makes the \((n+\tfrac12)\) scaling very transparent. In physical SI units, one multiplies by the actual angular
frequency \(\omega\). For example, if \(\omega\) is large, the spacing \(\hbar\omega\) becomes large, which is why
molecular vibrational spectra can show quantized energy bands.
Oscillator length scale
A useful length scale for the problem is
\[
a=\sqrt{\frac{\hbar}{m\omega}}.
\]
This quantity controls the natural spatial spread of the ground-state wave function. Larger mass or larger frequency
makes the oscillator more tightly localized, while smaller mass or smaller frequency spreads it out.
Physical meaning
The quantum harmonic oscillator appears throughout physics. It models molecular vibrations, small oscillations around
equilibrium, phonons in solids, quantum fields decomposed into modes, and many approximate bound-state systems near a
stable minimum. Because it is exactly solvable, it is also one of the main bridges between abstract quantum formalism
and concrete physical intuition.
At a more advanced level, the same system is described elegantly using ladder operators \(a\) and \(a^\dagger\), which
raise or lower the quantum number by one step. That formalism explains immediately why the spacing is uniform and why
the ground state cannot be lowered below \(E_0=\frac12\hbar\omega\).
| Concept |
Main relation |
Meaning |
| Potential energy |
\(V(x)=\frac12 kx^2=\frac12 m\omega^2x^2\) |
Quadratic restoring potential |
| Frequency relation |
\(\omega=\sqrt{k/m}\) |
Links spring constant and mass |
| Energy levels |
\(E_n=\hbar\omega(n+\frac12)\) |
Quantized equally spaced levels |
| Zero-point energy |
\(E_0=\frac12\hbar\omega\) |
Lowest state is not zero |
| Reduced coordinate |
\(\xi=\sqrt{m\omega/\hbar}\,x\) |
Natural coordinate for oscillator wave functions |
| Wave functions |
\(\psi_n(\xi)=\frac{1}{\sqrt{2^n n!\sqrt{\pi}}}H_n(\xi)e^{-\xi^2/2}\) |
Hermite-Gaussian stationary states |