The Harmonic Potential

The quantum harmonic oscillator is arguably the single most important exactly-solvable model in all of quantum mechanics. It appears everywhere — in vibrating molecules, lattice phonons, the electromagnetic field, and as the local approximation to any smooth potential near its minimum. Before solving it we need to understand the potential energy it is built on.

The classical harmonic oscillator

Classically, a mass $m$ attached to a spring of constant $k$ obeys a restoring force $F = -kx$ . Its potential energy is

$V(x) = \tfrac{1}{2}kx^2.$

The constant $k$ is related to the angular frequency of oscillation $\omega$ by $k = m\omega^2$ , so the potential is equivalently written

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

This parabolic bowl centred at $x = 0$ is the defining feature of a harmonic oscillator: the force is always directed back toward the origin, and its magnitude grows linearly with the displacement.

Why parabolas are universal near equilibrium

Consider any smooth potential $V(x)$ that has a minimum at $x_0$ . A Taylor expansion around that minimum gives

$V(x) \approx V(x_0) + \underbrace{V'(x_0)}_{=\,0}\,(x-x_0) + \tfrac{1}{2}V''(x_0)(x-x_0)^2 + \cdots$

The linear term vanishes because $x_0$ is a minimum, and for a stable equilibrium $V''(x_0) > 0$ . Setting $k = V''(x_0)$ and measuring energy from $V(x_0)$ , the lowest-order approximation is exactly $\tfrac{1}{2}k(x-x_0)^2$ — a harmonic potential. This is why the harmonic oscillator is not merely a textbook toy: every bound-state problem looks like a harmonic oscillator for small oscillations.

Setting up the quantum problem

To pass from classical mechanics to quantum mechanics we replace the classical energy with the Hamiltonian operator. The total classical energy is kinetic plus potential:

$E = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2.$

In quantum mechanics the momentum $p$ becomes the operator $\hat{p} = -i\hbar\,\partial/\partial x$ , and $x$ becomes the multiplication operator $\hat{x}$ . The Hamiltonian operator is therefore

= -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2.$$ The time-independent Schrödinger equation $\hat{H}\psi = E\psi$ then reads $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^2\,\psi = E\,\psi.$$ This is the equation we need to solve to find the allowed energy levels and wave functions of the quantum harmonic oscillator. <Callout type="tip"> Notice that the potential $V(x) = \tfrac{1}{2}m\omega^2 x^2$ grows without bound as $|x| \to \infty$. This means the particle is always confined, so all energy levels are discrete — there are no scattering states for the harmonic oscillator. </Callout> ## Natural scales of the problem The Schrödinger equation above contains three parameters: $\hbar$, $m$, and $\omega$. It is useful to identify the natural length and energy scales they define. The only combination of $\hbar$, $m$, and $\omega$ with dimensions of length is $$x_0 = \sqrt{\frac{\hbar}{m\omega}},$$ called the **zero-point length** or characteristic length of the oscillator. Similarly, $\hbar\omega$ has dimensions of energy and sets the natural energy scale. When we solve the eigenvalue problem in the next lesson, we will find that all energy levels are integer multiples of $\hbar\omega$ offset by a zero-point energy of $\tfrac{1}{2}\hbar\omega$ — a direct consequence of the uncertainty principle. ## Physical examples The harmonic oscillator model is not an abstraction. A few concrete cases illustrate its range: - **Diatomic molecules.** The potential energy between two atoms bonded in a molecule has a minimum near the equilibrium bond length. For small vibrations the parabolic approximation is excellent, and infrared spectroscopy measures the resulting equally-spaced vibrational energy levels. - **Phonons in crystals.** Atoms in a solid vibrate about their lattice sites. The collective normal modes are again described by harmonic oscillators, and "phonons" are their quantised excitations. - **Electromagnetic field modes.** Each mode of the radiation field in a cavity is mathematically identical to a harmonic oscillator; photons are the quanta of those modes. In every case the essential physics is captured by $\hat{H} = \hat{p}^2/(2m) + \tfrac{1}{2}m\omega^2\hat{x}^2$ — the setup we have just established.