Simple harmonic motion is exceptionally important in classical physics, where it serves as a prototype for more complex oscillations. As you might expect, a microscopic oscillator—the quantum harmonic oscillator—is equally important as a model of oscillations at the atomic level.

The defining characteristic of simple harmonic motion is a linear restoring force: $F=-kx$, where $k$ is the spring constant. The corresponding potential-energy function, as you learned in Chapter 10, is

where we’ll assume that the equilibrium position is $x_{\text{e}}=0$. The potential energy of a harmonic oscillator is shown in FIGURE 40.20. It is a potential-energy well with curved sides.

FIGURE 40.20 The potential energy of a harmonic oscillator.

A classical particle of mass $m$ oscillates with angular frequency

between the two turning points where the energy line crosses the parabolic potential-energy curve. As you’ve learned, this classical description fails if $m$ represents an atomic particle, such as an electron or an atom. In that case, we need to solve the Schrödinger equation to find the wave functions.

The Schrödinger equation for a quantum harmonic oscillator is

where we used $U\left(x\right)=\frac{1}{2}k{x}^2$. We will assert, without deriving them, that the wave functions of the first three states are

where

The constant $b$ has dimensions of length. We will leave it as a homework problem for you to show that $b$ is the classical turning point of an oscillator in the $n=1$ ground state. The constants $A_1\text{,}{A}_2$, and $A_3$ are normalization constants. For example, $A_1$ can be found by requiring

The completion of this calculation also will be left as a homework problem.

As expected, stationary states of a quantum harmonic oscillator exist only for certain discrete energy levels, the quantum states of the oscillator. The allowed energies are given by the simple equation

where $\omega$ is the classical angular frequency, Equation 40.43, and $n$ is the quantum number.

FIGURE 40.21 shows the first three energy levels and wave functions of a quantum harmonic oscillator. Notice that the energy levels are equally spaced by $\text{Δ}E=\text{ℏ}\omega$. This result differs from the particle in a box, where the energy levels get increasingly farther apart. Also notice that the wave functions, like those of the finite potential well, extend beyond the turning points into the classically forbidden region.

FIGURE 40.21 The first three energy levels and wave functions of a quantum harmonic oscillator.

FIGURE 40.22 shows the probability density $\left|\psi \left(x\right)\right|{}^2$ for the $n=11$ state of a quantum harmonic oscillator. Notice how the node spacing and the amplitude both increase as the particle moves away from the equilibrium position at $x=0$. This is consistent with item 3 of Tactics Box 40.1. The particle slows down as it moves away from the origin, causing its de Broglie wavelength and the probability of finding it to increase.

FIGURE 40.22 The quantum and classical probability densities for the $n=11$ state of a quantum harmonic oscillator.

Section 40.5 introduced the classical probability density $P_{\text{class}}\left(x\right)$ and noted that a classical particle is most likely to be found where it is moving the slowest. Figure 40.22 shows $P_{\text{class}}\left(x\right)$ for a classical particle with the same total energy as the $n=11$ quantum state. You can see that on average the quantum probability density $\left|\psi \left(x\right)\right|{}^2$ mimics the classical probability density. This is just what the correspondence principle leads us to expect.