Quantum harmonic oscillator

The quantum harmonic oscillator is a key idea in the study of quantum-mechanical systems. It is like the classical harmonic oscillator, which describes things like a swinging pendulum or a vibrating spring, but it works at the tiny scale of atoms and particles.

Some trajectories of a harmonic oscillator according to Newton's laws of classical mechanics (A–B), and according to the Schrödinger equation of quantum mechanics (C–H). In A–B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C–H, some solutions to the Schrödinger equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. C, D, E, F, but not G, H, are energy eigenstates. H is a coherent state—a quantum state that approximates the classical trajectory.

Because many smooth forces around us can be thought of as a harmonic potential near a balanced point, or equilibrium point, this model is very useful in quantum mechanics. It helps scientists understand how tiny particles behave in many different situations.

One special thing about the quantum harmonic oscillator is that scientists can find exact answers using math, called an analytical solution. This makes it one of the few quantum systems where everything can be figured out precisely, which is very helpful for learning and research.

One-dimensional harmonic oscillator

Wavefunction representations for the first eight bound eigenstates, n = 0 to 7. The horizontal axis shows the position x.

The quantum harmonic oscillator is an important model in quantum mechanics. It is the quantum version of the classical harmonic oscillator, which describes things like springs or pendulums. This model is useful because many smooth potentials can be approximated as harmonic near stable points.

The quantum harmonic oscillator has exact solutions, making it a good example for understanding quantum systems. Its energy levels are quantized, meaning they can only take certain values. These levels are evenly spaced, and the lowest energy state, called the ground state, has a small, non-zero energy known as zero-point energy. This is different from classical mechanics, where the lowest energy could be zero.

N-dimensional isotropic harmonic oscillator

The one-dimensional harmonic oscillator can be expanded to N dimensions, where N can be 1, 2, 3, and so on. In one dimension, the position of a particle is shown by a single coordinate, called x. In N dimensions, this becomes N position coordinates, which we name x₁, ..., x_N. Each position coordinate has a matching momentum, named p₁, ..., p_N.

Schrödinger 3D spherical harmonic orbital solutions in 2D density plots; the Mathematica source code that used for generating the plots is at the top

The energy of this system depends on both the position and momentum of the particle. For a specific set of numbers { n } which are { n₁, n₂, … , n_N }, the energy levels can be found using methods similar to the one-dimensional case.

The energy levels of the system are given by E = ℏω[ (n₁ + ⋯ + n_N) + N/2 ]. Here, n_i = 0, 1, 2, … represents the energy level in each dimension i.

Unlike the one-dimensional case, where each energy level has only one state, in N-dimensions (except for the lowest energy state), energy levels can have several states with the same energy. This is called degeneracy. For example, in three dimensions, the number of states with the same energy can be calculated using a simple formula.

Applications

Harmonic oscillators lattice: phonons

Molecular vibrations

Main article: Molecular vibration

The way two atoms vibrate around each other can be described using the harmonic oscillator model. This helps us understand how molecules behave.
This same idea is used to study how atoms move and interact in solids.
Other systems, like a charged particle in a magnetic field, can also be modeled similarly.
The harmonic oscillator gives a good first approximation for how atoms in molecules move, especially for simpler cases.

Another illustration of the time propagation of the common wave function for three different atoms emphasizes the effect of the angular momentum on the distribution behavior

Hooke's law

The Hooke's atom is a simple way to model an atom using the harmonic oscillator idea.
Hooke's law describes how a spring behaves — the force pulling it back is proportional to how far it is stretched.
This helps us understand how things like springs and elastic materials work.
In such systems, energy constantly shifts between moving and stored forms, but the total energy stays the same.

The inverted harmonic oscillator

The inverted harmonic oscillator has been studied by various researchers.

The Dirac oscillator

The harmonic oscillator has also been explored in connection to the Dirac equation by Lorella M. Jones.