KAHIBARO
Table of Contents
Why this oscillator matters in quantum mechanics
The quantum harmonic oscillator is one of the most important models in physics. It describes a particle moving in a restoring force that is proportional to displacement, like a mass on a spring near equilibrium. In classical mechanics, such motion is simple harmonic motion. In quantum mechanics, the same system behaves very differently in detail, but still keeps a very elegant mathematical structure.
This model is important for two reasons. First, many real systems behave approximately like harmonic oscillators when displaced only a little from equilibrium. Second, the quantum harmonic oscillator can be solved exactly, so it becomes a key example for learning how quantum systems work.
The potential energy
The oscillator is defined by the potential energy
$$
V(x) = \frac{1}{2} m \omega^2 x^2
$$
where $m$ is the mass of the particle and $\omega$ is the angular frequency.
This potential is shaped like a parabola. The minimum is at $x = 0$, which is the equilibrium position. As the particle moves farther from the center, the potential energy increases quadratically.
The Schrödinger equation for the oscillator
To find the allowed quantum states, we use the time independent Schrödinger equation:
$$
-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^2 \psi = E\psi
$$
Here, $\psi(x)$ is the wave function and $E$ is the energy of the state.
Unlike classical mechanics, the particle cannot have just any energy. Solving this equation gives a discrete set of allowed energies.
For the quantum harmonic oscillator, the allowed energies are
$$
E_n = \left(n + \frac{1}{2}\right)\hbar\omega, \qquad n=0,1,2,3,\dots
$$
The energies are quantized and equally spaced.
Quantized energy levels
The energy levels form a ladder. Each higher level is separated from the next by the same amount:
$$
\Delta E = \hbar\omega
$$
This equal spacing is a special and very useful feature of the harmonic oscillator.
A striking result is that the lowest possible energy is not zero. For the ground state, with $n=0$,
$$
E_0 = \frac{1}{2}\hbar\omega
$$
This is called the zero point energy.
The oscillator can never have both exactly zero position and exactly zero momentum. Because of this, its ground state energy is
$$
E_0 = \frac{1}{2}\hbar\omega
$$
not zero.
The wave functions
Each allowed energy level has a corresponding wave function $\psi_n(x)$. These wave functions are localized around the equilibrium point, but spread out over a range of positions.
The ground state wave function has the form
$$
\psi_0(x) = A e^{-m\omega x^2/(2\hbar)}
$$
where $A$ is a normalization constant. This is a Gaussian curve centered at $x=0$.
Higher states have the general form
$$
\psi_n(x) = N_n H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)e^{-m\omega x^2/(2\hbar)}
$$
where $H_n$ are Hermite polynomials and $N_n$ is a normalization constant.
For beginners, the important idea is that all oscillator wave functions are made from a polynomial times a Gaussian.
Nodes and shape of the states
As the quantum number $n$ increases, the wave function becomes more spread out and develops more nodes. A node is a point where the wave function is zero.
The state with quantum number $n$ has exactly $n$ nodes.
| Quantum number $n$ | Energy $E_n$ | Number of nodes |
|---|---|---|
| 0 | $\frac{1}{2}\hbar\omega$ | 0 |
| 1 | $\frac{3}{2}\hbar\omega$ | 1 |
| 2 | $\frac{5}{2}\hbar\omega$ | 2 |
| 3 | $\frac{7}{2}\hbar\omega$ | 3 |
Probability density
The physical meaning of the wave function comes from the probability density
$$
|\psi_n(x)|^2
$$
This tells us where the particle is likely to be found if a position measurement is made.
For the ground state, the probability density is greatest at the center. For excited states, the density has multiple peaks and zeros at the nodes.
In classical motion, a particle spends more time near the turning points because it moves more slowly there. In quantum mechanics, the probability pattern depends on the state, but for large $n$ it begins to resemble the classical picture more closely.
Ground state uncertainty
The ground state is the minimum energy state, but it is not a state of rest in the classical sense. The particle still has unavoidable quantum uncertainty.
For the ground state, the position and momentum uncertainties satisfy the minimum uncertainty relation:
$$
\Delta x \, \Delta p = \frac{\hbar}{2}
$$
This means the ground state is a minimum uncertainty state.
The ground state of the harmonic oscillator is special because it satisfies
$$
\Delta x \, \Delta p = \frac{\hbar}{2}
$$
This is the smallest uncertainty product allowed by quantum mechanics.
Operator method and ladder operators
A very elegant way to solve the oscillator uses operators called ladder operators, also called creation and annihilation operators. They are defined by
$$
a = \sqrt{\frac{m\omega}{2\hbar}} \left(x + \frac{i}{m\omega}p\right),
\qquad
a^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \left(x - \frac{i}{m\omega}p\right)
$$
where $p$ is the momentum operator.
These operators lower or raise the energy level:
$$
a \psi_n \propto \psi_{n-1}, \qquad a^\dagger \psi_n \propto \psi_{n+1}
$$
So, if we know one state, we can generate the others step by step.
The Hamiltonian can be written as
$$
\hat H = \hbar\omega\left(a^\dagger a + \frac{1}{2}\right)
$$
This form makes the energy spectrum almost immediate.
The ladder operator form of the Hamiltonian is
$$
\hat H = \hbar\omega\left(a^\dagger a + \frac{1}{2}\right)
$$
This directly leads to the quantized energies
$$
E_n = \left(n + \frac{1}{2}\right)\hbar\omega
$$
Classical and quantum comparison
The quantum harmonic oscillator is related to the classical oscillator, but the two pictures are not identical.
| Feature | Classical oscillator | Quantum oscillator |
|---|---|---|
| Allowed energies | Any positive value | Discrete values |
| Lowest energy | $0$ | $\frac12 \hbar\omega$ |
| Position | Exact at each moment | Described by probability |
| Motion | Definite trajectory | No definite classical trajectory in a stationary state |
In the limit of large quantum number, quantum results begin to approach classical expectations. This is an example of the correspondence principle.
Physical importance
The quantum harmonic oscillator appears in many areas of physics. Vibrations of atoms in molecules, vibrations of atoms in solids, and quantized modes of fields can all be modeled as harmonic oscillators, at least approximately.
Near a stable equilibrium point, many potentials can be approximated by a quadratic form. That is why the harmonic oscillator appears so often.
If a general potential $V(x)$ has a minimum at $x=x_0$, then near that point it can often be approximated as
$$
V(x) \approx V(x_0) + \frac{1}{2}k(x-x_0)^2
$$
which has the same mathematical form as a harmonic oscillator potential.
Key ideas to remember
The quantum harmonic oscillator is a system with potential
$$
V(x)=\frac12 m\omega^2 x^2
$$
Its energies are discrete and equally spaced:
$$
E_n=\left(n+\frac12\right)\hbar\omega
$$
Its ground state has nonzero energy, called zero point energy. The wave functions are Gaussian based functions with nodes that increase with $n$. The ladder operator method provides a powerful and elegant way to understand the whole system. Because many physical systems behave approximately like oscillators near equilibrium, this model is central to quantum mechanics.