4.5 Quantum harmonic oscillators

Quantum harmonic oscillator basics

The quantum harmonic oscillator is one of the most important exactly solvable models in physics. It describes any system where a particle experiences a restoring force proportional to its displacement from equilibrium. In statistical mechanics, it's the starting point for modeling everything from vibrating molecules to lattice vibrations in solids.

What makes the quantum version special is that energy comes in discrete chunks, and even the lowest-energy state has nonzero energy. These features have real thermodynamic consequences that show up clearly when you work through the statistical mechanics.

Harmonic potential energy

The potential energy of a harmonic oscillator is parabolic:

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

where $k$ is the spring constant and $x$ is the displacement from equilibrium. The restoring force is proportional to displacement (this is just Hooke's law, $F = -kx$). The symmetry of this parabolic potential is what makes the Schrödinger equation analytically solvable for this system.

Schrödinger equation for the oscillator

The time-independent Schrödinger equation for the quantum harmonic oscillator is:

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$

Solving this equation gives you both the quantized energy levels and the corresponding wavefunctions. The boundary condition is that $\psi(x) \to 0$ as $x \to \pm\infty$, since the particle can't have infinite displacement. This boundary condition is what forces the energy to be quantized: only specific values of $E$ produce physically acceptable solutions.

Zero-point energy

The lowest possible energy of the quantum harmonic oscillator is not zero. It's:

$E_0 = \frac{1}{2}\hbar\omega$

where $\omega = \sqrt{k/m}$ is the angular frequency. This zero-point energy is a direct consequence of the Heisenberg uncertainty principle: you can't simultaneously pin down both the position and momentum of the particle to zero, so the oscillator always retains some minimum energy. A classical oscillator sitting at rest at the bottom of the potential well has zero energy, but quantum mechanics forbids this. Zero-point energy has measurable consequences, including vacuum fluctuations and the Casimir effect.

Energy levels and eigenstates

Unlike a classical oscillator that can vibrate with any amplitude (and therefore any energy), the quantum harmonic oscillator has a discrete set of allowed energies. Each energy level has a corresponding eigenstate, a specific wavefunction that describes the particle's spatial probability distribution.

Quantized energy spectrum

The energy levels are:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega$

where $n = 0, 1, 2, 3, \ldots$ is the quantum number. A striking feature: the levels are equally spaced, with a gap of $\hbar\omega$ between consecutive levels. This uniform spacing is unique to the harmonic potential and is what makes the partition function so tractable. In spectroscopy, this equal spacing explains the characteristic absorption lines of molecular vibrations.

Hermite polynomials

The wavefunctions of the quantum harmonic oscillator involve Hermite polynomials $H_n(\xi)$, which are a family of orthogonal polynomials. The first few are:

• $H_0(\xi) = 1$
• $H_1(\xi) = 2\xi$
• $H_2(\xi) = 4\xi^2 - 2$

Each successive polynomial introduces one more node (zero crossing) in the wavefunction. Orthogonality of these polynomials is what ensures the eigenstates are orthogonal, which matters when you compute transition probabilities and selection rules.

Wavefunction representation

The full normalized wavefunctions are:

$\psi_n(x) = \frac{1}{\sqrt{2^n n!}}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/(2\hbar)}\, H_n\!\left(\sqrt{\frac{m\omega}{\hbar}}\,x\right)$

The structure here is a Gaussian envelope $e^{-m\omega x^2/(2\hbar)}$ multiplied by the Hermite polynomial. The Gaussian ensures the wavefunction decays at large $x$, while the Hermite polynomial creates $n$ nodes. The probability density $|\psi_n(x)|^2$ tells you where the particle is likely to be found. For the ground state ($n=0$), this is a simple Gaussian centered at the origin. For higher $n$, the probability spreads out and develops oscillatory structure.

Ladder operators

Ladder operators are an algebraic approach to the harmonic oscillator that avoids solving the differential equation directly. They're not just a mathematical convenience; the same formalism carries over into quantum field theory, where particles are created and destroyed.

Creation and annihilation operators

The annihilation operator $a$ and creation operator $a^\dagger$ are defined in terms of the position and momentum operators:

$a = \sqrt{\frac{m\omega}{2\hbar}}\left(x + \frac{ip}{m\omega}\right), \qquad a^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\left(x - \frac{ip}{m\omega}\right)$

Their action on energy eigenstates is:

• $a|n\rangle = \sqrt{n}\,|n-1\rangle$ (lowers the state by one quantum)
• $a^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle$ (raises the state by one quantum)

Note that $a|0\rangle = 0$: you can't annihilate below the ground state. This is actually another way to derive the zero-point energy.

Number operator

The number operator is defined as:

$\hat{N} = a^\dagger a$

It satisfies $\hat{N}|n\rangle = n|n\rangle$, so its eigenvalue is just the quantum number $n$. The Hamiltonian can be rewritten compactly as:

$H = \hbar\omega\left(\hat{N} + \frac{1}{2}\right)$

In many-particle contexts, $\hat{N}$ counts the number of quanta (e.g., phonons) in a given mode, which is central to the second quantization formalism.

Commutation relations

The fundamental commutation relation is:

$[a, a^\dagger] = 1$

This single relation, combined with the definition of $\hat{N}$, is enough to derive the entire energy spectrum and the matrix elements of $a$ and $a^\dagger$. It's also directly connected to the position-momentum commutator $[x, p] = i\hbar$. The same algebraic structure generalizes to other quantum systems, including angular momentum and spin.

Quantum vs classical oscillators

The differences between quantum and classical harmonic oscillators aren't just academic. They produce measurable thermodynamic effects, especially at low temperatures.

Uncertainty principle implications

The Heisenberg uncertainty relation constrains the oscillator:

$\Delta x\, \Delta p \geq \frac{\hbar}{2}$

A classical oscillator at rest has $x = 0$ and $p = 0$ simultaneously, giving zero energy. Quantum mechanics forbids this: the particle must have some spread in both position and momentum, which is why the ground state energy is $\frac{1}{2}\hbar\omega$ rather than zero.

Energy level spacing

• Quantum: energy levels are discrete, separated by $\hbar\omega$
• Classical: energy is continuous, any amplitude is allowed

At high quantum numbers (large $n$), the spacing $\hbar\omega$ becomes small relative to the total energy $E_n \approx n\hbar\omega$, so the spectrum looks nearly continuous. This is the correspondence principle: quantum mechanics reproduces classical results in the appropriate limit.

Ground state properties

The ground state ($n = 0$) has several distinctive features:

• Energy $E_0 = \frac{1}{2}\hbar\omega$ (nonzero)
• Gaussian probability distribution for both position and momentum
• Finite spatial width, meaning the particle has a nonzero probability of being found away from equilibrium

This zero-point motion is physically real. In crystals, atoms vibrate even at absolute zero, which affects lattice constants and material properties.

Thermodynamic properties

This is where the quantum harmonic oscillator connects to statistical mechanics. By computing the partition function, you can extract all the standard thermodynamic quantities.

Partition function derivation

Sum over all energy levels:

$Z = \sum_{n=0}^{\infty} e^{-\beta E_n} = \sum_{n=0}^{\infty} e^{-\beta\hbar\omega(n+1/2)}$

Factor out the zero-point energy term and evaluate the geometric series:

$Z = \frac{e^{-\beta\hbar\omega/2}}{1 - e^{-\beta\hbar\omega}} = \frac{1}{2\sinh(\beta\hbar\omega/2)}$

where $\beta = 1/(k_B T)$. The equal spacing of energy levels is what makes this geometric series work out so cleanly.

Average energy calculation

From the partition function:

$\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} = \frac{\hbar\omega}{2} + \frac{\hbar\omega}{e^{\beta\hbar\omega} - 1}$

The first term is the zero-point energy. The second term is the thermal contribution, which has the form of the Planck distribution (or equivalently, $\hbar\omega$ times the Bose-Einstein occupation number for a single mode).

• High temperature ($k_B T \gg \hbar\omega$): $\langle E \rangle \to k_B T$, recovering the classical equipartition result.
• Low temperature ($k_B T \ll \hbar\omega$): $\langle E \rangle \to \frac{1}{2}\hbar\omega$, dominated by zero-point energy with exponentially suppressed thermal excitations.

Heat capacity analysis

$C_V = \frac{\partial \langle E \rangle}{\partial T} = k_B (\beta\hbar\omega)^2 \frac{e^{\beta\hbar\omega}}{(e^{\beta\hbar\omega} - 1)^2}$

This is the Einstein heat capacity function. Its behavior:

• High $T$: $C_V \to k_B$ per oscillator, which is the Dulong-Petit limit.
• Low $T$: $C_V$ vanishes exponentially as $\sim (\beta\hbar\omega)^2 e^{-\beta\hbar\omega}$.

The classical prediction (constant $C_V = k_B$) fails at low temperatures. The quantum result correctly captures the "freezing out" of vibrational modes when $k_B T$ drops below the energy gap $\hbar\omega$. This was one of the early triumphs of quantum theory (Einstein, 1907).

Applications in statistical mechanics

Phonons in solids

Lattice vibrations in a crystal are decomposed into normal modes, each of which is a quantum harmonic oscillator. The quanta of these vibrations are called phonons. Phonons are bosons, so you apply Bose-Einstein statistics to them.

• The Einstein model treats all modes as having the same frequency $\omega$.
• The Debye model improves on this by using a realistic distribution of frequencies up to a cutoff $\omega_D$, giving the correct $T^3$ low-temperature heat capacity.
• Phonons also mediate electron-phonon coupling, which is the mechanism behind conventional (BCS) superconductivity.

Molecular vibrations

Diatomic and polyatomic molecules have vibrational modes that are well-approximated as quantum harmonic oscillators, at least for small displacements. Each vibrational mode contributes to the molecular partition function:

$Z_{\text{vib}} = \prod_i \frac{1}{2\sinh(\beta\hbar\omega_i/2)}$

where the product runs over all normal modes. This directly explains infrared absorption spectra: a photon is absorbed when its energy matches the spacing $\hbar\omega_i$ of a vibrational mode (subject to selection rules). Raman spectroscopy probes the same modes through a different mechanism.

Quantum field theory connections

In quantum field theory, each mode of a free field is mathematically identical to an independent quantum harmonic oscillator. The creation operator $a^\dagger$ creates a particle, and the annihilation operator $a$ destroys one. The vacuum state corresponds to the ground state of all oscillators, and its zero-point energy summed over all modes leads to the (divergent) vacuum energy, a deep issue in theoretical physics.

Coherent states

Coherent states are special quantum states of the harmonic oscillator that behave as classically as quantum mechanics allows. They're essential in quantum optics and provide insight into the quantum-classical boundary.

Definition and properties

A coherent state $|\alpha\rangle$ is an eigenstate of the annihilation operator:

$a|\alpha\rangle = \alpha|\alpha\rangle$

where $\alpha$ is a complex number. Key properties:

• They saturate the uncertainty bound: $\Delta x\, \Delta p = \hbar/2$
• They can be expanded as a superposition of number states: $|\alpha\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle$
• The probability of finding $n$ quanta follows a Poisson distribution with mean $|\alpha|^2$

Relation to classical limit

The expectation values $\langle x(t)\rangle$ and $\langle p(t)\rangle$ for a coherent state follow exactly the classical equations of motion. The wavepacket oscillates back and forth without changing shape (it doesn't spread). For large $|\alpha|$, the quantum fluctuations become negligible relative to the amplitude, and the state looks fully classical. This makes coherent states the natural bridge between quantum and classical oscillatory behavior.

Quantum optics applications

Coherent states describe the quantum state of laser light. A laser produces a coherent state of the electromagnetic field with a well-defined phase and Poissonian photon statistics. Beyond lasers:

• Squeezed states reduce uncertainty in one quadrature below the coherent-state level at the expense of the other, useful in gravitational wave detection (LIGO).
• Coherent states serve as a basis for quantum communication and quantum key distribution protocols.

Numerical methods

When the potential deviates from a perfect parabola (anharmonic terms, coupled oscillators, etc.), exact solutions no longer exist. Numerical methods let you handle these more realistic cases.

Finite difference approximations

1. Discretize the spatial coordinate $x$ onto a grid of $N$ points with spacing $\Delta x$.

2. Approximate the second derivative using finite differences: $\frac{d^2\psi}{dx^2} \approx \frac{\psi_{i+1} - 2\psi_i + \psi_{i-1}}{(\Delta x)^2}$.

3. This converts the Schrödinger equation into an $N \times N$ matrix eigenvalue problem.

4. Diagonalize the resulting matrix to obtain approximate energy levels and wavefunctions.

This approach works for arbitrary potentials, not just harmonic ones. Accuracy improves with finer grid spacing, at the cost of larger matrices.

Matrix diagonalization techniques

An alternative is to work in the basis of known harmonic oscillator eigenstates:

1. Choose a truncated basis $\{|0\rangle, |1\rangle, \ldots, |N_{\max}\rangle\}$.
2. Compute matrix elements $\langle m|H|n\rangle$ of the full Hamiltonian (including anharmonic terms) in this basis.
3. Diagonalize the resulting matrix numerically.

This is efficient when the system is close to harmonic, since a modest $N_{\max}$ captures the low-lying states well. The ladder operator algebra makes computing matrix elements of powers of $x$ and $p$ straightforward.

Perturbation theory approaches

For weakly anharmonic potentials like $V(x) = \frac{1}{2}kx^2 + \lambda x^3 + \mu x^4$, perturbation theory gives analytical corrections:

• First-order energy shift: $E_n^{(1)} = \langle n|V'|n\rangle$ where $V'$ is the anharmonic part.
• Second-order correction: involves sums over intermediate states and captures level shifts due to coupling between states.

Perturbation theory provides physical insight (which terms matter, how levels shift) that purely numerical methods don't. It breaks down when the anharmonic terms become large compared to $\hbar\omega$.