The quantum harmonic oscillator is a system in quantum mechanics whose restoring force is proportional to displacement from equilibrium. Its potential is quadratic, making it the quantum counterpart of the harmonic oscillator in classical mechanics. It is both an exactly solvable model and a useful approximation for small vibrations near stable equilibria. Its characteristic features include equally spaced energy levels, a nonzero minimum energy, and spatially extended states rather than definite particle trajectories. (ocw.mit.edu)
Hamiltonian and harmonic approximation
For a one-dimensional particle of mass (m), with equilibrium at (x=0) and angular frequency (\omega>0), the Hamiltonian operator is
[ \hat H=\frac{\hat p^{,2}}{2m}+\frac12m\omega^2\hat x^{,2}. ]
The first term represents kinetic energy and the second potential energy. Position and momentum satisfy ([\hat x,\hat p]=i\hbar), where (\hbar=h/(2\pi)) is the reduced Planck constant. In the position representation, (\hat p=-i\hbar,d/dx), and the stationary Schrödinger equation becomes
[ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} +\frac12m\omega^2x^2\psi=E\psi. ]
Physical bound-state solutions must be normalizable over the entire real line. (ocw.mit.edu)
The model’s broad applicability follows from the Taylor expansion of a smooth potential about a minimum (x_0):
[ V(x)=V(x_0)+\frac12V''(x_0)(x-x_0)^2+\cdots. ]
The linear term vanishes at equilibrium. If (V''(x_0)>0) and higher-order terms are sufficiently small, the motion is approximately harmonic, with (\omega=\sqrt{V''(x_0)/m}). The constant (V(x_0)) shifts all energies equally without changing the states. A flat minimum with vanishing quadratic curvature requires a different approximation. (ocw.mit.edu)
Energy spectrum and wave functions
The allowed energies are
[ E_n=\hbar\omega\left(n+\frac12\right), \qquad n=0,1,2,\ldots. ]
Each level is nondegenerate in one dimension, and adjacent levels differ by (\hbar\omega). This energy quantization contrasts with the continuous energies permitted classically. The ground-state energy (E_0=\hbar\omega/2), measured relative to the potential minimum, is the oscillator’s zero-point energy. (ocw.mit.edu)
Defining the characteristic length (\ell=\sqrt{\hbar/(m\omega)}), the normalized wave functions are
[ \psi_n(x)= \frac{H_n(x/\ell)e^{-x^2/(2\ell^2)}} {\pi^{1/4}\sqrt{2^n n!,\ell}}, ]
where (H_n) denotes a physicists’ Hermite polynomial. The state (\psi_n) has (n) nodes and parity ((-1)^n): even-numbered states are symmetric, and odd-numbered states antisymmetric. These functions form an orthonormal basis for the oscillator’s Hilbert space, allowing other normalizable states to be expanded in them. (ocw.mit.edu)
By the Born rule, (|\psi_n(x)|^2) is the position probability density. It extends beyond the classical turning points, where (V(x)=E_n), into regions inaccessible to a classical particle of that energy. Nevertheless, an energy eigenstate has a time-independent probability density: its time dependence is only the phase factor (e^{-iE_nt/\hbar}). (ocw.mit.edu)
Ladder operators
An algebraic solution uses creation and annihilation operators:
[ \hat a=\sqrt{\frac{m\omega}{2\hbar}}\hat x +\frac{i\hat p}{\sqrt{2m\hbar\omega}}, \qquad \hat a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\hat x -\frac{i\hat p}{\sqrt{2m\hbar\omega}}. ]
They satisfy ([\hat a,\hat a^\dagger]=1), and
[ \hat H=\hbar\omega\left(\hat a^\dagger\hat a+\frac12\right). ]
The number operator (\hat N=\hat a^\dagger\hat a) has eigenvalue (n) on the state (|n\rangle). The ladder actions are
[ \hat a|n\rangle=\sqrt n,|n-1\rangle,\qquad \hat a^\dagger|n\rangle=\sqrt{n+1},|n+1\rangle. ]
The condition (\hat a|0\rangle=0) identifies the ground state. Repeated raising generates every excited state and derives the spectrum without directly solving the differential equation. (ocw.mit.edu)
Uncertainty and dynamics
The ground state has a Gaussian position density. Its position and momentum uncertainties obey
[ (\Delta x)^2=\frac{\hbar}{2m\omega}, \qquad (\Delta p)^2=\frac{m\hbar\omega}{2}, \qquad \Delta x,\Delta p=\frac{\hbar}{2}. ]
It therefore saturates the uncertainty principle. A state with exactly zero displacement and momentum is impossible; the finite spreads explain why the minimum energy is not zero. In every energy eigenstate, the mean kinetic and potential energies are equal. (damtp.cam.ac.uk)
A general superposition evolves as
[ |\psi(t)\rangle=\sum_n c_n e^{-iE_nt/\hbar}|n\rangle. ]
Unlike a single energy eigenstate, such a state can have time-dependent position probabilities. For the isolated oscillator, the position and momentum expectation values follow the classical oscillator equations, although these averages do not define a definite microscopic trajectory. (damtp.cam.ac.uk)
Thermal behavior and applications
In statistical mechanics, an oscillator at temperature (T) has the partition function
[ Z=\frac{e^{-\beta\hbar\omega/2}} {1-e^{-\beta\hbar\omega}}, \qquad \beta=\frac{1}{k_BT}, ]
where (k_B) is the Boltzmann constant. Its mean energy is
[ \langle E\rangle=\frac{\hbar\omega}{2} +\frac{\hbar\omega}{e^{\beta\hbar\omega}-1}. ]
At high temperatures this approaches (k_BT), consistent with classical equipartition. At low temperatures, thermal excitation is suppressed and the energy approaches the zero-point value. (damtp.cam.ac.uk)
Molecular vibrations can be approximated by harmonic modes, providing a basis for vibrational spectroscopy. Crystal vibrations are likewise decomposed into modes whose excitation quanta are phonons. Quantized electromagnetic modes have oscillator structure, with occupation numbers counting photons; oscillator methods consequently enter quantum field theory. (damtp.cam.ac.uk)
Real systems exhibit anharmonicity when higher-order potential terms become important. Their levels need not remain equally spaced, and modes may interact. The harmonic model is most accurate where the quadratic approximation holds; it cannot by itself describe molecular dissociation or strong departures from equilibrium. (ocw.mit.edu)