Quantum harmonic oscillator explained

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.^[1]

One-dimensional harmonic oscillator

Hamiltonian and energy eigenstates

The Hamiltonian of the particle is:$$\backslash hat\; H\; =\; \backslash frac\; +\; \backslash frac\; k\; ^2\; =\; \backslash frac\; +\; \backslash frac\; m\; \backslash omega^2\; ^2\; \backslash ,,$$where is the particle's mass, is the force constant, $\backslash omega\; =\; \backslash sqrt$ is the angular frequency of the oscillator,

\hat{x} is the position operator (given by in the coordinate basis), and \hat{p} is the momentum operator (given by \hat p = - i \hbar \partial / \partial x in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.

The time-independent Schrödinger equation is,$$\backslash hat\; H\; \backslash left|\; \backslash psi\; \backslash right\backslash rangle\; =\; E\; \backslash left|\; \backslash psi\; \backslash right\backslash rangle\; ~,$$where

E denotes a real number (which needs to be determined) that will specify a time-independent energy level, or eigenvalue, and the solution | \psi \rangle denotes that level's energy eigenstate.\langle x | \psi \rangle = \psi (x) , using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions,^[2] $$\backslash psi\_n(x)\; =\; \backslash frac\; \backslash left(\backslash frac\backslash right)^\; e^\; H\_n\backslash left(\backslash sqrt\; x\; \backslash right),\; \backslash qquad\; n\; =\; 0,1,2,\backslash ldots.$$

The functions H_n are the physicists' Hermite polynomials,$$H\_n(z)=(-1)^n~\; e^\backslash frac\backslash left(e^\backslash right).$$

The corresponding energy levels are$$E\_n\; =\; \backslash hbar\; \backslash omega\backslash bigl(n\; +\; \backslash tfrac\backslash bigr).$$The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be $\backslash langle\; \backslash hat\; \backslash rangle\; =\; 0$ and $\backslash langle\; \backslash hat\; \backslash rangle\; =\; 0$ owing to the symmetry of the problem, whereas:

\langle \hat{x} ^{2} \rangle = ( 2 n +1 ) (\hbar)/(2 m\omega) = \sigma ^{2} _{x} \langle \hat{p} ^{2} \rangle = ( 2 n +1 ) (m\hbar\omega)/(2) = \sigma ^{2} _{p}

The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of $\backslash sigma\_x\; \backslash sigma\_p\; =\; \backslash frac$ which is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction.

This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the state, called the ground state) is not equal to the minimum of the potential well, but above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle.

The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.

Ladder operator method

Notes and References

1. Web site: Rashid . Muneer A. . Munir Ahmad Rashid . Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian . . . 2006 . . 19 October 2010 . 3 March 2016 . https://web.archive.org/web/20160303233341/http://www.ncp.edu.pk/docs/12th_rgdocs/Munir-Rasheed.pdf . dead .
2. Book: Gbur, Gregory J. . Greg Gbur . Mathematical Methods for Optical Physics and Engineering . Cambridge University Press . 2011 . 978-0-521-51610-5 . 631–633.