Quantum harmonic oscillator

Because an arbitrary potential can 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.

Hamiltonian and energy eigenstates

In the one-dimensional harmonic oscillator problem, a particle of mass m is subject to a potential V(x) given by

V(x) = \frac{1}{2} m \omega^2 x^2 \, ,

where ω is the angular frequency of the oscillator. In classical mechanics, \displaystyle m \omega^2 = k is called the spring stiffness coefficient, force constant or spring constant, and \displaystyle \omega the angular frequency.

The Hamiltonian of the particle is:

\hat H = \frac{{\hat p}^2}{2m} + \frac{1}{2} m \omega^2 {\hat x}^2 \, ,

where \hat x = x is the position operator, and \hat p is the momentum operator, given by

\hat p = - i \hbar {\partial \over \partial x} \, .

The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents the potential energy in which it resides. In order to find the energy levels and the corresponding energy eigenstates, we must solve the time-independent Schrödinger equation,

 \hat H \left| \psi \right\rangle = E \left| \psi \right\rangle \, .

We can solve the differential equation in the coordinate basis, using a spectral method. It turns out that there is a family of solutions. In the position basis they are

  \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ - \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots.

The functions Hn are the physicists’ Hermite polynomials:

H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right)

The corresponding energy levels are

 E_n = \hbar \omega \left(n + {1\over 2}\right).



Modern Quantum Mechanics (2nd Edition)

Quantum Mechanics Non-Relativistic Theory, Third Edition: Volume 3

Introduction to Quantum Mechanics (2nd Edition)

Introductory Quantum Mechanics (4th Edition)

Perspectives of Modern Physics, Sec 8-7


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