Harmonic Oscillator

The references that inspired this chapter are all mentioned in the References section.

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The harmonic oscillator is classically defined as follows [B1]:

A harmonic oscillator is a system consisting of a particle of mass \(m\) elastically bound to a center \(x_0\), with a restoring force \(F = - K (x - x_0)\) proportional to the distance from the center. The coefficient \(K\) is the spring constant of the oscillator, and the potential energy reads \(V(x) = V_0 + K (x - x_0)^2 / 2\).

The total energy (kinetic + potential) of the classical particle is then

\[E = \half m \dot{x}^2 + \half m \omega^2 x^2\]

According to the correspondence principle the quantum formulation of the harmonic oscillator has the form:

\[\hat{H} = \frac{ \hatsubsup{p}{x}{2} }{2m} + \half m \omega^2 \hatsup{x}{2}\]

Analytic solution

The problem is to solve the eigenvalue equation

\[\left( - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \half m \omega^2 x^2 \right) \psi(x) = E \, \psi(x)\]

By introducing dimensionless quantities, an analytic solution involving Hermite functions can be derived, with the quantized energies

\[E_n = \left( n + \half \right) \hbar \omega\]

Algebraic solution

An alternative solution due to Dirac can be obtained by introducing the observables

\[\hat{X} = \hat{x} \sqrt{ \frac{m \omega}{\hbar} } , \quad \hat{P} = \frac{\hat{p}}{ \sqrt{ m \hbar \omega } }\]

and further the annihilation, creation operators

\[\begin{split}\hat{a} = \frac{1}{\sqrt{2}} \left( \hat{X} + i \hat{P} \right), \quad \hatsup{a}{\dagger} = \frac{1}{\sqrt{2}} \left( \hat{X} - i \hat{P} \right) \\\end{split}\]

and the number operator

\[\hat{N} = \hatsup{a}{\dagger} \hat{a}\]

By simple considerations about these operators and their commutation relations, it can be shown that

The eigenvalues of \(\hat{N}\) are the nonnegative integers only.

Recursive relations about the creation and annihilation operators such as

\[\hat{a} \ket{n} = \sqrt{n} \ket{n - 1}, \quad \hatsup{a}{\dagger} \ket{n} = \sqrt{n + 1} \ket{n + 1}\]

allow to reconstruct the eigenfunctions of the Hamiltonian. These operators transform a state of energy \((n + 1/2) \hbar \omega\) into a state \((n + 1/2 \mp 1) \hbar \omega\), hence their names.


  • “The One Dimensional Harmonic Oscillator”, section 4.2, and
    “Algebraic Solution”, section 7.5 [B1]