The Simple Harmonic Oscillator

The one-dimensional simple harmonic oscillator is described by the differential equation

\begin{equation}
m\frac{d^2x}{dt^2}+kx=0
\end{equation}

Here \(m\) is the mass of the particle and \(k\) is the constant characterizing the restoring force.

Solution of the simple harmonic oscillator

The solution satisfying the initial condition \(x(0)=A\), where \(A\) is the amplitude of the oscillations, is given by

\begin{equation}
x(t)=A\cos(\omega_0t)\hspace{0.5cm};\hspace{0.5cm} \omega_0=\left(k/m\right)^{1/2}
\end{equation}

The oscillator is called harmonic because it is characterized by a unique frequency \(\omega_0\) and the superior harmonics \(2\omega_0, 3\omega_0,\dots\) do not appear in the solution.

Total energy of the simple harmonic oscillator

Multiplying Eq.(1) by \(dx/dt\) and taking into account the relations

\begin{align*}
\frac{1}{2}\frac{d}{dt}\left(\frac{dx}{dt}\right)^2&=\frac{dx}{dt}\frac{d^2x}{dt^2}\\
\frac{1}{2}\frac{d}{dt}x^2&=x\frac{dx}{dt}\\
\end{align*}

Eq.(1) can be rewritten in the form

\begin{equation}
\frac{d}{dt}\left\{m\frac{1}{2}\left(\frac{dx}{dt}\right)^2+m\frac{1}{2}\omega_0^2x^2\right\}=0.
\end{equation}

The first integral can be easily calculated:

\begin{equation}
m\frac{1}{2}\left(\frac{dx}{dt}\right)^2+m\frac{1}{2}\omega_0^2x^2=E.
\end{equation}

In the latter equation the constant of integration \(E\) is the total energy of the oscillator. The total energy \(E=K+V\) is the sum of the kinetic energy \(K=(m/2)(dx/dt)^2\) and the potential energy \(V=(m\omega_0^2/2)x^2=(k/2)x^2\).

Substituting in Eq.(4) the solution given by Eq.(2) one finds that the total energy \(E\) is characterized by the square of the amplitude \(A\) and does not depend on the frequency of the oscillations:

\begin{equation}
E=\frac{1}{2}kA^2.
\end{equation}

Comparison with the quantum harmonic oscillator

The amplitude \(A\), and consequently the energy \(E\) can take any value. This is in contrast with the quantum harmonic oscillator in which the energy can take only discrete values, doesn’t depend on the amplitude, and is proportional to the frequency of the oscillations (see QHO).

Substituting Eq.(2) in the expression for the momentum \(p=mdx/dt\) one obtains

\begin{equation}
p(t)=-m\omega_0A\sin(\omega_0t).
\end{equation}

It is clear from the equations (2) and (6) that the position and the momentum of the classical simple harmonic oscillator can take well defined values simultaneously at each instant \(t\). This is also in contrast with the quantum harmonic oscillator in which these couple of dynamical variables satisfy the uncertainty relation at each instant \(t\) (see QHO).

Hamiltonian of the simple harmonic oscillator

The Hamiltonian for the harmonic oscillator can be written in the form

\begin{equation}
H=\frac{p^2}{2m}+\frac{1}{2}m\omega_0^2x^2
\end{equation}