Model

The one-dimensional harmonic oscillator is defined by a quadratic potential $V(x)$ that for convenience is chosen to be symmetric to the origin,

$$V(x)=\frac{1}{2}m\omega^2x^2.$$ (225)

Here, $m$ is the mass of the particle and $\omega$ the parameter that determines the shape of the parabola. We wish to determine the behaviour of a particle of mass $m$ in this potential. In classical (Newtonian) physics, all one would have to do would be to solve Newton's equations for a given initial position $x_0$ and a given initial momentum $p_0$ at time $t=0$ to determine $x(t)$ and $p(t)$ at a later time $t>0$. The total energy

$$E= \frac{p^2}{2m}+V(x)= \frac{p_0^2}{2m}+V(x_0)$$ (226)

is constant and determines an ellipse in phase space. The particle starts at the point $(x_0,p_0)$ on this ellipse and then moves on this ellipse. Of course, as a function of time $t$ we can easily solve for $x(t)$ by solving the differential equation (Newton's law) $m\ddot{x}=F(x)=-V'(x)=-m\omega^2x$.

In quantum mechanics, we have the total energy replaces by the Hamilton operator (Hamiltonian)

$$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2x^2,$$ (227)

and we have to solve the time-dependent Schrödinger equation

$$i\hbar \frac{\partial}{\partial t} \Psi(x,t) =\hat{H} \Psi(x,t)$$ (228)

for a given initial wave function $\Psi(x,0)$. We have learned that this can be achieved by first solving the stationary Schrödinger equation

$$\hat{H}\psi = E \psi \leadsto \left( -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{1}{2}m\omega^2x^2 \right) \psi(x) = E\psi(x),$$ (229)

which is an equation for the possible energy eigenvalues $E$ and eigenfunctions $\psi(x)$. The eigenfunctions are useful themselves as they provide inside into the possible states the particle can be in. We have also learned that the eigenfunctions provide a basis into which the initial wave function $\Psi(x)$ can be expanded and thus the time-evolution of an arbitrary initial wave function can be obtained.