The Harmonic Oscillator

Now consider a particle subject to a restoring force $F = -kx$, as might arise for a mass-spring system obeying Hooke's Law. The potential is then

$$V(x) = - \int_{-\infty}^{\infty} (-kx) dx$$ (118)

$$= V_0 + \frac{1}{2} kx^2$$

If we choose the energy scale such that $V_0 = 0$ then $V(x) = (1/2)kx^2$. This potential is also appropriate for describing the interaction of two masses connected by an ideal spring. In this case, we let $x$ be the distance between the masses, and for the mass $m$ we substitute the reduced mass $\mu$. Thus the harmonic oscillator is the simplest model for the vibrational motion of the atoms in a diatomic molecule, if we consider the two atoms as point masses and the bond between them as a spring. The one-dimensional Schrödinger equation becomes

$$- \frac{\hbar^2}{2 \mu} \frac{d^2\psi}{dx^2} + \frac{1}{2} kx^2 \psi(x) = E \psi(x)$$ (119)

After some effort, the eigenfunctions are

$$\psi_n(x) = N_n H_n(\alpha^{1/2} x) e^{-\alpha x^2 / 2} \hspace{1.0cm} n=0,1,2,\ldots$$ (120)

where $H_n$ is the Hermite polynomial of degree $n$, and $\alpha$ and $N_n$ are defined by

$$\alpha = \sqrt{\frac{k \mu}{\hbar^2}} \hspace{1.5cm} N_n = \frac{1}{\sqrt{2^n n!}} \left( \frac{\alpha}{\pi} \right)^{1/4}$$ (121)

The eigenvalues are

$$E_n = \hbar \omega (n + 1/2)$$ (122)

with $\omega = \sqrt{k/ \mu}$.