where $H_n(x)$ are the Hermite polynomials. The wave function crosses zero $n$ times and decays due to the exponential factor with a characteristic length $\xi = \sqrt{\frac{\hbar}{m\omega}}$. There is a peak in amplitude of the wave function near the classical turning point, $x_n = \sqrt{2E_n/k}$. The wave function can be written in terms of $\xi$,

Note that in this form it is clear that $\psi_n(x)$ has the units of m^{-1/2}. The code below will calculate the the value of the wave function $\psi_n(x)$.

The harmonic oscillator wave functions are plotted and tabulated below. $n=$ , $m=$ kg, $k=$ N/m, . The red dots are the classical turning points. The parameters for some diatomic molecules can be loaded with these buttons.

$\psi_n$ [m^{-1/2}]

$\psi^2_n$ [m^{-1}]

$\omega=$ rad/s
$\hbar\omega=$ eV
$E_n=$ eV
$\xi=$ Å
$x_n=$ Å

The energy of a harmonic oscillator is a sum of the kinetic energy and the potential energy,

$$E= \frac{mv^2}{2}+\frac{kx^2}{2}.$$

At the turning points where the particle changes direction, the kinetic energy is zero and the classical turning points for this energy are $x=\sqrt{2E/k}$. For large quantum numbers $n$, the probability of finding a particle at some location $|\psi|^2$ approximates the distribution you would expect for a classical particle which would also be more likely to be found near the classical turning points. This is an example of the correspondence principle.