The classical harmonic oscillator obeys an arcsine law in that the distribution of positions of the particle over a single time cycle is proportional to $\frac{1}{\sqrt{A^2-x^2}}$, $A$ being the amplitude.
There is an illustration which seems to be fairly common (I'm looking at figure 2.7b in Griffiths's book on QM) in which a high-$n$ energy eigenstate of the quantum harmonic oscillator is superimposed with the aforementioned distribution. The graphs of the two functions appear to be similar.
Is there a proof that they do coincide in some sense in some limit?
Best Answer
I am not sure about the $ \frac{1}{\sqrt{A^2-x^2}} $ part in you approximation. In the asymptotic limit $n \rightarrow \infty$,the Hermite polynomials behave as follows:
The cosine part relates to the oscillations present in wavefunction which are visible even in fig 2.7b in Griffiths.The $(1-\frac{x^2}{2n})^{\frac{-1}{4}}$ part is the classical behaviour and in this case the graphs seems to match.
References:
Hermite Polynomials on Wikipedia
See the asympotic behaviour part for the above expression.