- #1

- 1,481

- 4

For the sake of convenience I am copying here the text that follows the animation:

I have made this animation in order to present my little puzzle with the quantum harmonic oscillator. Think about a classical oscillator, a swing, a weight on a spring, a pendulum in a clock ... It has a certain characteristic spring constant, and a mass. For simplicity let us choose units so that these constants are numerically equal to 1. Let us also assume that the time scale is choosen so that the period is equal 2 pi. Given energy E, the classical oscillator will vibrate with an amplitude A. Its deviation x(t) from the equilibrium position x=0 is given by the formula

x(t) = A cos t

The relation between energy E and amplitude A is simple: square of A is 2E. For a classical oscillator the energy E can be any positive number.

For a quantum oscillator, assuming units in which the Planck constant hbar = 1, the energy can not be arbitrary, it is quantized: 2E(n) = 2n+1. To each energy level there corresponds a "quantum eigenstate", the "wave function". Using Mathematica notation it is given by the formula

psi[n_,x_] = 1/Sqrt[Sqrt[Pi] 2^n n!] Exp[-x^2/2] HermiteH[n, x]

This wave function (notice that it is real-valued) is normalized so that its square gives the probability density of finding the oscillating point (with energy E(n)) at the point x. Classically we will never find x beyond the turning ponts xmin = -A and xmax = A. But for our quantum oscillator there is always a nonzero probability of finding the point in a classically forbidden region. We say that that there is a non-zero tunneling probability.

Now the puzzle:

Here is a relevant sentence from one of the popular textbooks:

"The probability of being found in classically forbidden regions decreases quickly with increasing n, and vanishes entirely as n approaches inﬁnity, as we would expect from the correspondence principle."

"Quanta, Matter and Change. A molecular approach to physical chemistry", P.W. Atkins, J. de

Paula and R.S. Friedman, Oxford University Press, Oxford/New York, 2009, p. 66

So I used Wolfram's Mathematica to calculate these tunnelling probabilities for n = 0,...,512. The calculation have been done symbolically in order to avoid as much as possible numerical errors.

The animation shows the sequence of plots of probability densities, the classical limits, and also, in a sub-window, the tunelling probability for each n. There is also an U shaped curve representing the classical probability density of finding the swing at a given position given only its energy, but not knowing its phase. In the animation each graph is scaled so that the classical turning points are always at x=-1 and x=1. The vertical axis is also scaled so that the total probability (the area under the probability densities) is always 1.

It can be seen that indeed, the tunneling probability, at first, decreases rather quicly, but then its decrease rate is essentially slowing down with the increase of the quantum number n. This is puzzling. It is a surprise. From my correspondence with one of the authors of the quoted monograph I know that it is also a surprise for him. For instance, I could not find any estimate of what n is needed to have the tunneling probability smaller than, say, 1/1000000? Is it known?