Classically forbidden behavior of the quantum harmonic oscillator for large quantum numbers
James J. Diamond · American Journal of Physics · 1992
The probability that the quantum harmonic oscillator in quantum level n will penetrate into its classically forbidden region, has been calculated, via an algorithm based on a quadrature formula for the probability density, for values of the quantum number n ranging up to 250. These results have been analyzed numerically. The conjecture is made that the probability P(n) of the position of the oscillator in quantum state n being measured in the classically forbidden region follows the approximate formula P(n)≊A(n+1/2)−1/3+O(n+1/2)−1 where the constant A has the value A=0.1339706...; this asymptotic formula has a relative error of less than 2% for quantum numbers 250≳n≳7. The consequences of this formula in regard to the rate of approach to the classical limit are discussed.