Computing normalized oscillator states for very large N (Matlab)

[srihari83]

Hi everyone, I have a rather fundamental question about building oscillator wavefunctions numerically. I'm using Matlab. Since it's 1/√(2ⁿn!∏)*exp(-x²/2)*H_n(x), the normalization term tends to zero rapidly. So for very large N (N>=152 in Matlab) it is zero to machine precision! Though asymptotic expansions for H_n(x) exist in literature (Abromowitz&Stegun, Polyanin&Manzhirov etc), they never say whether these Hermite polynomials are unit normalized for large N. They don't seem to be, i.e these expressions are just H_n(x). Numerically is not unlikely to be able to unit normalize unless one takes a extremely large & dense grid. But it is ok for my calculations if these functions have a ||ψ_N||² <1, only if they are exactly zero, they drive certain matrices to singularity. so how do people calculate these polynomials without numbers getting exactly zero?? Any help/advice is greatly appreciated!

 

[azntoon]

The most common approach for calculating oscillator wavefunctions numerically is to use the recursion relations. These recurrence relations can be used to iteratively calculate the Hermite polynomials for large values of N. Once the Hermite polynomial is calculated, the wavefunction can then be obtained by multiplying the Hermite polynomial with the appropriate normalization constant. This approach will also ensure that the wavefunction is always properly normalized.