- #1
chafelix
- 26
- 0
I know how to do SHO propagator by computing the action. I was only trying to do it
via the eigenfunction expansion
K(x’,x;t)=sum_ i phi_i(x’) phi_i(x) exp(-iε_it/hbar )=(m omega/pi*hbar)
sum_i=-^infty h_i(y’) h_i(y) exp[-(y**2+y’**2)/2] [s(t)/2]**i
with s(t)=exp(-iomega t)
This looks close, but not quite there:
I can get the 1/i! from the Hermite polynomials h_i, and I can use the generating function, but that only applies to a single h, not a product. Am I missing something along this way? I also tried substituting the expression involving (y-d/dy)**i for the h_i, but cannot get it to work
via the eigenfunction expansion
K(x’,x;t)=sum_ i phi_i(x’) phi_i(x) exp(-iε_it/hbar )=(m omega/pi*hbar)
sum_i=-^infty h_i(y’) h_i(y) exp[-(y**2+y’**2)/2] [s(t)/2]**i
with s(t)=exp(-iomega t)
This looks close, but not quite there:
I can get the 1/i! from the Hermite polynomials h_i, and I can use the generating function, but that only applies to a single h, not a product. Am I missing something along this way? I also tried substituting the expression involving (y-d/dy)**i for the h_i, but cannot get it to work