Compute SHO Propagator: Eigenfunction Expansion

[chafelix]

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

 

[chafelix]

ok, need the Mehler formula

 

[Entropia]

.

It seems like you have a good understanding of the SHO propagator and the eigenfunction expansion method. However, your expression for the propagator is not quite correct. The correct expression is:

K(x',x;t) = (mω/πℏ)exp[-(mω/2ℏ)(x^2+x'^2)cos(ωt) + (mω/2ℏ)(xx'sin(ωt)]

To derive this expression, you can use the eigenfunction expansion method by expressing the propagator as a sum over the eigenfunctions of the SHO Hamiltonian. This will lead to the expression you have, but with the additional cosine and sine terms.

Alternatively, you can also use the Feynman path integral method to compute the SHO propagator. This method involves summing over all possible paths from the initial position x to the final position x' in a given time t. This will also lead to the correct expression for the propagator.

I suggest reviewing your calculations and double-checking your algebra to ensure that you have the correct expression. It's also a good idea to consult a textbook or other reliable source for guidance. Keep up the good work!