The Hamiltonian of an electron in solids is given by H. We know that H is an Hermitian operator, it satisfies the following eigenvalue equation:
H|Φn> = εn|Φn>
Let us define the following operators in terms of H as:
U = e^[(iHt)/ħ] , S = sin[(Ht)/ħ] , G = (ε - H)-1
a) Find the expectation value of U, G and S operators in state |Φn> as:
b) Now considr that electron is in state |Ψ> = e-α ∑ |Φn> (where ∑ denotes an infinite sum from n = 0 to n = ∞), where α is a complex variable. Calculate the expectation value of U, G, and S operators in state |Ψ>.
H = (-ħ/2m)*(d2/dx2)
H = iħ*(d/dt)
Hint from professor: ex = ∑xn/n!
The Attempt at a Solution
It's been a long time since I took the pre-requisite courses for this quantum course so I am having a lot of problems figuring out how to do this question. I saw online that H could be expressed as iħ*(d/dt), plugging that into the equations for U and S simplifies them to 1/e and sin(i) respectively. After being simplified they can be removed from the brackets leaving me with:
(1/e)*<Φn|Φn> and sin(i)<Φn|Φn>
For both of these expressions the term in brackets is equal to 1, leaving me with an expectation value of 1/e for U and sin(i) for S. Can someone tell me if my approach is correct? The textbook for the course only expresses H in position dependent form (first relevant equation) and not in the time dependent form I have used, so I feel as though I'm "cheating" somehow and I won't get full marks. We probably have to solve it using the hint (3rd relevant equation) somehow but I don't understand how. Aside from that I have no idea how to find the expectation value for G and I haven't even gotten to part b yet. Any help would be appreciated.