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## Homework Statement

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|Φ

_{n}> = ε

_{n}|Φ

_{n}>

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 |Φ

_{n}> as:

<Φ

_{n}|U|Φ

_{n}>

<Φ

_{n}|S|Φ

_{n}>

<Φ

_{n}|G|Φ

_{n}>

b) Now considr that electron is in state |Ψ> = e

^{-α}∑ |Φ

_{n}> (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>.

## Homework Equations

H = (-ħ/2m)*(d

^{2}/dx

^{2})

H = iħ*(d/dt)

Hint from professor: e

^{x}= ∑x

^{n}/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)*<Φ

_{n}|Φ

_{n}> and sin(i)<Φ

_{n}|Φ

_{n}>

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.