Finding expectation values for given operators

[phys-student]

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)*(d²/dx²)
H = iħ*(d/dt)
Hint from professor: e^x = ∑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)*<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.

 

[blue_leaf77]

|Ψ> = e-α ∑ |Φn>

The question asks you to calculate the expectation values with respect to $|\phi_n\rangle$, which is known to be an eigenstate of $H$, therefore you don't need to define your own state.

Hint from professor: ex = ∑xn/n!

You should consider this hint instead. This hint tells you that you can expand a smooth function of x in terms of its powers. Do the same with $U$, $G$, and $S$, namely expand each of these function operators in terms of powers of $H$.

 

[phys-student]

I've tried expressing U in terms of powers of H and got the following:

e^(iHt/ħ) = [e^(it/ħ)]^H = ∑(H^n)/n! = 1 + H + H²/2 + H³/6 +...

What exactly am I supposed to do with this now?... I think my main source of confusion is that I'm unfamiliar with Dirac notation (which unfortunately seems to be a big part of the course). Is the following statement valid?

<U> = <[e^(it/ħ)]^H>
= <∑Hⁿ/n!>
= <Φ_n|∑εⁿ/n!|Φ_n>
= ∑εⁿ/n!<Φ_n|Φ_n> = ∑εⁿ/n!

Am I allowed to pull an infinite sum outside the brackets like that? Am I allowed to simply substitute ε for H like I did in the 3rd line? Is it okay to leave the expectation value as an infinite sum? Sorry if these are stupid questions I'm just having a lot of trouble with this question

 

[blue_leaf77]

[e(it/ħ)]H = ∑(H^n)/n!

More correctly, it should be
$$
e^{iHt/\hbar} = \sum_{n=0} \frac{1}{n!} \left(\frac{iHt}{\hbar}\right)^n
$$

<Φn|∑εn/n!|Φn>

You must not use the same index between the states and the summation, make them different. Also don't forget to include the factor $\frac{it}{\hbar}$.

Am I allowed to pull an infinite sum outside the brackets like that?

As I have mentioned above, the kets should actually have different index from that of the sum. In other words, these kets were not originally part of the summation. Therefore you can bring them inside the summation sign.

Am I allowed to simply substitute ε for H like I did in the 3rd line?

That "substitution" actually follows from
$$
H^n|\phi_m\rangle = \epsilon_m^n |\phi_m\rangle
$$

Is it okay to leave the expectation value as an infinite sum?

Pay attention in the form of the series at the end of your calculation. Does it look similar to the series you have in the beginning when you expand the exponential operator?

 

[phys-student]

Thank you, I've been able to get through all of part a. I'm on part b now, and I'm not sure how to express <Ψ|U|Ψ>...

Given that |Ψ> = e^-α∑|Φ_n>

Can <Ψ|U|Ψ> be expressed as:

<Ψ|U|Ψ> = e^-α∑<Φ_n|U|Φ_n>

So then the expectation value of U in state Ψ, is the infinite sum of the expectation value found in part a (in state Φ_n) multiplied by the constant e^-α?

 

[blue_leaf77]

|Ψ> = e-α ∑ |Φn>

The form of $|\psi\rangle$ given above actually looks fishy to me. All basis in the series have equal probability, I'm not sure if such form is normalizable.
Despite of that, when calculating the expectation value of an operator with respect to states which are written as linear combination of some basis, you have to retain both sums. In this problem you want to calculate $\langle \psi |U| \psi \rangle$, with $|\psi \rangle = e^{-\alpha} \sum_n |\phi_n\rangle$. Therefore
$$
\langle \psi |U| \psi \rangle = e^{-\alpha^*} \sum_n \langle \phi_n| U e^{-\alpha} \sum_m |\phi_m\rangle = e^{-2\Re[\alpha]}\sum_m \sum_n \langle \phi_n |U| \phi_m \rangle
$$
Since you know how $U$ acts on $|\phi_m\rangle$, seeing what the next step is should be easy for you.