Finding expectation values for given operators

In summary, the homework statement asks you to calculate the expectation values of U, G, and S operators in state |Ψ> as:<Φn|U|Φn> <Φn|S|Φn> <Φn|G|Φn>
  • #1
phys-student
32
0

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|Φn> = εnn>

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:
n|U|Φn>
n|S|Φn>
n|G|Φn>

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

Homework Equations


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)*<Φnn> and sin(i)<Φnn>

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.
 
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  • #2
phys-student said:
|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.
phys-student said:
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##.
 
  • #3
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!>
= <Φn|∑εn/n!|Φn>
= ∑εn/n!<Φnn> = ∑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
 
  • #4
phys-student said:
[e(it/ħ)]H = ∑(H^n)/n!
More correctly, it should be
$$
e^{iHt/\hbar} = \sum_{n=0} \frac{1}{n!} \left(\frac{iHt}{\hbar}\right)^n
$$
phys-student said:
<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}##.
phys-student said:
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.
phys-student said:
Am I allowed to simply substitute ε for H like I did in the 3rd line?
That "substitution" actually follows from
$$
H^n|\phi_m\rangle = \epsilon_m^n |\phi_m\rangle
$$
phys-student said:
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?
 
  • #5
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∑|Φn>

Can <Ψ|U|Ψ> be expressed as:

<Ψ|U|Ψ> = e∑<Φn|U|Φn>

So then the expectation value of U in state Ψ, is the infinite sum of the expectation value found in part a (in state Φn) multiplied by the constant e?
 
  • #6
phys-student said:
|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.
 

Related to Finding expectation values for given operators

1. What is the purpose of finding expectation values for given operators?

The purpose of finding expectation values for given operators is to determine the average value of a physical quantity in a quantum mechanical system. This is important for understanding and predicting the behavior of particles and systems in the quantum realm.

2. How do you calculate the expectation value for a given operator?

The expectation value for a given operator can be calculated by taking the inner product of the state vector with the operator multiplied by the state vector, and then taking the complex conjugate of this value. This can be represented mathematically as:

⟨A⟩ = ⟨ψ|A|ψ⟩*

3. What is the significance of the expectation value being a complex number?

The expectation value being a complex number indicates that there is a probabilistic nature to quantum mechanics. The real part of the complex number represents the average value of the physical quantity, while the imaginary part represents the uncertainty or spread of the measurement.

4. Can the expectation value for a given operator be negative?

Yes, the expectation value for a given operator can be negative. This means that the average value of the physical quantity is negative, but it does not necessarily mean that the physical quantity itself is negative. This is due to the complex nature of quantum mechanics and the probabilistic interpretation of the expectation value.

5. How does the uncertainty principle relate to finding expectation values for given operators?

The uncertainty principle states that the position and momentum of a particle cannot be known simultaneously with absolute certainty. This is reflected in the expectation values for certain operators in quantum mechanics. For example, the expectation values for position and momentum operators will have a non-zero imaginary part, indicating the uncertainty in these values.

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