Show that an expected value of a vacuum state is equal to 1

mcas
Messages
22
Reaction score
5
Homework Statement
Using the following anticommutator relations of fermionic one-particle operators:
[itex]\{\hat{c}_{k\alpha},\hat{c}_{k'\beta} \}= \{ \hat{c}^\dagger_{k\alpha}, \hat{c}^\dagger_{k'\beta} \} = 0[/itex]
[itex]\{\hat{c}_{k\alpha},\hat{c}^\dagger_{k'\beta} \}=\delta_{kk'}\delta_{\alpha\beta}[/itex]

Show that the expected value for a vacuum state [itex]|\phi_0>[/itex] is:

[itex] <\phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0>=1
[/itex]
Relevant Equations
Given in the homework statement
<br /> \langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| - \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow} \hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| \hat{c}_{k \uparrow} \hat{c}^\dagger_{k \uparrow} \hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0|(1- \hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow} ) \hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0|\hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle - \langle \phi_0|\hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow}\hat{c}_{-k \downarrow}|\phi_0\rangle<br />

Then I changed \hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow} in the second term to (1- \hat{c}^\dagger_{k \uparrow} \hat{c}_{k \uparrow} ) and the result was \langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle which is exactly what I sarted from.

I don't know where to go from this so I would really appreciate any help!
 
Last edited:
Physics news on Phys.org
Where did you get this problem from?

Are the ##\hat{c}_{k\alpha}## usual annihilation operators?
How is the ##\ket{\phi_0}## state defined?

If ##\ket{\phi_0}## is the usual vacuum defined from the ##\hat{c}## operators then, if I'm not wrong, by definition ##\hat{c}_{-k \downarrow} \ket{\phi_0}=0## and the whole expectation value vanishes. So, either I'm confused by something, or the problem is wrong, or you have extra information that is crucial for solving the problem.
 
  • Like
Likes mcas and anuttarasammyak
Gaussian97 said:
Where did you get this problem from?

Are the ##\hat{c}_{k\alpha}## usual annihilation operators?
How is the ##\ket{\phi_0}## state defined?

If ##\ket{\phi_0}## is the usual vacuum defined from the ##\hat{c}## operators then, if I'm not wrong, by definition ##\hat{c}_{-k \downarrow} \ket{\phi_0}=0## and the whole expectation value vanishes. So, either I'm confused by something, or the problem is wrong, or you have extra information that is crucial for solving the problem.
It's a problem formulated by my professor.
After you post I contacted him and indeed, he made a mistake in the homework statement, as the last operator \hat{c}_{-k \downarrow} should be conjugated for the expected value to be one. In the original statement, the value will be 0.
Thank you!
 
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top