Using Creation and Annihilation Operators in a Fermionic 2-State System

[Urvabara]

Homework Statement

A fermionic 2-state system is $$\left\{\left|00\right\rangle, \left|10\right\rangle, \left|01\right\rangle, \left|11\right\rangle\right\}$$, where
$$\left|ab\right\rangle = \left|a\right\rangle_{1}\left|b\right\rangle_{2}$$ and $$\left\langle ab\left|cd\right\rangle = \delta_{ac}\delta_{bd}$$.

Homework Equations

What are the creation and annihilation operators $$f_{1}, f_{1}^{\dagger}, f_{2}, f_{2}^{\dagger}$$ in this base?

The Attempt at a Solution

I just do not know, how to get started. I just cannot find the theory and I do not know how to use it in this problem anyway. Can you give me some hints? Please, do not give right away the correct answers/results, just the hints to get started.

 

[Gokul43201]

How would you find the matrix elements <ab|f|cd> of an operator f?

 

[Urvabara]

How would you find the matrix elements <ab|f|cd> of an operator f?

So, I tried to calculate the matrix elements of an annihilation operator $$f_{1}$$. There are 16 of them, I think. Is this correct?
$$\left\langle 00\left|f_{1}\left|10\right\rangle = \left\langle 00\left|00\right\rangle = \delta_{00}\delta_{00} = 1\cdot 1 = 1.$$
$$\left\langle 00\left|f_{1}\left|01\right\rangle = 0.$$
$$\left\langle 00\left|f_{1}\left|11\right\rangle = \left\langle 00\left|01\right\rangle = \delta_{00}\delta_{01} = 1\cdot 0 = 0.$$
$$\left\langle 00\left|f_{1}\left|00\right\rangle = 0.$$
$$\left\langle 10\left|f_{1}\left|10\right\rangle = \left\langle 10\left|00\right\rangle = \delta_{10}\delta_{00} = 0\cdot 1 = 0.$$
$$\left\langle 10\left|f_{1}\left|01\right\rangle = 0.$$
$$\left\langle 10\left|f_{1}\left|11\right\rangle = \left\langle 10\left|01\right\rangle = \delta_{10}\delta_{01} = 0\cdot 0 = 0.$$
$$\left\langle 10\left|f_{1}\left|00\right\rangle = 0.$$
$$\left\langle 01\left|f_{1}\left|10\right\rangle = \left\langle 01\left|00\right\rangle = \delta_{00}\delta_{10} = 1\cdot 0 = 0.$$
$$\left\langle 01\left|f_{1}\left|01\right\rangle = 0.$$
$$\left\langle 01\left|f_{1}\left|11\right\rangle = \left\langle 01\left|01\right\rangle = \delta_{00}\delta_{11} = 1\cdot 1 = 1.$$
$$\left\langle 01\left|f_{1}\left|00\right\rangle = 0.$$
$$\left\langle 11\left|f_{1}\left|10\right\rangle = \left\langle 11\left|00\right\rangle = \delta_{10}\delta_{10} = 0\cdot 0 = 0.$$
$$\left\langle 11\left|f_{1}\left|01\right\rangle = 0.$$
$$\left\langle 11\left|f_{1}\left|11\right\rangle = \left\langle 11\left|01\right\rangle = \delta_{10}\delta_{11} = 0\cdot 1 = 0.$$
$$\left\langle 11\left|f_{1}\left|00\right\rangle = 0.$$

So, $$f_{1} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix}?$$

 

[Gokul43201]

You've determined the non-zero matrix elements correctly, but put them in the wrong positions. The row and column indices come from the left side of those equations. Also, you haven't determined possible normalization constants.

$$f_1|11 \rangle = c |01 \rangle$$

What is c?

 

[Urvabara]

You've determined the non-zero matrix elements correctly, but put them in the wrong positions. The row and column indices come from the left side of those equations.

So, something like this
$$f_{1} = \begin{pmatrix}0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\\end{pmatrix}?$$

Also, you haven't determined possible normalization constants.
$$f_1|11 \rangle = c |01 \rangle$$
What is c?

Hmm. I have no idea. Can you give a hint?

 

[Gokul43201]

So, something like this
$$f_{1} = \begin{pmatrix}0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\\end{pmatrix}?$$

Yes, that looks better.

Hmm. I have no idea. Can you give a hint?

Have you come across the number operator (e.g., $n_1=f_1^{\dagger}f_1$)?

 

[Urvabara]

Hi and thanks!

Yes, that looks better.
Have you come across the number operator (e.g., $n_1=f_1^{\dagger}f_1$)?

Yes, I think so. It gives the number of particles in the ground state. Right?

 

[olgranpappy]

Also, n.b., this is not a two-state system.

 

[Urvabara]

Also, n.b., this is not a two-state system.

Hmm. It is not? Oh boy. This was a exam problem. I failed badly in that exam.

 

[olgranpappy]

well, there are four states, |00>, |01>, |10>, |11>... So it's a four-state system.

 

[olgranpappy]

...two times two-states.

 

[Gokul43201]

I think the original question may have called it a "2-site" problem.