Antisymmetry in fermionic Fock space

[IanBerkman]

Dear all,

When we annihilate a particle at level ##k## in fermionic Fock space we use the relation
$$\hat{c}_k| \dots, 1_k, \dots \rangle = (-1)^{\sum_{i=1}^{k-1} n_i}|\dots,0_k,\dots\rangle.$$

Where the factor (##\pm1##) depends on the occupation numbers of all the levels below the level ##k##.
However, I do not see what the link is between antisymmetry when permuting two electrons, i.e.
$$\hat{c}_n^\dagger\hat{c}_m^\dagger|0\rangle = -\hat{c}_m^\dagger\hat{c}_n^\dagger|0\rangle,$$
and how this is incorporated by counting the number of electrons below level ##k##.

Thanks in advance,
Ian

 

[eljose79]

Dear Ian,

The relationship you have mentioned is known as the fermionic anticommutation relation. It is a fundamental property of fermions, which are particles that follow the Pauli exclusion principle and have half-integer spin. This relation is crucial in understanding the behavior of fermionic systems, such as electrons in a solid state material.

To understand the connection between the anticommutation relation and the annihilation of a particle at level ##k##, we need to look at the creation and annihilation operators in more detail. These operators are used to describe the creation and annihilation of particles in a quantum system. In the case of fermions, the operators have the property that they anticommute with each other, meaning that the order in which they are applied matters. This is what leads to the ##\pm1## factor in the relation you mentioned.

Now, when we annihilate a particle at level ##k##, we are essentially removing it from the system. This means that the occupation number of that level changes from ##1## to ##0##. However, this also affects the occupation numbers of all the levels below ##k##, as the particles are now rearranged in the system. This is why we need to take into account the occupation numbers of all the levels below ##k## when we annihilate a particle at that level. This is also why the relation includes the sum over all the occupation numbers below ##k##.

In summary, the fermionic anticommutation relation and the annihilation of a particle at level ##k## are intimately connected because they both arise from the fundamental property of fermions to anticommute. By counting the number of electrons below level ##k##, we are taking into account the rearrangement of particles in the system, which is necessary to properly describe the behavior of fermions.

I hope this clarifies your doubts. Let me know if you have any further questions.