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## Main Question or Discussion Point

I have a doubt on the second quantization formalism. Suppose that we have two spin-1/2 fermions which can have just two possible quantum number, 1 and 2. Consider the wave function:

$$

\psi(r_1,r_2)=\frac{1}{\sqrt{2}}\left(\psi_1(r_1)\psi_2(r_2)-\psi_1(r_2)\psi_2(r_1)\right).

$$

The second quantized field is defined as:

$$

\Psi(r)=\sum_{k=1}^2a_k\psi_k(r),

$$

where [itex]a_k[/itex] are the annihilation operators for fermions, i.e. anti-commuting with each other.

What's the action of [itex]\Psi(r)[/itex] on the wave function [itex]\psi(r_1,r_2)[/itex]?

$$

\psi(r_1,r_2)=\frac{1}{\sqrt{2}}\left(\psi_1(r_1)\psi_2(r_2)-\psi_1(r_2)\psi_2(r_1)\right).

$$

The second quantized field is defined as:

$$

\Psi(r)=\sum_{k=1}^2a_k\psi_k(r),

$$

where [itex]a_k[/itex] are the annihilation operators for fermions, i.e. anti-commuting with each other.

What's the action of [itex]\Psi(r)[/itex] on the wave function [itex]\psi(r_1,r_2)[/itex]?