# Second quantization operators

## 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 $a_k$ are the annihilation operators for fermions, i.e. anti-commuting with each other.

What's the action of $\Psi(r)$ on the wave function $\psi(r_1,r_2)$?

## Answers and Replies

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Bill_K
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 $a_k$ are the annihilation operators for fermions, i.e. anti-commuting with each other.

What's the action of $\Psi(r)$ on the wave function $\psi(r_1,r_2)$?
In second quantization, the field operator doesn't act on a wavefunction at all, it acts on an abstract state vector in Fock space.

Be careful to keep straight the meaning of the subscripts in the two examples. In your first-quantized wavefunction, the subscripts refer to particle 1 or particle 2. But in the expression for the second-quantized field operator, the subscript k is used to indicate a single-particle state, not a particle.

Got it! Thank you