Editing Second quantization (section)

==== Definition ====

The fermion creation (annihilation) operator is a linear operator, whose action on a ''N''-particle first-quantized wave function <math>\Psi</math> is defined as
:<math>c_\alpha^\dagger \Psi = \frac{1}{\sqrt{N+1}}\psi_\alpha\otimes_-\Psi,</math>
:<math>c_\alpha\Psi = \frac{1}{\sqrt{N}}\psi_\alpha\oslash_-\Psi,</math>
where <math>\psi_\alpha\otimes_-</math> inserts the single-particle state <math>\psi_\alpha</math> in <math>N+1</math> possible insertion positions anti-symmetrically, and <math>\psi_\alpha\oslash_-</math> deletes the single-particle state <math>\psi_\alpha</math> from <math>N</math> possible deletion positions anti-symmetrically.

It is particularly instructive to view the results of creation and annihilation operators on states of two (or more) fermions, because they demonstrate the effects of exchange. A few illustrative operations are given in the example below.  The complete algebra for creation and annihilation operators on a two-fermion state can be found in ''Quantum Photonics''.<ref name="Pearsall2020">{{cite book
 | author = Pearsall, Thomas P.
 | title = Quantum Photonics
 | publisher = Springer
 | location = Cham, Switzerland
 | edition = 2nd
 | series = Graduate Texts in Physics
 | isbn = 978-3-030-47325-9
 | doi = 10.1007/978-3-030-47325-9
 | year = 2020
 | pages = 301–302
| bibcode = 2020quph.book.....P
 }}</ref>

===== Examples =====
Hereinafter the tensor symbol <math>\otimes</math> between single-particle states is omitted for simplicity. Take the state <math>|1_1,1_2\rangle=(\psi_1\psi_2-\psi_2\psi_1)/\sqrt{2}</math>, attempt to create one more fermion on the occupied <math>\psi_1</math> state will quench the whole many-body wave function,
:<math>\begin{array}{rl}c_1^\dagger|1_1,1_2\rangle=&\frac{1}{\sqrt{2}}(c_1^\dagger\psi_1\psi_2-c_1^\dagger\psi_2\psi_1)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}\psi_1\otimes_-\psi_1\psi_2-\frac{1}{\sqrt{3}}\psi_1\otimes_-\psi_2\psi_1\right)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}(\psi_1\psi_1\psi_2-\psi_1\psi_1\psi_2+\psi_1\psi_2\psi_1)-\frac{1}{\sqrt{3}}(\psi_1\psi_2\psi_1-\psi_2\psi_1\psi_1+\psi_2\psi_1\psi_1)\right)\\=&0.\end{array}</math>
Annihilate a fermion on the <math>\psi_2</math> state,
take the state <math>|1_1,1_2\rangle=(\psi_1\psi_2-\psi_2\psi_1)/\sqrt{2}</math>,
:<math>\begin{array}{rl}c_2|1_1,1_2\rangle=&\frac{1}{\sqrt{2}}(c_2\psi_1\psi_2-c_2\psi_2\psi_1)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}\psi_2\oslash_-\psi_1\psi_2-\frac{1}{\sqrt{2}}\psi_2\oslash_-\psi_2\psi_1\right)\\=&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}(0-\psi_1)-\frac{1}{\sqrt{2}}(\psi_1-0)\right)\\=&-\psi_1\\=&-|1_1,0_2\rangle.\end{array}</math>
The minus sign (known as the fermion sign) appears due to the anti-symmetric property of the fermion wave function.