Operator of creation

In Quantum physics, in Second quantification, a operator of creation is an operator who acts on the Espace of Fock by changing a state with $N$ particles into a state with $N+1$ particles.

In the case of the Boson S, the operator of creation $a^\; \backslash \; dagger\_i$ which creates one particle in the state $i$ is such as:

$a^\backslash dagger\_i\; \backslash ;|\; N\_1,\; N\_2,\; \backslash \; ldots,\; N\_i,\; \backslash \; ldots\; \backslash \; rangle\; =\; \backslash \; sqrt\; \{N\_i+1\}\; \backslash ;\; |\; N\_1,\; N\_2,\; \backslash \; ldots,\; N\_i+1,\; \backslash \; ldots\; \backslash \; rangle$

In addition, the operators of creation commutate between them:

$=0$

A standardized state of the space of Fock bosonic is thus written:

$|\; N\_1,\; N\_2,\; \backslash \; ldots\; \backslash \; rangle\; =\; \backslash \; prod\_k\; \backslash \; frac\; \{(a\_k^\; \backslash \; dagger)\; ^\; \{N\_k\}\}\; \{\backslash \; sqrt\; \{N\_k!\}\}\; \backslash ;\; |\; 0\; \backslash \; rangle$, where $|0\; \backslash \; rangle$ indicates the vacuum.

In the case of the Fermion S, because of the Principle of exclusion of Pauli, it is not possible to create two fermions in the same state, so that $(c^\; \backslash \; dagger\_i)\; ^2=0$. The action of the operator of creation is thus defined by:

$c^\backslash dagger\_i\; \backslash ;|\; N\_1,\; N\_2,\; \backslash \; ldots,\; 0\_i,\; \backslash \; ldots\; \backslash \; rangle\; =\; |\; N\_1,\; N\_2,\; \backslash \; ldots\; 1\_i,\; \backslash \; ldots\; \backslash \; rangle$

Moreover, the operators of creation of fermions anticommutent: $c^\backslash dagger\_i\; c^\backslash dagger\_j\; =-c^\backslash dagger\_j\; c^\backslash dagger\_i$

One can write the standardized states of the space of Fock in the form:

$|N\_1,\; N\_2,\; \backslash \; ldots\; \backslash \; rangle\; =\; (c^\; \backslash \; dagger\_\; \{i\_1\})\; ^\; \{N\_\; \{i\_1\}\}\; (c^\; \backslash \; dagger\_\; \{i\_2\})\; ^\; \{N\_\; \{i\_2\}\}\; \backslash \; ldots\; |0\; \backslash \; rangle$

But because of anticommutation of the operators of creation of fermions, it is necessary to choose a particular convention for the order in which the operators of creation must act.

The combined square one of an operator of creation is a Opérateur of annihilation.

In the case of the bosons, the operators of creation and the operators of annihilation satisfy relations of commutation: $=\; \backslash \; delta\_\; \{I,\; J\}$, which make that the algebra of the operators of creation and annihilation of bosons is identical to the algebra of the operators who generate the spectrum of the Oscillateur quantum harmonic. That involves that the Phonons and the photons can be treated like bosons.

In the case of the fermions, the operators of creation and annihilation satisfy relations of anticommutation:

$\backslash \; \{c^\; \backslash \; dagger\_i,\; c\_j\; \backslash \}\; =\; \backslash \; delta\_\; \{ij\}$.

References

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