Editing Second quantization (section)

==== Action on Fock states ====
Starting from the single-mode vacuum state <math>|0_\alpha\rangle=1</math>, applying the creation operator <math>b_\alpha^\dagger</math> repeatedly, one finds
:<math>b_\alpha^\dagger|0_\alpha\rangle=\psi_\alpha\otimes_+ 1=\psi_\alpha=|1_\alpha\rangle,</math>
:<math>b_\alpha^\dagger|n_\alpha\rangle=\frac{1}{\sqrt{n_\alpha+1}}\psi_\alpha\otimes_+ \psi_\alpha^{\otimes n_\alpha}=\sqrt{n_\alpha+1}\psi_\alpha^{\otimes (n_\alpha+1)}=\sqrt{n_\alpha+1}|n_\alpha+1\rangle.</math>
The creation operator raises the boson occupation number by 1. Therefore, all the occupation number states can be constructed by the boson creation operator from the vacuum state
:<math>|n_\alpha\rangle=\frac{1}{\sqrt{n_\alpha!}}(b_{\alpha}^\dagger)^{n_\alpha}|0_\alpha\rangle.</math>
On the other hand, the annihilation operator <math>b_\alpha</math> lowers the boson occupation number by 1
:<math>b_\alpha|n_\alpha\rangle=\frac{1}{\sqrt{n_\alpha}}\psi_\alpha\oslash_+ \psi_\alpha^{\otimes n_\alpha}=\sqrt{n_\alpha}\psi_\alpha^{\otimes (n_\alpha-1)}=\sqrt{n_\alpha}|n_\alpha-1\rangle.</math>
It will also quench the vacuum state <math>b_\alpha|0_\alpha\rangle=0</math> as there has been no boson left in the vacuum state to be annihilated. Using the above formulae, it can be shown that
:<math>b_\alpha^\dagger b_\alpha|n_\alpha\rangle=n_\alpha|n_\alpha\rangle,</math>
meaning that <math>\hat{n}_\alpha = b_\alpha^\dagger b_\alpha</math> defines the boson number operator.

The above result can be generalized to any Fock state of bosons.
:<math>b_\alpha^\dagger|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= \sqrt{n_\alpha+1}|\cdots,n_\beta,n_\alpha+1,n_\gamma,\cdots\rangle.</math>
:<math>b_\alpha|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= \sqrt{n_\alpha}|\cdots,n_\beta,n_\alpha-1,n_\gamma,\cdots\rangle.</math>
These two equations can be considered as the defining properties of boson creation and annihilation operators in the second-quantization formalism. The complicated symmetrization of the underlying first-quantized wave function is automatically taken care of by the creation and annihilation operators (when acting on the first-quantized wave function), so that the complexity is not revealed on the second-quantized level, and the second-quantization formulae are simple and neat.