For each single-particle state
of the single-particle Hilbert space
a Boson or Fermion creation operator
is defined by its action on any symmetrized or antisymmetrized state
of the Hilbert space of
Bosons,
, or
Fermions,
, as follows:
![$\displaystyle a_{\lambda}^{+}\vert\lambda_{1}...\lambda_{N}\}=\vert\lambda\lambda_{1}...\lambda_{N}\}$](img1149.png) |
(A.4) |
The creation operators
do not operate within one space
or
, but rather operate from any space
or
to
or
. It is useful to define the Fock space as the direct sum of the
Boson or Fermion spaces
|
|
![$\displaystyle \mathcal{B}=\oplus_{n=0}^{\infty}\mathcal{B}_{n},$](img1154.png) |
(A.5) |
|
|
![$\displaystyle \mathcal{F}=\oplus_{n=0}^{\infty}\mathcal{F}_{n},$](img1155.png) |
|
where by definition:
|
|
![$\displaystyle \mathcal{B}_{0}=\mathcal{F}_{0}=\vert\rangle,$](img1156.png) |
(A.6) |
|
|
![$\displaystyle \mathcal{B}_{1}=\mathcal{F}_{1}=\mathcal{H}.$](img1157.png) |
|
It can be easily shown that for Bosons the creation operators commute:
![$\displaystyle a_{\lambda}^{+}a_{\mu}^{+}-a_{\mu}^{+}a_{\lambda}^{+}=0,$](img1158.png) |
(A.7) |
whereas they anticommute for Fermions:
![$\displaystyle a_{\lambda}^{+}a_{\mu}^{+}+a_{\mu}^{+}a_{\lambda}^{+}=0.$](img1159.png) |
(A.8) |
The annihilation operators
are defined as the adjoints of the creation operators
. The commutation and anticommutation
relations of annihilation operators follow from (A.7) and (A.8), respectively. They commute for Bosons:
![$\displaystyle a_{\lambda}a_{\mu}-a_{\mu}a_{\lambda}=0,$](img1162.png) |
(A.9) |
whereas they anticommute for Fermions:
![$\displaystyle a_{\lambda}a_{\mu}+a_{\mu}a_{\lambda}=0.$](img1163.png) |
(A.10) |
The action of the annihilation operator on a many particle state is given for Bosons as
![$\displaystyle a_{\lambda}\vert\alpha_{1}...\alpha_{n}\}=\sum_{i=1}^{n}\delta_{\lambda\alpha_{i}}\vert\alpha_{1}...\hat{\alpha}_{i}...\alpha_{n}\},$](img1164.png) |
(A.11) |
while for Fermions it reads:
![$\displaystyle a_{\lambda}\vert\alpha_{1}...\alpha_{n}\}=\sum_{i=1}^{n}(-1)^{i-1}\delta_{\lambda\alpha_{i}}\vert\alpha_{1}...\hat{\alpha}_{i}...\alpha_{n}\}.$](img1165.png) |
(A.12) |
Here
shows that the state
has been removed from the mani-particle state
.
The commutation rules for the creation and annihilation operators are:
If the orthonormal basis
transforms into another basis
, the creation and annihilation operators transform as
follows:
|
|
![$\displaystyle a_{\widetilde{\alpha}}^{+}=\sum_{\alpha}\langle\alpha\vert\widetilde{\alpha}\rangle a_{\alpha}^{+},$](img1175.png) |
(A.14) |
|
|
![$\displaystyle a_{\widetilde{\alpha}}=\sum_{\alpha}\langle\widetilde{\alpha}\vert\alpha\rangle a_{\alpha}.$](img1176.png) |
|
Of particular importance is the coordinate basis
. In this case the creation and annihilation operators are traditionally denoted by
and
and are called field operators. From (A.15) it follows:
|
|
![$\displaystyle \hat{\psi}^{+}(\vec{x})=\sum_{\alpha}\phi_{\alpha}^{*}(\vec{x})a_{\alpha}^{+},$](img1180.png) |
(A.15) |
|
|
![$\displaystyle \hat{\psi}(\vec{x})=\sum_{\alpha}\phi_{\alpha}(\vec{x})a_{\alpha},$](img1181.png) |
|
where
is the coordinate representation wave function of the state
.
It can be shown that
-body operators (A.3) can be expressed through the creation and annihilation operators in a simple form:
![$\displaystyle \hat{U}=\frac{1}{n!}\sum_{\lambda_{1}...\lambda_{n}}\sum_{\mu_{1}...
......\mu_{n}) a_{\lambda_{1}}^{+}...a_{\lambda_{n}}^{+}a_{\mu_{n}}...a_{\mu_{1}}.$](img1184.png) |
(A.16) |
For example using the coordinate representation, the kinetic energy operator
![$\displaystyle \hat{T}=\sum_{i}\frac{\hat{\vec{p}}^{2}_{i}}{2m}$](img1186.png) |
(A.17) |
may be rewritten in second quantized form as:
![$\displaystyle \hat{T}=-\frac{\hbar^{2}}{2m}\int\,d^{3}x\hat{\psi}^{+}(\vec{x})\nabla^{2}\hat{\psi}(\vec{x}).$](img1187.png) |
(A.18) |
S. Smirnov: