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{ \large \bf Lecture 3:
 Introduction into second quantization. }

Review of Quantum Mechanics for a finite number of particles.

Motivation for the introduction of  a
tensor product  Hilbert space $H_0\bigoplus H_1 \bigoplus ... $
to accomodate an indefinite number of particles.


Second quantization as a new notation for old concepts.

$
 \{ \alpha _1 ...\alpha _N \left. {} \right\rangle  = a_{\alpha _1 }^\dag  ...a_{\alpha _N }^\dag  \left| 0 \right\rangle \,\,\,\,\,(F)
 \\
$

$
 \{ \alpha _1 ...\alpha _N \left. {} \right\rangle  = \prod\limits_\lambda  {\frac{1}{{\sqrt {n_\lambda  } }}} a_{\alpha _1 }^\dag  ...a_{\alpha _N }^\dag  \left| 0 \right\rangle \,\,\,\,\,(B) \\
$

$
 a_{\alpha _{} }^\dag  \{ \alpha _1 ...\alpha _N \left. {} \right\rangle  = \{ \alpha \alpha _1 ...\alpha _N \left. {} \right\rangle \,\,(F) \\
$

$
 {\rm if}\,\,\,\{ \alpha _1 ...\alpha _N \left. {} \right\rangle _U  \equiv a_{\alpha _1 }^\dag  ...a_{\alpha _N }^\dag  \left| 0 \right\rangle \,\,\,(B) \\
$

$
 a_{\alpha _{} }^\dag  \{ \alpha _1 ...\alpha _N \left. {} \right\rangle _U  = \sqrt {n_\lambda   + 1} \,\,\{ \alpha \alpha _1 ...\alpha _N \left. {} \right\rangle _U \,\,\,\,(B) \\
 $

 $
 \{ \alpha _1 ...\alpha _N \left. {} \right\rangle  = \sum\limits_p {} \frac{sign P}{{\sqrt {N!} }}P\left| {\alpha _1 } \right\rangle ...\left| {\alpha _N } \right\rangle \,\,\,(F) \\
 $

 $
 \{ \alpha _1 ...\alpha _N \left. {} \right\rangle  = \sum\limits_p {} \frac{1}{{\sqrt {N!} }}P\left| {\alpha _1 } \right\rangle ...\left| {\alpha _N } \right\rangle \,\,\,(B) \\
 $

changes of basis $ Q_\beta ^\dag   = \sum\limits_\alpha  {}
a_{\alpha _{} }^\dag  \left\langle {\alpha |\beta }
\right\rangle  $

{ \large \bf  Second quantized notation for matrix elements of one and two body 
operators }%

$
 T = \sum\limits_{\alpha \beta } {} \left\langle {\alpha |T|\beta } \right\rangle a_{\alpha _{} }^\dag  a_\beta  ;\,\,\,T = \sum\limits_i {} T(i) \\
$

$
 V = \frac{1}{2}\sum\limits_{\alpha \beta \gamma \delta } {} \left\langle {\alpha \beta |T|\gamma \delta } \right\rangle a_{\alpha _{} }^\dag  a_{\beta _{} }^\dag  a_\delta  a_\gamma  ;\,\,\,V = \sum\limits_{i < j} {} T(ij) \\
$

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