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.

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

$\{ \alpha _1 ...\alpha _N \left. {} \right\rangle = \prod\limits_\lambda {\fr... ...a _1 }^\dag ...a_{\alpha _N }^\dag\left\vert 0 \right\rangle (B) \\$

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

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

$a_{\alpha _{} }^\dag\{ \alpha _1 ...\alpha _N \left. {} \right\rangle _U = \s... ... \{ \alpha \alpha _1 ...\alpha _N \left. {} \right\rangle _U (B) \\$

$\{ \alpha _1 ...\alpha _N \left. {} \right\rangle = \sum\limits_p {} \frac{si... ...pha _1 } \right\rangle ...\left\vert {\alpha _N } \right\rangle (F) \\$

$\{ \alpha _1 ...\alpha _N \left. {} \right\rangle = \sum\limits_p {} \frac{1}... ...pha _1 } \right\rangle ...\left\vert {\alpha _N } \right\rangle (B) \\$

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

Second quantized notation for matrix elements of one and two body operators

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

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