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\chapter{REVIEW of QUANTUM MECHANICS} 

Key Concepts:

Hilbert space of states;

Observables;

Evaluation operator;

Along a contour;

Imaginary time;

Chronological Order;

Schr\"{o}dinger-Heisenberg interaction;

Picture representation;

Schr\"{o}edinger picture, states evolve in time according:

$$i{\partial {|{\psi _{s}(t)}>}}{\over {\partial t}\hfill} = H_{s}(t)|{\psi _{s}(t)>}$$

$$|{\psi_{s}(t)}> = U_{s}(t,t_{0})|{\psi_{s}(t_{0})>$$

$U_{s}(t,t_{0})$ evelution operator plays an essential  role  in the theroy.

$$i{\partial {U(t,t_{0}}{\over {\partial t}\hfill} = H_{s}(t)U(t,t_{0})$$

$$U(t,t_{0}) = I$$

For mixed states, use density matrix:

$$i{{d\rho _{s}}{{\over dt}\hfill} = [H_{s}, {\rho _{s}}]$$

Formal solution of the evolution equation:

$U(t,t_{0}) = Te^{{1\over i}\int\limit_{t_{0}}^tH_{s}(\tau)d\tau}$

$           = \sum_{n=0}^{\infty}{{(1\over i)^n} \int_{t_{0}}^t 
\ldots\int_{t_{0}}^{t_{n-1}}H_{s}(t_{1})\ldots H_{s}(t_{n})dt_{1}\ldots dt_{n}$

The operator
%$
%\[T(A(t_{1}), B(t_{2}))\equiv \left\{ \begin{array}{lll}                                                       A(t_{1}) &\mbox{B(t_{2})}&\mbox{t_{1}>t_{2}}\                                  B(t_{2}) &\mbox{A(t_{1})}&\mbox{t_{2} > t_{1}}\                                \end{array}                                                                  \right. \]

%$
orders operatorrs putting the latter ones on the right.

Eqn.( )--( ) can be in fact formulated for an arbituary oriented contour {\cal C} in the condex plane if we assume that \( H_{s}\) is defined there.

If {\cal C} is a contour going from $Z_{1}$ to $Z_{2}$ :

\begin {figure}
\vspace {3in}
\caption { }
\label{1}

\end {figure}
 

$$U(Z_{2}, Z_{1})\equiv T_{\cal C}e^{{1\over i}\int_{\cal C} H_{s}(z)\,dz}$$

obeys

$$i{\partial U(t_{2}, t_{1})}{\over {\partial Z_{2}}\hfill}=H_{s}(Z_{2})U(Z_{2}, Z_{1})$$


$T_{\cal C}$ orders operators along the contour {\cal C} putting latter operators on the right.

Heisenberg picture.

Here one performs a unitary rotation to keep the state vectors fixed,{\it i.e.}

$$|{\psi _{H}(t)>}=U(t,t_{0})^{-1}|{\psi _{s}(t)>}$$

$t_{0}$ is an arbituary time where all the pictures ? made. The observables evolve
according to $O_{H}(t)=U(t,t_{0})^{-1}O_{s}(t)U(t,t_{0})$

$$\.{O}_{H}= \partial {O_{H}}{\over {\partial t}\hfill}+{\frac {1}{i}}[O_{H},H_{H}]$$

where ${\partial {O_{H}}{\over {\partial {t}}\hfill}\equiv U(t,t_{0})_{-1}{\partial {O_{S}}{\over {\partial {t}}\hfill}}U(t,t_{0})$

The density matrix obeys $d\rho _{H}{\over {dt}\hfill}=0$

The notion of evolution along a contour is useful because it allows to unify
different formulations of many body methods. A particular useful contour is 
[0, -i\beta], {[\it i.e.]} along the imaging axis. (see Fig. 2)

\begin {figure}
\vspace{4.0in}
\caption{figure 2}
\label {2}
\end {figure}

Example: The evolution operator from $0$ to $-i\beta $along the contour shown
in Fig. $2$ is given by:

$$U_{S}(-i\beta,0)=T_{\cal C}e^{{\frac {1}{i}}\int_{\cal C}H_{S}(\tau)\,d\tau^{'}}}$$


and equals to the Boltzmann density matrix $e^{-\beta H_{S}}$ if $H_{S}$ 
does not depend on time.

Interaction picture:

This representation is defined with reference to a ?thing of the hamitonian 
into two parts $H=H_{0}+V$. Roughly speaking, the onservables and evolved which is usually simple while the state vectors are driven by $V$.

$$|\psi _{I}(t)> = e^{iH_{0}(t_{1}-t_{0})}|\psi _{S}(t)>$$

The observables evolve according to $V_{I}(t) = e^{iH_{0}(t-t_{0})}V_{S}(t)\e^{-iH_{0}(t-t_{0})}$. The evolution operator in this picture is given by:

$$U_{I}(t_{i},t)=e^{iH_{0}(t_{1}-t_{0})}U_{S}(t_{1},t)e^{-iH_{0}(t-t_{0})}$$

and obeys:

$$i{\partial {U_{I}(t_{1},t_{2})}{\over {\partial t}\hfill}}=V_{I}(t_{1})U_{I}(t_{1},t_{2})$$

$$U(t,t)=I$$

The formal solution:

$$U_{I}(t_{1},t_{2})=Te^{{\frac {1}{i}}\int_{t_{2}}^{t_{1}}V_{I}(\tau)\,d\tau}$$

similarly:

$$i{\.{O}}_{I}=[O_{I},H_{0}]$$

$$i{\.{\rho}}_{I}=[V_{I},{\rho}_{I}]$$


Application:

$$t_{0}=0$$

$$t_{1}=-i\beta$$

$$t=0$$

$$U_{I}(t_{1},0)=e^{iH_{0}t_{1}}U_{s}(t_{1},0)$$

$$U_{I}(t_{1},0)=T_{\cal C}e^{{1\over {i}}\int_{\cal C}V_{I}(z)\,dz$$

$$U_{S}(-i\beta,0)=e^{-\beta H}$$

$$e^{-\beta H}=e^{-\beta H_{0}}T_{\cal C}exp({-\int_{0}^{\beta}e^{-H_{0}\tau}Ve^{H_{0}\tau}\,d\tau})$$

which expresses the density matrix of an interacting system in terms of $H_{0}$and a ? $V$.


{\bf Problems}

1) Let {\cal C} be a oriented contour in ? define an antichronological order operator by $\tilde{\tau}(A(Z_{1})B(Z_{2}))

=\left\{ \begin{array}{ll}                                                        A(Z_{1})B(Z_{2})}& if Z_{1} occurs before Z_{2}\                               B(Z_{2})A(Z_{1})} \end{array}\right.$


a) Prove $\dag {T(A,B)}=\tilde {T}(\dag {A}\dag {B})$

b) What equation ? $\tilde {T} exp({1\over {i}}\int_{\cal C}H(Z)\,dz})\equiv U(z_{2},Z_{1})$ obeys.




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