Representation of interaction

The representation of interaction or representation of Dirac of quantum mechanics is a manner of dealing with the problems depending on time.

Condition for application of the representation of interaction

In the representation of interaction, one applies the following assumptions:

One considers a Hamiltonien having the form according to:

$\backslash \; hat\; H\; =\; \backslash \; hat\; H\_0\; +\; \backslash \; hat\; V\; (T)$

where $\backslash \; hat\; H\_0$ is constant in time and $\backslash \; hat\; V\; (T)$ describes a perturbative interaction which can depend on time.

• the clean states are dependant on time
• the operators are also dependant on time
• the dynamics of the states is described according to the representation of Schrödinger while the dynamics of the operators is described according to the Représentation of Heisenberg.
• the representation of Dirac applies effectively only to certain problems. The example more speaking is that of the disturbances depending on time.

Propagators

In order to recognize that one works in the representation of interaction, the states and the operators will be followed index " I" (like interaction). The direction of this representation holds in what the dependence in time had with $\backslash \; hat\; H\_0$ will be taken into account in the explicit dependence of observable according to time and the dependence in time which had with $\backslash \; hat\; V\; (T)$ in the development of the function of wave. It is another way of describing same physics. This means that the significant physical sizes are unchanged.

There are two operators of evolution in time:

• the operator " normal" relating to the Hamiltonian complet$\backslash \; hat\; H$

$\backslash \; hat\; U\; (T,\; t\_0)\; =e^\; \{-\; I\; \backslash \; hat\; H\; (t-t\_0)/\backslash \; hbar\}$

• the operator relating to the nondisturbed Hamiltonian $\backslash \; hat\; H\_0$

$\backslash \; hat\; U\_0\; (T,\; t\_0)\; =e^\; \{-\; I\; \backslash \; hat\; H\_0\; (t-t\_0)/\backslash \; hbar\}$

Definition of the Hamiltonians and function of wave of interaction

The operator depend on time $\backslash \; hat\; A\_I\; (T)$ is written as in the representation of Heisenberg

$A\_I\; (T)\; =\; \backslash \; hat\; U\_0^\; \{\backslash \; dagger\}\; (T,\; t\_0)\; \backslash \; hat\; A\_S\; (t\_0)\; \backslash \; hat\; U\_0\; (T,\; t\_0)\; =\; \{\backslash \; rm\; E\}\; ^\; \{\backslash \; frac\; \{I\; \backslash ,\; \backslash \; hat\; H\_0\; (t-t\_0)\}\{\backslash \; hbar\}\}\; \backslash \; hat\; A\_S\; (t\_0)\; \{\backslash \; rm\; E\}\; ^\; \{-\; \backslash \; frac\; \{I\; \backslash ,\; \backslash \; hat\; H\_0\; (t-t\_0)\}\{\backslash \; hbar\}\}\; \backslash .$

the state depend on time $|\backslash \; psi\; (T)\; \backslash \; rangle\_I$ is accessible only indirectly, by reduction (in the representation of Schrödinger) of the state of dynamics supplements $|\backslash \; psi\; (T)\; \backslash \; rangle\_\; \{\backslash \; rm\; S\}$, in order to define.

$|\backslash \; psi\; (T)\; \backslash \; rangle\_I\; =\; \backslash \; hat\; U\_0^\; \{\backslash \; dagger\}\; (T,\; t\_0)|\backslash \; psi\; (T)\; \backslash \; rangle\_S\; =\; \{\backslash \; rm\; E\}\; ^\; \{\backslash \; frac\; \{I\; \backslash ,\; \backslash \; hat\; H\_0\; (t-t\_0)\}\{\backslash hbar\}\}|\backslash \; psi\; (T)\; \backslash \; rangle\_S\; \backslash .$

From there we define also the operator depend on time $H\_I\; (T)$:

$\backslash \; hat\; H\_I\; (T)\; =\; \{\backslash \; rm\; E\}\; ^\; \{\backslash \; frac\; \{I\; \backslash ,\; \backslash \; hat\; H\_0\; (t-t\_0)\}\{\backslash \; hbar\}\}\; \backslash \; hat\; H\_0\; \{\backslash \; rm\; E\}\; ^\; \{-\; \backslash \; frac\; \{I\; \backslash ,\; \backslash \; hat\; H\_0\; (t-t\_0)\}\{\backslash \; hbar\}\}\; \backslash .$

Equations of evolution of the function of wave and the observable ones

The evolution of the function of state is written in this representation:

$I\; \backslash \; hbar\; \backslash \; frac\; \{D\}\; \{dt\}\; |\; \backslash \; psi\_\; \{I\}\; (T)\; \backslash \; row\; =\; \backslash \; hat\; H\_0\; |\; \backslash \; psi\_\; \{I\}\; (T)\; \backslash \; row$.

This equation is known under the name of equation of Schwinger - Tomonaga . The evolution of the physical size represented by the operator has is written:

$i\; \backslash ,\; \backslash \; hbar\; \backslash \; frac=\; \backslash \; left\; A\_\; \{\backslash \; rm\; I\}\; (T),\; \backslash \; hat\; H\_0\; \backslash \; right\; +\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; hat\; A\_I\}\; \{\backslash \; partial\; T\}$

See too

• A. Messiah, Mechanical Quantum (Dunod)
• J.L. Basdevant, quantum Course of mechanics of the polytechnic school (Ellipses)
• J.J. Sakurai and S.F. Tuan, Modern Mechanics Quantum, Benjamin-Cummings 1985, Reading, Addison-Wesley 2003

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