Interaction picture

The interaction picture (also referred to as interaction representation or, according to Paul Dirac, as Dirac picture or Dirac representation ) is a model in quantum mechanics for dealing with time-dependent problems, taking interactions into account .

It is largely equivalent to the Heisenberg and Schrödinger images ; H. all physically relevant quantities ( scalar products , eigenvalues , etc.) remain the same (see also the mathematical structure of quantum mechanics ).

To indicate that the interaction picture is used, states and operators are sometimes given the index "I" (like interaction ) or "D" (like Dirac picture ): or ${\displaystyle |\psi _{\rm {D}}(t)\rangle }$ ${\displaystyle {\hat {A}}_{\rm {D}}(t)\,.}$

history

The interaction picture was introduced into quantum mechanics by Paul Dirac in 1926. In connection with quantum electrodynamics , the interaction picture was also introduced by Tomonaga , Dirac and (in an unpublished work as a student at the City College of New York ) by Julian Schwinger (1934). The treatment of the relativistic quantum field theory in the interaction picture with second quantization found its way into the standard textbooks.

Assumptions

The following assumptions apply to the interaction diagram:

• The Hamilton operator of the system is given by , where ${\displaystyle {\hat {H}}={\hat {H}}_{0}+{\hat {H}}_{1}}$
  • ${\displaystyle {\hat {H}}_{0}=\operatorname {const} }$ is the time-independent Hamilton operator of the undisturbed system
  • ${\displaystyle {\hat {H}}_{1}}$describes the disturbance caused by the interaction , which can be time-dependent.

It can be useful to bring about such a formal splitting of the Hamilton operator even if there is no interaction.

• States are time-dependent:, their dynamics are described by the adapted Schrödinger equation . ${\displaystyle |\psi \rangle =|\psi (t)\rangle }$
• Operators are also time-dependent:, their dynamics are given by the adapted Heisenberg equation of motion . ${\displaystyle {\hat {A}}={\hat {A}}(t)}$
• Only certain calculations are easier to carry out in the Dirac image. The best example here is the derivation of the time-dependent perturbation theory .

description

The basic idea of ​​the interaction picture is to put the temporal development of the system, which is caused by, into the temporal dependence of the operators, while the temporal dependence caused by enters into the development of the state. ${\displaystyle {\hat {H}}_{0}}$ ${\displaystyle {\hat {H}}_{1}}$

For this purpose, two time evolution operators are defined:

• the "normal", with which - as explained in the time evolution operator - is defined: ${\displaystyle {\hat {H}}}$

${\displaystyle {\hat {U}}(t,t_{0})={\hat {T}}\left[\exp \left(-{\frac {i}{\hbar }}\int _{t_{0}}^{t}{\hat {H}}(t^{\prime })dt^{\prime }\right)\right]}$

with the timing operator ${\displaystyle {\hat {T}}}$

• the time evolution operator generated only by : ${\displaystyle {\hat {H}}_{0}}$

${\displaystyle {\hat {U}}_{0}(t,t_{0})=\exp \left(-{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})\right).}$

The expected value  a of the operator must be the same in all three images (Heisenberg image: index , Schrödinger image: index , Dirac): ${\displaystyle {\hat {A}}}$ ${\displaystyle _{H}}$ ${\displaystyle _{S}}$

${\displaystyle a=\langle \psi _{\text{S}}(t)|{\hat {A}}_{\text{S}}(t)|\psi _{\text{S}}(t)\rangle =\langle \psi _{\text{S}}(t)|\underbrace {{\hat {U}}_{0}(t,t_{0}){\hat {U}}_{0}^{\dagger }(t,t_{0})} _{1}\,{\hat {A}}_{\text{S}}(t)\,\underbrace {{\hat {U}}_{0}(t,t_{0}){\hat {U}}_{0}^{\dagger }(t,t_{0})} _{1}\,|\psi _{\text{S}}(t)\rangle }$

${\displaystyle a=\langle \underbrace {{\hat {U}}_{0}^{\dagger }(t,t_{0})\psi _{\text{S}}(t)} _{\psi _{\text{D}}(t)}|\underbrace {{\hat {U}}_{0}^{\dagger }(t,t_{0})\,{\hat {A}}_{\text{S}}(t)\,{\hat {U}}_{0}(t,t_{0})} _{{\hat {A}}_{\text{D}}(t)}\,|\underbrace {{\hat {U}}_{0}^{\dagger }(t,t_{0})\psi _{\text{S}}(t)} _{\psi _{\text{D}}(t)}\rangle =\langle \psi _{\text{D}}(t)|{\hat {A}}_{\text{D}}(t)|\psi _{\text{D}}(t)\rangle }$

As in the Heisenberg picture, the time-dependent operator is given by: ${\displaystyle {\hat {A}}_{\rm {D}}(t)}$

${\displaystyle {\begin{aligned}{\hat {A}}_{\rm {D}}(t)&={\hat {U}}_{0}^{\dagger }(t,t_{0})\,{\hat {A}}_{\rm {S}}(t)\,{\hat {U}}_{0}(t,t_{0})\\&={\rm {e}}^{{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}\,{\hat {A}}_{\rm {S}}(t)\,{\rm {e}}^{-{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}\end{aligned}}}$

The time-dependent state can only be defined indirectly - via the reduction of the state that completely describes the dynamics (in the Schrödinger picture) by the part of its time development caused by: ${\displaystyle |\psi _{\rm {D}}(t)\rangle }$ ${\displaystyle |\psi _{\rm {S}}(t)\rangle }$ ${\displaystyle {\hat {H}}_{0}}$

${\displaystyle |\psi _{\rm {D}}(t)\rangle ={\hat {U}}_{0}^{\dagger }(t,t_{0})\,|\psi _{\rm {S}}(t)\rangle ={\rm {e}}^{{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}\,|\psi _{\rm {S}}(t)\rangle }$

This can be used to define the operator : ${\displaystyle {\hat {H}}_{1{\rm {D}}}(t)}$

${\displaystyle {\hat {H}}_{1{\rm {D}}}(t)={\hat {U}}_{0}^{\dagger }(t,t_{0})\,{\hat {H}}_{1{\rm {S}}}(t)\,{\hat {U}}_{0}(t,t_{0})={\rm {e}}^{{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}\,{\hat {H}}_{1{\rm {S}}}(t)\,{\rm {e}}^{-{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}}$

The time-independent part of the Hamilton operator is identical in the interaction picture to that in the Schrödinger picture: ${\displaystyle {\hat {H}}_{0}}$

${\displaystyle {\hat {H}}_{0{\rm {D}}}(t)={\hat {H}}_{0{\rm {S}}}}$

The dynamics of the states are described (similar to the Schrödinger picture) by the equation:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi _{\rm {D}}(t)\rangle ={\hat {H}}_{1{\rm {D}}}(t)\,|\psi _{\rm {D}}(t)\rangle }$

The dynamics of the operators is described (as in the Heisenberg picture) by the Heisenberg equation of motion, with the non-time-dependent Hamilton operator that describes the undisturbed system: ${\displaystyle {\hat {H}}_{0}}$

${\displaystyle i\hbar {\frac {{\rm {d}}{\hat {A}}_{\rm {D}}}{{\rm {d}}t}}=\left[{\hat {A}}_{\rm {D}}(t),{\hat {H}}_{0}\right]+i\hbar {\frac {\partial {\hat {A}}_{\rm {D}}}{\partial t}}}$

With the Dirac picture goes into the Heisenberg picture. ${\displaystyle {\hat {H}}_{1{\rm {S}}}={\hat {H}}_{1{\rm {D}}}=0}$

At the time, all three images match: ${\displaystyle t_{0}}$

${\displaystyle {\hat {A}}_{\text{D}}(t_{0})={\hat {A}}_{\text{H}}(t_{0})={\hat {A}}_{\text{S}}(t_{0})}$

${\displaystyle |\psi _{\text{D}}(t_{0})\rangle =|\psi _{\text{H}}(t_{0})\rangle =|\psi _{\text{S}}(t_{0})\rangle }$

Derivation of the equations of motion

In preparation, the time derivatives of and are determined: ${\displaystyle {\hat {U}}_{0}}$ ${\displaystyle {\hat {U}}_{0}^{\dagger }}$

${\displaystyle {\begin{aligned}&{\frac {\partial }{\partial t}}{\hat {U}}_{0}(t,t_{0})={\frac {\partial }{\partial t}}\operatorname {e} ^{-{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}=\operatorname {e} ^{-{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}\left(-{\frac {i}{\hbar }}{\hat {H}}_{0}\right)=-{\frac {i}{\hbar }}{\hat {U}}_{0}(t,t_{0})\,{\hat {H}}_{0}=-{\frac {i}{\hbar }}{\hat {H}}_{0}\,{\hat {U}}_{0}(t,t_{0})\\&{\frac {\partial }{\partial t}}{\hat {U}}_{0}^{\dagger }(t,t_{0})={\frac {\partial }{\partial t}}\operatorname {e} ^{{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}=\operatorname {e} ^{{\frac {i}{\hbar }}{\hat {H}}_{0}(t-t_{0})}\left({\frac {i}{\hbar }}{\hat {H}}_{0}\right)={\frac {i}{\hbar }}{\hat {U}}_{0}^{\dagger }(t,t_{0})\,{\hat {H}}_{0}={\frac {i}{\hbar }}{\hat {H}}_{0}\,{\hat {U}}_{0}^{\dagger }(t,t_{0})\end{aligned}}}$

Equation of motion for the states:

${\displaystyle {\begin{aligned}i\hbar {\frac {\partial }{\partial t}}|\psi _{\text{D}}(t)\rangle &=i\hbar {\frac {\partial }{\partial t}}{\hat {U}}_{0}^{\dagger }(t,t_{0})\,|\psi _{\text{S}}(t)\rangle =\underbrace {\left(i\hbar {\frac {\partial }{\partial t}}{\hat {U}}_{0}^{\dagger }(t,t_{0})\right)} _{{\hat {U}}_{0}^{\dagger }(t,t_{0})(-{\hat {H}}_{0})}\,|\psi _{\text{S}}(t)\rangle +{\hat {U}}_{0}^{\dagger }(t,t_{0})\underbrace {\left(i\hbar {\frac {\partial }{\partial t}}\,|\psi _{\text{S}}(t)\rangle \right)} _{({\hat {H}}_{0}+{\hat {H}}_{1{\rm {S}}})|\psi _{\text{S}}(t)\rangle }\\&={\hat {U}}_{0}^{\dagger }(t,t_{0}){\hat {H}}_{1{\text{S}}}(t)\,|\psi _{\text{S}}(t)\rangle =\underbrace {{\hat {U}}_{0}^{\dagger }(t,t_{0}){\hat {H}}_{1{\text{S}}}(t){\hat {U}}_{0}(t,t_{0})} _{{\hat {H}}_{1{\rm {D}}}(t)}\underbrace {{\hat {U}}_{0}^{\dagger }(t,t_{0})\,|\psi _{\text{S}}(t)\rangle } _{|\psi _{\text{D}}(t)\rangle }={\hat {H}}_{1D}(t)|\psi _{\text{D}}(t)\rangle \end{aligned}}}$

Equation of motion for the operators:

${\displaystyle {\begin{aligned}i\hbar {\frac {{\text{d}}{\hat {A}}_{\text{D}}}{{\text{d}}t}}&=i\hbar {\frac {\text{d}}{{\text{d}}t}}\left({\hat {U}}_{0}^{\dagger }\,{\hat {A}}_{\text{S}}\,{\hat {U}}_{0}\right)=\underbrace {\left(i\hbar {\frac {\text{d}}{{\text{d}}t}}{\hat {U}}_{0}^{\dagger }\,\right)} _{-{\hat {H}}_{0}{\hat {U}}_{0}^{\dagger }}{\hat {A}}_{\text{S}}\,{\hat {U}}_{0}+{\hat {U}}_{0}^{\dagger }\,{\hat {A}}_{\text{S}}\underbrace {\left(i\hbar {\frac {\text{d}}{{\text{d}}t}}\,{\hat {U}}_{0}\right)} _{{\hat {U}}_{0}{\hat {H}}_{0}}+i\hbar \underbrace {{\hat {U}}_{0}^{\dagger }\left({\frac {\partial }{\partial t}}{\hat {A}}_{\text{S}}\right)\,{\hat {U}}_{0}} _{\frac {\partial {\hat {A}}_{\text{D}}}{\partial t}}\\&=-{\hat {H}}_{0}\underbrace {{\hat {U}}_{0}^{\dagger }\,{\hat {A}}_{\text{S}}\,{\hat {U}}_{0}} _{{\hat {A}}_{\text{D}}}+\underbrace {{\hat {U}}_{0}^{\dagger }\,{\hat {A}}_{\text{S}}\,{\hat {U}}_{0}} _{{\hat {A}}_{\text{D}}}{\hat {H}}_{0}+i\hbar {\frac {\partial {\hat {A}}_{\text{D}}}{\partial t}}=\left[{\hat {A}}_{\text{D}},{\hat {H}}_{0}\right]+i\hbar {\frac {\partial {\hat {A}}_{\text{D}}}{\partial t}}\end{aligned}}}$

literature

• Nolting: Basic course in theoretical physics. Vol. 5/1: Quantum Mechanics . Springer, Berlin
• Cohen-Tannoudji: Quantum Mechanics 1/2 . de Gruyter, Berlin

Individual evidence

1. ↑ Dirac On the theory of quantum mechanics , Proc. Roy. Soc. A 112, 1926, 661; he also used it in Dirac The quantum theory of the emission and absorption of radiation , Proc. Roy. Soc. A 114, 1927, 243. Historical information based on the representation in Charles Enz Not time to be brief. A scientific biography of Wolfgang Pauli , Oxford University Press 2002, p. 176
2. ↑ Tomonaga On a relativistically invariant formulation of the quantum theory of wave fields , Progress of Theoretical Physics, 1, 1946, 27
3. ↑ Dirac, Wladimir Fock , Boris Podolsky On Quantum Electrodynamics , Phys. Z. Soviet Union, 2, 1932, 468
4. ↑ Mehra, Milton Climbing the Mountain. The Scientific Biography of Julian Schwinger , Oxford University Press 2000, p. 14. He later used it in his first published work on quantum field theory (starting with Schwinger Quantum Electrodynamics I , Physical Review, 74, 1948, 1439)