 Interaction picture

Quantum mechanics
Introduction
Mathematical formulationsFundamental conceptsExperimentsDoubleslit · Davisson–Germer
Stern–Gerlach · Bell's inequality
Popper · Schrödinger's cat
Elitzur–Vaidman bomb tester
Quantum eraser
Delayed choice quantum eraser
Wheeler's delayed choiceFormulationsEquationsInterpretationsde Broglie–Bohm
Consciousnesscaused
Consistent histories · Copenhagen
Ensemble · Hidden variables
Manyworlds · Objective collapse
Pondicherry · Quantum logic
Relational · Stochastic
TransactionalScientistsBell · Bohm · Bohr · Born · Bose
de Broglie · Dirac · Ehrenfest
Everett · Feynman · Heisenberg
Jordan · Kramers · von Neumann
Pauli · Planck · Schrödinger
Sommerfeld · Wien · WignerIn quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables.
Equations that include operators acting at different times, which hold in the interaction picture, don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because timedependent unitary transformations relate operators in one picture to the analogous operators in the others. Not all textbooks and articles make explicit which picture each operator comes from, which can lead to confusion and mistakes.
Contents
Definition
Operators and state vectors in the interaction picture are related by a change of basis (unitary transformation) to those same operators and state vectors in the Schrödinger picture.
To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts, H_{S} = H_{0,S} + H_{1,S}. (Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so that H_{0,S} is well understood and exactly solvable, and H_{1,S} contains some hardertoanalyze perturbation to this system.)
If the Hamiltonian has explicit timedependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly timedependent terms with H_{1,S}, leaving H_{0,S} timeindependent. We will proceed assuming that this is the case. (If there is a context in which it makes sense to have H_{0,S} be timedependent, then one can proceed by replacing by the corresponding timeevolution operator in the definitions below.)
State vectors
A state vector in the interaction picture is defined as^{[1]}
(where is the same state vector in the Schrödinger picture.)
Operators
An operator in the interaction picture is defined as
(Note that the A_{S}(t) will typically not depend on t, and can be rewritten as just A_{S}. It only depends on t if the operator has "explicit time dependence", for example due to its dependence on an applied, external, timevarying electric field.)
Hamiltonian operator
For the operator H_{0} itself, the interaction picture and Schrödinger picture are the same:
(this can be proved using the fact that operators commute with differentiable functions of themselves.) This particular operator can thus be called H_{0} with no ambiguity.
For the perturbation Hamiltonian H_{1,I}, we have:
where the interaction picture perturbation Hamiltonian becomes a timedependent Hamiltonian (unless [H_{1,s},H_{0,s}] = 0).
It is possible to obtain the interaction picture for a timedependent Hamiltonian H_{0,s}(t) as well but the exponentials need to be replaced by the unitary propagator for the evolution due to H_{0,s}(t) or more explicitly with a timeordered exponential integral.
Density matrix
The density matrix can be shown to transform to the interaction picture in the same way as any other operator. In particular, let ρ_{I} and ρ_{S} be the density matrix in the interaction picture and the Schrödinger picture, respectively. If there is probability p_{n} to be in the physical state , then
Timeevolution equations in the interaction picture
Timeevolution of states
Transforming the Schrödinger equation into the interaction picture gives:
This equation is referred to as the SchwingerTomonaga equation.
Timeevolution of operators
If the operator A_{S} is time independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for A_{I}(t) is given by:
In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian H' = H_{0}.
Timeevolution of the density matrix
Transforming the SchwingerTomonaga equation into the language of the density matrix (or equivalently, transforming the von Neumann equation into the interaction picture) gives:
Use of interaction picture
The purpose of the interaction picture is to shunt all the time dependence due to H_{0} onto the operators, leaving only H_{1, I} affecting the timedependence of the state vectors.
The interaction picture is convenient when considering the effect of a small interaction term, H_{1, S}, being added to the Hamiltonian of a solved system, H_{0, S}. By switching into the interaction picture, you can use timedependent perturbation theory to find the effect of H_{1, I}.
References
 Townsend, John S. (2000). A Modern Approach to Quantum Mechanics, 2nd ed.. Sausalito, CA: University Science Books. ISBN 1891389130.
 ^ The Interaction Picture, lecture notes from New York University
See also
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
interaction picture — sąveikos atvaizdas statusas T sritis fizika atitikmenys: angl. interaction picture vok. Wechselwirkungsbild, n rus. представление взаимодействия, n pranc. représentation d’interaction, f … Fizikos terminų žodynas
Quartic interaction — In quantum field theory, a quartic interaction is a theory about a scalar field phi; which contains an interaction term phi^4, and is considered by many teachers and students to be the simplest example of interacting fields. This theory consists… … Wikipedia
Heisenberg picture — In physics, the Heisenberg picture is that formulation of quantum mechanics where the operators (observables and others) are time dependent and the state vectors are time independent. It stands in contrast to the Schrödinger picture in which… … Wikipedia
Schrödinger picture — In quantum mechanics, a state function is a linear combination (a superposition) of eigenstates. In the Schrödinger picture, the state of a system evolves with time, where the evolution for a closed quantum system is brought about by a unitary… … Wikipedia
représentation d’interaction — sąveikos atvaizdas statusas T sritis fizika atitikmenys: angl. interaction picture vok. Wechselwirkungsbild, n rus. представление взаимодействия, n pranc. représentation d’interaction, f … Fizikos terminų žodynas
Weak interaction — Standard model of particle physics … Wikipedia
Exchange interaction — In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap.… … Wikipedia
Valency interaction formula — The Valency Interaction Formula, or VIF is a method for drawing molecular structural formulas based in quantum mechanics. The mathematical basis for VIF was formulated by Oktay Sinanoglu in a series of five papers published in 1984. [ Sinanoğlu,… … Wikipedia
Rotating wave approximation — The rotating wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian which oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic… … Wikipedia
Mathematical formulation of quantum mechanics — Quantum mechanics Uncertainty principle … Wikipedia