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quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the interaction picture (also known as the Dirac picture after
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
) is an intermediate
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
between the
Schrödinger picture In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
and the
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, bu ...
. Whereas in the other two pictures either the state vector or the
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
carry time dependence, in the interaction picture both carry part of the time dependence of
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
s. The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. Most field-theoretical calculations use the interaction representation because they construct the solution to the many-body Schrödinger equation as the solution to the free-particle problem plus some unknown interaction parts. 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 time-dependent unitary transformations relate operators in one picture to the analogous operators in the others. The interaction picture is a special case of
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
applied to the Hamiltonian and state vectors.


Definition

Operators and state vectors in the interaction picture are related by a change of basis (
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
) to those same operators and state vectors in the Schrödinger picture. To switch into the interaction picture, we divide the Schrödinger picture
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
into two parts: 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, while ''H''1,S contains some harder-to-analyze perturbation to this system. If the Hamiltonian has ''explicit time-dependence'' (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 time-dependent terms with ''H''1,S, leaving ''H''0,S time-independent. We proceed assuming that this is the case. If there ''is'' a context in which it makes sense to have ''H''0,S be time-dependent, then one can proceed by replacing \mathrm^ by the corresponding
time-evolution operator Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
in the definitions below.


State vectors

Let , \psi_\text(t)\rangle = \mathrm^, \psi(0)\rangle be the time-dependent state vector in the Schrödinger picture. A state vector in the interaction picture, , \psi_\text(t)\rangle, is defined with an additional time-dependent unitary transformation.


Operators

An operator in the interaction picture is defined as Note that ''A''S(''t'') will typically not depend on and can be rewritten as just ''A''S. It only depends on if the operator has "explicit time dependence", for example, due to its dependence on an applied external time-varying electric field. Another instance of explicit time dependence may occur when ''A''S(''t'') is a density matrix (see below).


Hamiltonian operator

For the operator H_0 itself, the interaction picture and Schrödinger picture coincide: :H_(t) = \mathrm^ H_ \mathrm^ = H_. This is easily seen through the fact that operators
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with differentiable functions of themselves. This particular operator then can be called H_0 without ambiguity. For the perturbation Hamiltonian H_, however, :H_(t) = \mathrm^ H_ \mathrm^, where the interaction-picture perturbation Hamiltonian becomes a time-dependent Hamiltonian, unless 'H''1,S, ''H''0,S= 0. It is possible to obtain the interaction picture for a time-dependent Hamiltonian ''H''0,S(''t'') as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by ''H''0,S(''t''), or more explicitly with a time-ordered exponential integral.


Density matrix

The
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
can be shown to transform to the interaction picture in the same way as any other operator. In particular, let and be the density matrices in the interaction picture and the Schrödinger picture respectively. If there is probability to be in the physical state , ''ψ''''n''⟩, then :\begin \rho_\text(t) &= \sum_n p_n(t) \left, \psi_(t)\right\rang \left\lang \psi_(t)\ \\ &= \sum_n p_n(t) \mathrm^ \left, \psi_(t)\right\rang \left\lang \psi_(t)\ \mathrm^ \\ &= \mathrm^ \rho_\text(t) \mathrm^. \end


Time-evolution


Time-evolution of states

Transforming the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
into the interaction picture gives : \mathrm \hbar \frac , \psi_\text(t)\rang = H_(t) , \psi_\text(t)\rang, which states that in the interaction picture, a quantum state is evolved by the interaction part of the Hamiltonian as expressed in the interaction picture. A proof is given in Fetter and Walecka.


Time-evolution 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 : \mathrm\hbar\fracA_\text(t) = _\text(t),H_ In the interaction picture the operators evolve in time like the operators in the
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, bu ...
with the Hamiltonian .


Time-evolution of the density matrix

The evolution of the
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
in the interaction picture is : \mathrm\hbar \frac \rho_\text(t) = _(t), \rho_\text(t) in consistency with the Schrödinger equation in the interaction picture.


Expectation values

For a general operator A, the expectation value in the interaction picture is given by : \langle A_\text(t) \rangle = \langle \psi_\text(t) , A_\text(t) , \psi_\text(t) \rangle = \langle \psi_\text(t) , e^ e^ \, A_\text \, e^ e^ , \psi_\text(t) \rangle = \langle A_\text(t) \rangle. Using the density-matrix expression for expectation value, we will get :\langle A_\text(t) \rangle = \operatorname\big(\rho_\text(t) \, A_\text(t)\big).


Schwinger–Tomonaga equation

The term interaction representation was invented by Schwinger. In this new mixed representation the state vector is no longer constant in general, but it is constant if there is no coupling between fields. The change of representation leads directly to the Tomonaga–Schwinger equation: :ihc \frac = \hat(x)\Psi(\sigma) : \hat(x) = - \frac j_(x) A^(x) Where the Hamiltonian in this case is the QED interaction Hamiltonian, but it can also be a generic interaction, and \sigma is a space like surface that is passing through the point x. The derivative formally represents a variation over that surface given x fixed. It is difficult to give a precise mathematical formal interpretation of this equation. This approach is called by Schwinger the differential and field approach opposed to the integral and particle approach of the Feynman diagrams. Schwinger and Feynman The core idea is that if the interaction has a small coupling constant (i.e. in the case of electromagnetism of the order of the fine structure constant) successive perturbative terms will be powers of the coupling constant and therefore smaller.


Use

The purpose of the interaction picture is to shunt all the time dependence due to ''H''0 onto the operators, thus allowing them to evolve freely, and leaving only ''H''1,I to control the time-evolution 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 utilizing the interaction picture, one can use
time-dependent perturbation theory In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for whic ...
to find the effect of ''H''1,I, e.g., in the derivation of
Fermi's golden rule In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of ...
, or the
Dyson series In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams. ...
in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
: in 1947,
Shin'ichirō Tomonaga , usually cited as Sin-Itiro Tomonaga in English, was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian ...
and
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
appreciated that covariant perturbation theory could be formulated elegantly in the interaction picture, since
field operators In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible. Historically, this was not quite ...
can evolve in time as free fields, even in the presence of interactions, now treated perturbatively in such a Dyson series.


Summary comparison of evolution in all pictures

For a time-independent Hamiltonian ''H''S, where ''H''0,S is the free Hamiltonian,


References


Further reading

* *


See also

*
Bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
*
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
*
Haag's theorem While working on the mathematical physics of an interacting, relativistic, quantum field theory, Rudolf Haag developed an argument against the existence of the interaction picture, a result now commonly known as Haag’s theorem. Haag’s origina ...
{{Quantum mechanics topics Quantum mechanics es:Imagen de evolución temporal