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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the Heisenberg picture (also called the Heisenberg representation) is a
formulation Formulation is a term used in various senses in various applications, both the material and the abstract or formal. Its fundamental meaning is the putting together of components in appropriate relationships or structures, according to a formul ...
(largely due to
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
in 1925) of
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, ...
in which 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 ...
(
observables 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 p ...
and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory. It stands in contrast to 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 ma ...
in which the operators are constant, instead, and the states evolve in time. The two pictures only differ by a basis change with respect to time-dependency, which corresponds to the difference between active and passive transformations. The Heisenberg picture is the formulation of
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal. It further serves to define a third, hybrid, picture, the
interaction picture In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state ...
.


Mathematical details

In the Heisenberg picture of quantum mechanics the state vectors , ''ψ''⟩ do not change with time, while observables satisfy where "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, is the Hamiltonian and denotes the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of two operators (in this case and ). Taking expectation values automatically yields the
Ehrenfest theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of the ...
, featured in the
correspondence principle In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it say ...
. By the
Stone–von Neumann theorem In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named afte ...
, the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. In some sense, the Heisenberg picture is more natural and convenient than the equivalent Schrödinger picture, especially for relativistic theories.
Lorentz invariance In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation In physics, the Lorentz transformations are a six-parameter famil ...
is manifest in the Heisenberg picture, since the state vectors do not single out the time or space. This approach also has a more direct similarity to
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
: by simply replacing the commutator above by the
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
, the Heisenberg equation reduces to an equation in
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
.


Equivalence of Heisenberg's equation to the Schrödinger equation

For the sake of pedagogy, the Heisenberg picture is introduced here from the subsequent, but more familiar,
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 ma ...
. The
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of an observable ''A'', which is a
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
, for a given Schrödinger state , ''ψ''(''t'')⟩, is given by \lang A \rang _t = \lang \psi (t) , A , \psi(t) \rang. In the Schrödinger picture, the state , ''ψ''(''t'')⟩ at time is related to the state , ''ψ''(0)⟩ at time 0 by a unitary
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 di ...
, , , \psi(t)\rangle = U(t) , \psi(0)\rangle. In the Heisenberg picture, all state vectors are considered to remain constant at their initial values , ''ψ''(0)⟩, whereas operators evolve with time according to A(t) := U^(t) A U(t) \, . The Schrödinger equation for the time-evolution operator is \frac U(t) = -\frac U(t) where ''H'' is the Hamiltonian and ''ħ'' is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
. It now follows that \begin \frac A(t) & = \frac U^(t) H A U(t) + U^(t) \left(\frac\right) U(t) + \frac U^(t) A (-H) U(t) \\ & = \frac U^(t) H U(t) U^(t) A U(t) + U^(t) \left(\frac\right) U(t) - \fracU^(t) A U(t) U^(t) H U(t) \\ & = \frac \left( H(t) A(t) - A(t) H(t) \right) + U^(t) \left(\frac\right) U(t) , \end where differentiation was carried out according to the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
. Note that the Hamiltonian that appears in the final line above is the Heisenberg Hamiltonian ''H''(''t''), which may differ from the Schrödinger Hamiltonian. An important special case of the equation above is obtained if the Hamiltonian does not vary with time. Then the time-evolution operator can be written as U(t) = e^ , Therefore, \lang A \rang _t = \lang \psi (0) , e^ A e^ , \psi(0) \rang . and, \begin \frac A(t) & = \frac H e^ A e^ + e^ \left(\frac\right) e^ + \frac e^ A \cdot (-H) e^ \\ & = \frac e^ \left( H A - A H \right) e^ + e^ \left(\frac\right) e^ \\ & = \frac \left( H A(t) - A(t) H \right) + e^ \left(\frac\right) e^ . \end Here is the time derivative of the initial ''A'', not the ''A''(''t'') operator defined. The last equation holds since commutes with . The equation is solved by the ''A''(''t'') defined above, as evident by use of the standard operator identity, = A + ,A+ \frac ,[B,A_+_\frac[B,_,[B,A.html" ;"title=",A.html" ;"title=",[B,A">,[B,A + \frac[B, ,[B,A">,A.html" ;"title=",[B,A">,[B,A + \frac[B, ,[B,A+ \cdots\, , which implies A(t) = A + \frac[H,A] + \frac\left(\frac\right)^2 [H,[H,A + \frac \left(\frac\right)^3 [H,[H,[H,A] + \cdots This relation also holds for classical mechanics, the classical limit of the above, given the correspondence between
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
s and
commutators In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
, ,H\quad \longleftrightarrow \quad i\hbar\ In classical mechanics, for an ''A'' with no explicit time dependence, \ = \frac~, so again the expression for ''A''(''t'') is the Taylor expansion around ''t'' = 0. In effect, the arbitrary rigid Hilbert space basis , ''ψ''(0)⟩ has receded from view, and is only considered at the final step of taking specific expectation values or matrix elements of observables.


Commutator relations

Commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators. For example, consider the operators and . The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator, H = \frac + \frac , the evolution of the position and momentum operators is given by: \frac x(t) = \frac H , x(t) = \frac , \frac p(t) = \frac H , p(t) = -m \omega^2 x . Differentiating both equations once more and solving for them with proper initial conditions, \dot(0) = -m \omega^2 x_0 , \dot(0) = \frac , leads to x(t) = x_0 \cos(\omega t) + \frac\sin(\omega t) , p(t) = p_0 \cos(\omega t) - m \omega x_0 \sin(\omega t) . Direct computation yields the more general commutator relations, (t_1), x(t_2)= \frac \sin\left(\omega t_2 - \omega t_1\right) , (t_1), p(t_2)= i\hbar m\omega \sin\left(\omega t_2 - \omega t_1\right) , (t_1), p(t_2)= i\hbar \cos\left(\omega t_2 - \omega t_1\right) . For t_1 = t_2, one simply recovers the standard canonical commutation relations valid in all pictures.


Summary comparison of evolution in all pictures

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


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. Mathem ...
*
Interaction picture In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state ...
*
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 ma ...
* Heisenberg–Langevin equations *
Phase space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...


References

* *
Albert Messiah Albert Messiah (23 September 1921, Nice – 17 April 2013, Paris) was a French physicist. He studied at the Ecole Polytechnique. He spent the Second World War in the Free France forces: he embarked on 22 June 1940 at Saint-Jean-de-Luz for Engla ...
, 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. * Merzbacher E., ''Quantum Mechanics'' (3rd ed., John Wiley 1998) p. 430-1
Online copy
* R. Shankar (1994); ''Principles of Quantum Mechanics'', Plenum Press, . *
J. J. Sakurai was a Japanese-American particle physicist and theorist. While a graduate student at Cornell University, Sakurai independently discovered the V-A theory of weak interactions. He authored the popular graduate text ''Modern Quantum Mechanics'' ( ...
(1993); ''
Modern Quantum Mechanics ''Modern Quantum Mechanics'', often called ''Sakurai'' or ''Sakurai and Napolitano'', is a standard graduate-level quantum mechanics textbook written originally by J. J. Sakurai and edited by San Fu Tuan in 1985, with later editions coauthored ...
'' (Revised Edition), .


External links


Pedagogic Aides to Quantum Field Theory
Click on the link for Chap. 2 to find an extensive, simplified introduction to the Heisenberg picture. * Some expanded derivations and an example of the harmonic oscillator in the Heisenberg pictur

* The original Heisenberg paper translated (although difficult to read, it contains an example for the anharmonic oscillator): Sources of Quantum mechanics B.L. Van Der Waerde

* The computations for the hydrogen atom in the Heisenberg representation originally from a paper of Paul

{{DEFAULTSORT:Heisenberg Picture Quantum mechanics Werner Heisenberg