In
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 r ...
, the Heisenberg picture (also called the Heisenberg representation
) is a
formulation (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 breakthrough paper. In the subsequent series ...
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 physi ...
and others) incorporate a dependency on time, but the
state vectors are time-independent, an arbitrary fixed
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
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 may ...
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 transformation
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
s. 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.
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
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 ...
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, a ...
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 th ...
, featured in the
correspondence principle.
By the
Stone–von Neumann theorem, 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
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 breakthrough paper. In the subsequent series ...
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. A Lorentz scalar may be generated from e.g., the scalar product of ...
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. Th ...
, 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 may ...
.
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 ...
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 mapping V \to W between two vector spaces that pre ...
, for a given Schrödinger state , ''ψ''(''t'')⟩, is given by
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 disc ...
, ,
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
The Schrödinger equation for the time-evolution operator is
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 equivale ...
.
It now follows that
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
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 ...
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
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 ...
does not vary with time. Then the time-evolution operator can be written as
Therefore,
and,
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,
which implies
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. Th ...
s and
commutators,
In classical mechanics, for an ''A'' with no explicit time dependence,
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,
the evolution of the position and momentum operators is given by:
Differentiating both equations once more and solving for them with proper initial conditions,
leads to
Direct computation yields the more general commutator relations,
For
, 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
*
Interaction picture
*
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 ...
*
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
Ramamurti Shankar (born April 28, 1947) is the Josiah Willard Gibbs professor of Physics at Yale University, in New Haven, Connecticut.
Education
He received his B. Tech in electrical engineering from the Indian Institute of Technology Madras, I ...
(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 TheoryClick 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