Heisenberg picture

Short description: Formulation of quantum mechanics

Part of a series on

Quantum mechanics
[math]\displaystyle{ i \hbar \frac{\partial}{\partial t} | \psi (t) \rangle = \hat{H} | \psi (t) \rangle }[/math]

Schrödinger equation

* Introduction
* Glossary
* History

Background

* Classical mechanics
* Old quantum theory
* Bra–ket notation

* Hamiltonian
* Interference

Fundamentals

* Coherence
* Decoherence
* Complementarity
* Energy level
* Entanglement
* Hamiltonian
* Uncertainty principle
* Ground state
* Interference
* Measurement
* Nonlocality
* Observable
* Operator
* Quantum
* Quantum fluctuation
* Quantum foam
* Quantum levitation
* Quantum number
* Quantum noise
* Quantum realm
* Quantum state
* Quantum system
* Quantum teleportation
* Qubit
* Spin
* Superposition
* Symmetry
* (Spontaneous) symmetry breaking
* Vacuum state
* Wave propagation
* Wave function
* Wave function collapse
* Wave–particle duality
* Matter wave

Effects

* Zeeman effect
* Stark effect
* Aharonov–Bohm effect
* Landau quantization
* Quantum Hall effect
* Quantum Zeno effect
* Quantum tunnelling
* Photoelectric effect
* Casimir effect

Experiments

* Afshar
* Bell's inequality
* Davisson–Germer
* Double-slit
* Elitzur–Vaidman
* Franck–Hertz
* Leggett–Garg inequality
* Mach–Zehnder
* Popper

* Quantum eraser (delayed-choice)

* Schrödinger's cat
* Quantum suicide and immortality
* Stern-Gerlach
* Wheeler's delayed-choice

Formulations

* Overview

* Heisenberg
* Interaction
* Matrix
* Phase-space
* Schrödinger
* Sum-over-histories (path-integral)
* Hellmann–Feynman theorem

Equations

* Dirac
* Klein–Gordon
* Pauli
* Rydberg
* Schrödinger

Interpretations

* Overview

* Consistent histories
* Copenhagen
* de Broglie–Bohm
* Ensemble
* Hidden-variable
* Many-worlds
* Objective collapse
* Bayesian
* Quantum logic
* Relational
* Stochastic
* Scale relativity
* Transactional

Advanced topics

* Quantum annealing
* Quantum chaos
* Quantum computing
* Density matrix
* Quantum field theory
* Fractional quantum mechanics
* Quantum gravity
* Quantum information science
* Quantum machine learning
* Perturbation theory (quantum mechanics)
* Relativistic quantum mechanics
* Scattering theory
* Spontaneous parametric down-conversion
* Quantum statistical mechanics

Scientists

* Aharonov
* Bell
* Blackett
* Bloch
* Bohm
* Bohr
* Born
* Bose
* de Broglie
* Candlin
* Compton
* Dirac
* Davisson
* Debye
* Ehrenfest
* Einstein
* Everett
* Fock
* Fermi
* Feynman
* Glauber
* Gutzwiller
* Heisenberg
* Hilbert
* Jordan
* Kramers
* Pauli
* Lamb
* Landau
* Laue
* Moseley
* Millikan
* Onnes
* Planck
* Rabi
* Raman
* Rydberg
* Schrödinger
* Sommerfeld
* von Neumann
* Weyl
* Wien
* Wigner
* Zeeman
* Zeilinger
* Goudsmit
* Uhlenbeck
* Yang

* v
* t
* e

In physics, the Heisenberg picture or Heisenberg representation[1] 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, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.

It stands in contrast to the Schrödinger picture 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 in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal.

It further serves to define a third, hybrid, picture, the interaction picture.

## Contents

* 1 Mathematical details
* 2 Equivalence of Heisenberg's equation to the Schrödinger equation
* 3 Commutator relations
* 4 Summary comparison of evolution in all pictures
* 5 See also
* 6 References
* 7 External links

## Mathematical details

In the Heisenberg picture of quantum mechanics the state vectors |ψ⟩ do not change with time, while observables A satisfy

[math]\displaystyle{ \frac{d}{dt}A_\text{H}(t)=\frac{i}{\hbar}[H_\text{H}(t),A_\text{H}(t)]+\left( \frac{\partial A_\text{S}}{\partial t} \right)_\text{H} , }[/math]

where "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, H is the Hamiltonian and [·,·] denotes the commutator of two operators (in this case H and A). Taking expectation values automatically yields the Ehrenfest theorem, 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 some sense, the Heisenberg picture is more natural and convenient than the equivalent Schrödinger picture, especially for relativistic theories. Lorentz invariance 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: by simply replacing the commutator above by the Poisson bracket, the Heisenberg equation reduces to an equation in Hamiltonian mechanics.

## 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.

According to Schrödinger's equation, the quantum state at time [math]\displaystyle{ t }[/math] is [math]\displaystyle{ |\psi(t)\rangle = U(t) |\psi(0)\rangle }[/math], where [math]\displaystyle{ U(t)=T e^{-\frac{i}{\hbar}\int_0^t ds H_{{\rm S}} (s)} }[/math]is the time-evolution operator induced by a Hamiltonian [math]\displaystyle{ H_{\rm S}(t) }[/math] that could depend on time, and [math]\displaystyle{ |\psi(0) \rangle }[/math] is the initial state. [math]\displaystyle{ T }[/math] refers to time-ordering, ħ is the reduced Planck constant, and i is the imaginary unit. The expectation value of an observable [math]\displaystyle{ A_{\rm S} (t) }[/math] in the Schrödinger picture, which is a Hermitian linear operator that could also be time-dependent, in the state [math]\displaystyle{ |\psi(t)\rangle }[/math] is given by [math]\displaystyle{ \lang A \rang _t = \lang \psi (t) | A_{\rm S}(t) | \psi(t) \rang. }[/math]

In the Heisenberg picture, the quantum state is assumed to remain constant at its initial value [math]\displaystyle{ |\psi(0) \rangle }[/math], whereas operators evolve with time according to the definition [math]\displaystyle{ A_{\rm H} (t) := U^{\dagger}(t) A_{\rm S} (t) U(t) \, . }[/math] This readily implies [math]\displaystyle{ \langle A \rangle_t =\langle \psi(0) | A_{\rm H}(t) | \psi(0) \rangle }[/math], so the same expectation value can be obtained by working in either picture. The Schrödinger equation for the time-evolution operator is [math]\displaystyle{ \frac{d}{dt} U(t) = -\frac{i}{\hbar} H_{\rm S}(t) U(t) . }[/math] It follows that [math]\displaystyle{ \begin{align} \frac{d}{dt} A_{\rm H} (t) &= \left( \frac{d}{dt} U^\dagger (t) \right) A_{\rm S}(t) U(t) + U^\dagger(t) A_{\rm S} (t) \left( \frac{d}{dt} U (t) \right) + U^\dagger (t) \left(\frac{\partial A_{\rm S}} {\partial t}\right) U(t) \\\ & = \frac{i}{\hbar} U^{\dagger}(t) H_{\rm S}(t) A_{{\rm S}} (t) U(t) - \frac{i}{\hbar} U^{\dagger}(t) A_{\rm S}(t) H_{\rm S} (t) U(t) + U^{\dagger}(t) \left(\frac{\partial A_{\rm S}}{\partial t}\right) U(t)\\\ & = \frac{i}{\hbar} U^{\dagger}(t) H_{\rm S} (t) U(t) U^\dagger(t) A_{\rm S}(t) U(t) - \frac{i}{\hbar} U^{\dagger}(t) A_{\rm S} (t) U(t) U^\dagger(t) H_{\rm S}(t) U(t) + \left(\frac{\partial A_{\rm S} }{\partial t}\right)_{\rm H}\\\ & = \frac{i}{\hbar} [H_{\rm H}(t), A_{\rm H} (t)] + \left(\frac{\partial A_{\rm S}}{\partial t} \right) _{\rm H} , \end{align} }[/math] where differentiation was carried out according to the product rule. This is Heisenberg's equation of motion. Note that the Hamiltonian that appears in the final line above is the Heisenberg Hamiltonian [math]\displaystyle{ H_{\rm H}(t) }[/math], which may differ from the Schrödinger Hamiltonian [math]\displaystyle{ H_{\rm S} (t) }[/math].

An important special case of the equation above is obtained if the Hamiltonian [math]\displaystyle{ H_{\rm S} }[/math] does not vary with time. Then the time-evolution operator can be written as [math]\displaystyle{ U(t) = e^{-\frac{i} {\hbar} t H_{\rm S}} , }[/math] and hence [math]\displaystyle{ H_{\rm H} \equiv H_{\rm S}\equiv H }[/math] since [math]\displaystyle{ U(t) }[/math] now commutes with [math]\displaystyle{ H }[/math]. Therefore, [math]\displaystyle{ \lang A \rang _t = \lang \psi (0) | e^{\frac i \hbar t H} A_{{\rm S}} (t) e^{-\frac i \hbar t H} | \psi(0) \rang }[/math] and following the previous analyses, [math]\displaystyle{ \begin{align} \frac{d}{dt} A_{\rm H}(t) & = \frac{i}{\hbar} [H,A_{\rm H}(t)]+ e^{\frac i \hbar t H}\left(\frac{\partial A_{\rm S}}{ \partial t} \right) e^{-\frac i \hbar t H}. \end{align} }[/math]

Furthermore, if [math]\displaystyle{ A_{{\rm S}} \equiv A }[/math] is also time-independent, then the last term vanishes and

[math]\displaystyle{ \frac {d}{dt} A_{\rm H}(t)=\frac {i}{\hbar} [H, A_{\rm H} (t)] , }[/math]

where [math]\displaystyle{ A_{{\rm H}}(t) \equiv A(t)= e^{\frac {i} {\hbar} t H} A e^{- \frac {i} {\hbar} t H} }[/math] in this particular case. The equation is solved by use of the standard operator identity,[math]\displaystyle{ {e^B A e^{-B}} = A + [B,A] + \frac{1}{2!} [B,[B,A]] + \frac{1}{3!}[B,[B,[B,A]]] + \cdots\, , }[/math] which implies [math]\displaystyle{ A(t) = A + \frac{i t}{\hbar}[H,A] + \frac{1}{2!}\left(\frac{i t}{\hbar}\right)^2 [H,[H,A]] + \frac{1}{3!} \left(\frac{i t}{\hbar}\right)^3 [H,[H,[H,A]]] + \cdots }[/math]

A similar relation also holds for classical mechanics, the classical limit of the above, given by the correspondence between Poisson brackets and commutators: [math]\displaystyle{ [A,H] \quad \longleftrightarrow \quad i\hbar\\{A,H\\}. }[/math] In classical mechanics, for an A with no explicit time dependence, [math]\displaystyle{ \\{A,H\\} = \frac{dA}{dt}~, }[/math] so again the expression for A(t) is the Taylor expansion around t = 0.

In effect, the initial state of the quantum system has receded from view, and is only considered at the final step of taking specific expectation values or matrix elements of observables that evolved in time according to the Heisenberg equation of motion. A similar analysis applies if the initial state is mixed.

The time evolved state [math]\displaystyle{ |\psi(t)\rangle }[/math] in the Schrödinger picture is sometimes written as [math]\displaystyle{ |\psi_{\rm S}(t)\rangle }[/math] to differentiate it from the evolved state [math]\displaystyle{ |\psi_{\rm I}(t)\rangle }[/math] that appears in the different interaction picture.

## Commutator relations

Commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators. For example, consider the operators x(t1), x(t2), p(t1) and p(t2). The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator, [math]\displaystyle{ H = \frac{p^2}{2m} + \frac{m\omega^2 x^2}{2} , }[/math] the evolution of the position and momentum operators is given by: [math]\displaystyle{ \frac{d}{dt} x(t) = \frac{i}{\hbar} [ H , x(t) ] = \frac {p}{m}, }[/math] [math]\displaystyle{ \frac{d}{dt} p(t) = \frac{i}{\hbar} [ H , p(t) ] = -m \omega^2 x . }[/math]

Note that the Hamiltonian is time independent and hence [math]\displaystyle{ x(t),p(t) }[/math] are the position and momentum operators in the Heisenberg picture. Differentiating both equations once more and solving for them with proper initial conditions, [math]\displaystyle{ \dot{p}(0) = -m \omega^2 x_0 , }[/math] [math]\displaystyle{ \dot{x}(0) = \frac{p_0}{m} , }[/math] leads to [math]\displaystyle{ x(t) = x_0 \cos(\omega t) + \frac{p_0}{\omega m}\sin(\omega t) , }[/math] [math]\displaystyle{ p(t) = p_0 \cos(\omega t) - m \omega x_0 \sin(\omega t) . }[/math]

Direct computation yields the more general commutator relations, [math]\displaystyle{ [x(t_1), x(t_2)] = \frac{i\hbar}{m\omega} \sin\left(\omega t_2 - \omega t_1\right) , }[/math] [math]\displaystyle{ [p(t_1), p(t_2)] = i\hbar m\omega \sin\left(\omega t_2 - \omega t_1\right) , }[/math] [math]\displaystyle{ [x(t_1), p(t_2)] = i\hbar \cos\left(\omega t_2 - \omega t_1\right) . }[/math]

For [math]\displaystyle{ t_1 = t_2 }[/math], one simply recovers the standard canonical commutation relations valid in all pictures.

## Summary comparison of evolution in all pictures

For a time-independent Hamiltonian HS, where H0,S is the free Hamiltonian,

## See also

* Bra–ket notation
* Interaction picture
* Schrödinger picture
* Heisenberg–Langevin equations
* Phase space formulation

## References

1. ↑ "Heisenberg representation". Encyclopedia of Mathematics. http://www.encyclopediaofmath.org/index.php/Heisenberg_representation.

* Cohen-Tannoudji, Claude; Bernard Diu; Frank Laloe (1977). Quantum Mechanics (Volume One). Paris: Wiley. pp. 312–314. ISBN 0-471-16433-X.
* Albert Messiah, 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) pp. 430–431 ISBN:0-471-88702-1
* L.D. Landau, E.M. Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1. Online copy
* R. Shankar (1994); Principles of Quantum Mechanics, Plenum Press, ISBN:978-0306447907.
* J. J. Sakurai (1993); Modern Quantum Mechanics (Revised Edition), ISBN:978-0201539295.

## 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 picture [1]
* 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 Waerden [2]
* The computations for the hydrogen atom in the Heisenberg representation originally from a paper of Pauli [3]

* v
* t
* e

Quantum mechanics

Background|

* Introduction
* History
* timeline
* Glossary
* Classical mechanics
* Old quantum theory

Fundamentals|

* Bra–ket notation
* Casimir effect
* Complementarity
* Density matrix
* Energy level
* ground state
* excited state
* degenerate levels
* Vacuum state
* Zero-point energy
* QED vacuum
* QCD vacuum
* Hamiltonian
* Operator
* Quantum coherence
* Quantum decoherence
* Measurement
* Quantum
* Quantum realm
* Quantum system
* Quantum state
* Quantum number
* Quantum entanglement
* Quantum superposition
* Quantum nonlocality
* Quantum tunnelling
* Quantum levitation
* Quantum fluctuation
* Quantum annealing
* Quantum foam
* Quantum noise
* Heisenberg uncertainty principle
* Spontaneous parametric down-conversion
* Von Neumann entropy
* Spin
* Scattering theory
* Symmetry in quantum mechanics
* Symmetry breaking
* Spontaneous symmetry breaking
* Wave propagation
* Quantum interference
* Wave function
* Wave function collapse
* Wave–particle duality
* Matter wave
* Qubit
* Qutrit
* Observable
* Probability distribution

Formulations|

* Formulations
* Heisenberg
* Interaction
* Matrix mechanics
* Schrödinger
* Path integral formulation
* Phase space

Equations|

* Dirac
* Klein–Gordon
* Pauli
* Rydberg
* Schrödinger

Interpretations|

* Interpretations
* Bayesian
* Consistent histories
* Copenhagen
* de Broglie–Bohm
* Ensemble
* Hidden variables
* Many worlds
* Objective collapse
* Quantum logic
* Relational
* Stochastic
* Transactional
* Cosmological

Experiments|

* Afshar
* Bell's inequality
* Cold Atom Laboratory
* Davisson–Germer
* Delayed-choice quantum eraser
* Double-slit
* Franck–Hertz experiment
* Leggett–Garg inequality
* Mach-Zehnder inter.
* Elitzur–Vaidman
* Popper
* Quantum eraser
* Schrödinger's cat
* Quantum suicide and immortality
* Stern–Gerlach
* Wheeler's delayed choice

Science|

* Q-analog
* List
* Quantum algebra
* Quantum calculus
* Quantum differential calculus
* Quantum geometry
* Quantum group
* Quantum Bayesianism
* Quantum biology
* Quantum chemistry
* Quantum chaos
* Quantum cognition
* Quantum cosmology
* Quantum dynamics
* Quantum economics
* Quantum evolution
* Quantum finance
* Quantum game theory
* Quantum measurement problem
* Quantum mind
* Quantum probability
* Quantum social science
* Quantum stochastic calculus
* Quantum spacetime

Technology|

* Quantum algorithms
* Quantum amplifier
* Quantum cellular automata
* Quantum finite automata
* Quantum electronics
* Quantum logic gates
* Quantum clock
* Quantum channel
* Quantum bus
* Quantum circuit
* Phase qubit
* Matrix isolation
* Quantum dot
* Quantum dot display
* Quantum dot solar cell
* Quantum dot cellular automaton
* Quantum dot single-photon source
* Quantum dot laser
* Quantum well
* Quantum computing
* Timeline
* Quantum cryptography
* Post-quantum cryptography
* Quantum error correction
* Quantum imaging
* Quantum image processing
* Quantum information
* Quantum key distribution
* Quantum machine
* Quantum machine learning
* Quantum metamaterial
* Quantum metrology
* Quantum network
* Quantum neural network
* Quantum optics
* Quantum programming
* Quantum sensors
* Quantum simulator
* Quantum teleportation
* Quantum levitation
* Time travel
* Quantum complexity theory

Extensions|

* Quantum statistical mechanics
* Relativistic quantum mechanics
* Quantum field theory
* Axiomatic quantum field theory
* Quantum field theory in curved spacetime
* Thermal quantum field theory
* Topological quantum field theory
* Local quantum field theory
* Conformal field theory
* Two-dimensional conformal field theory
* Liouville field theory
* History
* Quantum gravity

* Category
* Physics Portal
* Physics WikiProject
* Commons

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Heisenberg picture. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Heisenberg_picture&oldid=3281342"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template