HOME
The Info List - Heisenberg Picture





In physics, the Heisenberg picture (also called the 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
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[edit] In the Heisenberg picture of quantum mechanics the state vectors, ψ(t)〉, do not change with time, while observables A satisfy

d

d t

A ( t ) =

i ℏ

[ H , A ( t ) ] +

(

∂ A

∂ t

)

H

,

displaystyle frac d dt A(t)= frac i hbar [H,A(t)]+left( frac partial A partial t right)_ H ,

where 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
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
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[edit] For the sake of pedagogy, the Heisenberg picture is introduced here from the subsequent, but more familiar, Schrödinger picture. The expectation value of an observable A, which is a Hermitian linear operator, for a given Schrödinger state ψ(t)〉, is given by

⟨ A

t

= ⟨ ψ ( t )

A

ψ ( t ) ⟩ .

displaystyle langle Arangle _ t =langle psi (t)Apsi (t)rangle .

In the Schrödinger picture, the state ψ(t)〉at time t is related to the state ψ(0)〉at time 0 by a unitary time-evolution operator, U(t),

ψ ( t ) ⟩ = U ( t )

ψ ( 0 ) ⟩ .

displaystyle psi (t)rangle =U(t)psi (0)rangle .

If the Hamiltonian does not vary with time, then the time-evolution operator can be written as

U ( t ) =

e

− i H t

/

,

displaystyle U(t)=e^ -iHt/hbar ,

where H is the Hamiltonian and ħ is the reduced Planck constant. Therefore,

⟨ A

t

= ⟨ ψ ( 0 )

e

+ i H t

/

A

e

− i H t

/

ψ ( 0 ) ⟩ .

displaystyle langle Arangle _ t =langle psi (0)e^ +iHt/hbar Ae^ -iHt/hbar psi (0)rangle .

Peg all state vectors to a rigid basis of ψ(0)〉, and then define

A ( t ) :=

e

+ i H t

/

A

e

− i H t

/

  .

displaystyle A(t):=e^ +iHt/hbar Ae^ -iHt/hbar .

It now follows that

d

d

t

A ( t )

=

i ℏ

H

e

i H t

/

A

e

− i H t

/

+

e

+ i H t

/

(

∂ A

∂ t

)

e

− i H t

/

+

i ℏ

e

+ i H t

/

A ⋅ ( − H )

e

− i H t

/

=

i ℏ

e

i H t

/

(

H A − A H

)

e

− i H t

/

+

e

+ i H t

/

(

∂ A

∂ t

)

e

− i H t

/

=

i ℏ

(

H A ( t ) − A ( t ) H

)

+

e

+ i H t

/

(

∂ A

∂ t

)

e

− i H t

/

.

displaystyle begin aligned operatorname d over operatorname d !t A(t)&= i over hbar He^ iHt/hbar Ae^ -iHt/hbar +e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar + i over hbar e^ +iHt/hbar Acdot (-H)e^ -iHt/hbar \&= i over hbar e^ iHt/hbar left(HA-AHright)e^ -iHt/hbar +e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar \&= i over hbar left(HA(t)-A(t)Hright)+e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar .end aligned

Differentiation was according to the product rule, while ∂A/∂t is the time derivative of the initial A, not the A(t) operator defined. The last equation holds since exp(−i H t/ħ) commutes with H. Thus

d

d

t

A ( t ) =

i ℏ

[ H , A ( t ) ] +

e

+ i H t

/

(

∂ A

∂ t

)

e

− i H t

/

,

displaystyle operatorname d over operatorname d !t A(t)= i over hbar [H,A(t)]+e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar ,

and hence emerges the above Heisenberg equation of motion, since the convective functional dependence on x(0) and p(0) converts to the same dependence on x(t), p(t), so that the last term converts to ∂A(t)/∂t . [X, Y] is the commutator of two operators and is defined as [X, Y] := XY − YX. The equation is solved by the A(t) defined above, as evident by use of the standard operator identity,

e

B

A

e

− B

= A + [ B , A ] +

1

2 !

[ B , [ B , A ] ] +

1

3 !

[ B , [ B , [ B , A ] ] ] + ⋯   .

displaystyle e^ B Ae^ -B =A+[B,A]+ frac 1 2! [B,[B,A]]+ frac 1 3! [B,[B,[B,A]]]+cdots .

which implies

A ( t ) = A +

i t

[ H , A ] +

1

2 !

(

i t

)

2

[ H , [ H , A ] ] +

1

3 !

(

i t

)

3

[ H , [ H , [ H , A ] ] ] + …

displaystyle A(t)=A+ frac it hbar [H,A]+ frac 1 2! left( frac it hbar right)^ 2 [H,[H,A]]+ frac 1 3! left( frac it hbar right)^ 3 [H,[H,[H,A]]]+dots

This relation also holds for classical mechanics, the classical limit of the above, given the correspondence between Poisson brackets and commutators,

[ A , H ]

i ℏ

A , H

displaystyle [A,H]quad longleftrightarrow quad ihbar A,H

In classical mechanics, for an A with no explicit time dependence,

A , H

=

d

A

d

t

  ,

displaystyle A,H = frac operatorname d !A operatorname d !t ~,

so again the expression for A(t) is the Taylor expansion around t = 0. In effect, the arbitrary rigid Hilbert space
Hilbert space
basis ψ(0)〉 has receded from view, and is only considered at the very last step of taking specific expectation values or matrix elements of observables. Commutator relations[edit] 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,

H =

p

2

2 m

+

m

ω

2

x

2

2

displaystyle H= frac p^ 2 2m + frac momega ^ 2 x^ 2 2

,

the evolution of the position and momentum operators is given by:

d

d t

x ( t ) =

i ℏ

[ H , x ( t ) ] =

p m

displaystyle d over dt x(t)= i over hbar [H,x(t)]= frac p m

,

d

d t

p ( t ) =

i ℏ

[ H , p ( t ) ] = − m

ω

2

x

displaystyle d over dt p(t)= i over hbar [H,p(t)]=-momega ^ 2 x

.

Differentiating both equations once more and solving for them with proper initial conditions,

p ˙

( 0 ) = − m

ω

2

x

0

,

displaystyle dot p (0)=-momega ^ 2 x_ 0 ,

x ˙

( 0 ) =

p

0

m

,

displaystyle dot x (0)= frac p_ 0 m ,

leads to

x ( t ) =

x

0

cos ⁡ ( ω t ) +

p

0

ω m

sin ⁡ ( ω t )

displaystyle x(t)=x_ 0 cos(omega t)+ frac p_ 0 omega m sin(omega t)

,

p ( t ) =

p

0

cos ⁡ ( ω t ) − m ω

x

0

sin ⁡ ( ω t )

displaystyle p(t)=p_ 0 cos(omega t)-momega !x_ 0 sin(omega t)

.

Direct computation yields the more general commutator relations,

[ x (

t

1

) , x (

t

2

) ] =

i ℏ

m ω

sin ⁡ ( ω

t

2

− ω

t

1

)

displaystyle [x(t_ 1 ),x(t_ 2 )]= frac ihbar momega sin(omega t_ 2 -omega t_ 1 )

,

[ p (

t

1

) , p (

t

2

) ] = i ℏ m ω sin ⁡ ( ω

t

2

− ω

t

1

)

displaystyle [p(t_ 1 ),p(t_ 2 )]=ihbar momega sin(omega t_ 2 -omega t_ 1 )

,

[ x (

t

1

) , p (

t

2

) ] = i ℏ cos ⁡ ( ω

t

2

− ω

t

1

)

displaystyle [x(t_ 1 ),p(t_ 2 )]=ihbar cos(omega t_ 2 -omega t_ 1 )

.

For

t

1

=

t

2

displaystyle t_ 1 =t_ 2

, one simply recovers the standard canonical commutation relations valid in all pictures. Summary comparison of evolution in all pictures[edit]

Evolution Picture

of: Heisenberg Interaction Schrödinger

Ket state constant

ψ

I

( t ) ⟩ =

e

i

H

0 , S

  t

/

ψ

S

( t ) ⟩

displaystyle psi _ I (t)rangle =e^ iH_ 0,S ~t/hbar psi _ S (t)rangle

ψ

S

( t ) ⟩ =

e

− i

H

S

  t

/

ψ

S

( 0 ) ⟩

displaystyle psi _ S (t)rangle =e^ -iH_ S ~t/hbar psi _ S (0)rangle

Observable

A

H

( t ) =

e

i

H

S

  t

/

A

S

e

− i

H

S

  t

/

displaystyle A_ H (t)=e^ iH_ S ~t/hbar A_ S e^ -iH_ S ~t/hbar

A

I

( t ) =

e

i

H

0 , S

  t

/

A

S

e

− i

H

0 , S

  t

/

displaystyle A_ I (t)=e^ iH_ 0,S ~t/hbar A_ S e^ -iH_ 0,S ~t/hbar

constant

Density matrix constant

ρ

I

( t ) =

e

i

H

0 , S

  t

/

ρ

S

( t )

e

− i

H

0 , S

  t

/

displaystyle rho _ I (t)=e^ iH_ 0,S ~t/hbar rho _ S (t)e^ -iH_ 0,S ~t/hbar

ρ

S

( t ) =

e

− i

H

S

  t

/

ρ

S

( 0 )

e

i

H

S

  t

/

displaystyle rho _ S (t)=e^ -iH_ S ~t/hbar rho _ S (0)e^ iH_ S ~t/hbar

See also[edit]

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

References[edit]

^ " Heisenberg representation". Encyclopedia of Mathematics. Retrieved 3 September 2013. 

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) p. 430-1 ISBN 0-471-88702-1 L.D. Landau, E.M. Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. 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
Quantum mechanics
(Revised Edition), ISBN 978-0201539295.

External links[edit]

Pedagogic Aides to Quantum Field Theory Click on the link for Chap. 2 to find an extensive, simplified introduction to the Heisenberg picture.

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 Photon entanglement 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
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

Quantum Bayesianism Quantum biology Quantum calculus Quantum chemistry Quantum chaos Quantum cognition Quantum cosmology Quantum differential calculus Quantum dynamics Quantum evolution Quantum geometry Quantum group Quantum measurement problem Quantum mind Quantum probability 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 complexity theory 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

Extensions

Quantum statistical mechanics Relativistic quantum mechanics Fractional 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 Portal:Phys

.