Heisenberg picture explained

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 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.

Mathematical details

In the Heisenberg picture of quantum mechanics the state vectors |ψ⟩ do not change with time, while observables satisfywhere "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, is the Hamiltonian and denotes the commutator of two operators (in this case and). 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

t is |\psi(t)\rangle = U(t) |\psi( 0 )\rangle , where U(t) = T e ^{- (i)/(\int ^{t} _{0} ds H _{{\rm} \hbar)} (s)} is the time-evolution operator induced by a Hamiltonian H _{\rm} (t) that could depend on time, and |\psi( 0 ) \rangle is the initial state. T refers to time-ordering, ħ is the reduced Planck constant, and i is the imaginary unit. The expectation value of an observable A _{\rm} (t) in the Schrödinger picture, which is a Hermitian linear operator that could also be time-dependent, in the state |\psi(t)\rangle is given by$$\backslash lang\; A\; \backslash rang\; \_t\; =\; \backslash lang\; \backslash psi\; (t)\; |\; A\_(t)\; |\; \backslash psi(t)\; \backslash rang.$$

In the Heisenberg picture, the quantum state is assumed to remain constant at its initial value

|\psi( 0 ) \rangle , whereas operators evolve with time according to the definition $$A\_\; (t)\; :=\; U^(t)\; A\_\; (t)\; U(t)\; \backslash ,\; .$$This readily implies \langle A \rangle _{t} = \langle \psi( 0 ) | A _{\rm} (t) | \psi( 0 ) \rangle , so the same expectation value can be obtained by working in either picture. The Schrödinger equation for the time-evolution operator is$$\backslash frac\; U(t)\; =\; -\backslash frac\; H\_(t)\; U(t)\; .$$ It follows thatwhere 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 H _{\rm} (t) , which may differ from the Schrödinger Hamiltonian H _{\rm} (t) .H _{\rm} does not vary with time. Then the time-evolution operator can be written as$$U(t)\; =\; e^,$$and hence H _{\rm} \equiv H _{\rm} \equiv H since U(t) now commutes with H . Therefore,$$\backslash lang\; A\; \backslash rang\; \_t\; =\; \backslash lang\; \backslash psi\; (0)\; |\; e^\; A\_\; (t)\; e^\; |\; \backslash psi(0)\; \backslash rang$$and following the previous analyses,

Furthermore, if

A _{{\rm} } \equiv A is also time-independent, then the last term vanishes and(d)/(dt) A _{\rm} (t) = (i)/(\hbar) [H, A _{\rm} (t)] ,

where

A _{{\rm} }(t) \equiv A(t)= e^ A e^ in this particular case. The equation is solved by use of the standard operator identity,$$=\; A\; +\; [B,A]\; +\; \backslash frac\; [B,[B,A]]\; +\; \backslash frac[B,[B,[B,A]]]\; +\; \backslash cdots\backslash ,,$$which implies$$A(t)\; =\; A\; +\; \backslash frac[H,A]\; +\; \backslash frac\backslash left(\backslash frac\backslash right)^2\; [H,[H,A]]\; +\; \backslash frac\; \backslash left(\backslash frac\backslash right)^3\; [H,[H,[H,A]]]\; +\; \backslash cdots$$

A similar relation also holds for classical mechanics, the classical limit of the above, given by the correspondence between Poisson brackets and commutators:$$[A,H]\; \backslash quad\; \backslash longleftrightarrow\; \backslash quad\; i\backslash hbar\backslash .$$In classical mechanics, for an A with no explicit time dependence,$$\backslash \; =\; \backslash frac~,$$ 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

|\psi(t)\rangle in the Schrödinger picture is sometimes written as |\psi _{\rm} (t)\rangle to differentiate it from the evolved state |\psi _{\rm} (t)\rangle 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 and . The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator,$$H\; =\; \backslash frac\; +\; \backslash frac,$$the evolution of the position and momentum operators is given by:$$\backslash frac\; x(t)\; =\; \backslash frac\; [H,\; x(t)\; ]\; =\; \backslash frac,$$$$\backslash frac\; p(t)\; =\; \backslash frac\; [H,\; p(t)\; ]\; =\; -m\; \backslash omega^2\; x\; .$$

Note that the Hamiltonian is time independent and hence

x(t),p(t) are the position and momentum operators in the Heisenberg picture. Differentiating both equations once more and solving for them with proper initial conditions,$$\backslash dot(0)\; =\; -m\; \backslash omega^2\; x\_0,$$$$\backslash dot(0)\; =\; \backslash frac,$$ leads to$$x(t)\; =\; x\_0\; \backslash cos(\backslash omega\; t)\; +\; \backslash frac\backslash sin(\backslash omega\; t),$$$$p(t)\; =\; p\_0\; \backslash cos(\backslash omega\; t)\; -\; m\; \backslash omega\; x\_0\; \backslash sin(\backslash omega\; t)\; .$$

Direct computation yields the more general commutator relations,$$[x(t\_1),\; x(t\_2)]\; =\; \backslash frac\; \backslash sin\backslash left(\backslash omega\; t\_2\; -\; \backslash omega\; t\_1\backslash right),$$$$[p(t\_1),\; p(t\_2)]\; =\; i\backslash hbar\; m\backslash omega\; \backslash sin\backslash left(\backslash omega\; t\_2\; -\; \backslash omega\; t\_1\backslash right),$$$$[x(t\_1),\; p(t\_2)]\; =\; i\backslash hbar\; \backslash cos\backslash left(\backslash omega\; t\_2\; -\; \backslash omega\; t\_1\backslash right)\; .$$

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
• Interaction picture
• Schrödinger picture
• Heisenberg–Langevin equations
• Phase space formulation

References

1. Web site: Heisenberg representation. Encyclopedia of Mathematics. 3 September 2013.

• Book: Cohen-Tannoudji , Claude . Claude Cohen-Tannoudji . Bernard Diu . Frank Laloe . Quantum Mechanics (Volume One) . Wiley . 1977 . Paris . 312–314 . 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
• Book: . 1977 . Quantum Mechanics: Non-Relativistic Theory . 3rd . 3 . . 978-0-08-020940-1. Online copy
• R. Shankar (1994); Principles of Quantum Mechanics, Plenum Press, .
• J. J. Sakurai (1993); Modern Quantum Mechanics (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 picture https://courses.physics.illinois.edu/phys580/fa2013/heisenberg.pdf
• 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 https://physicstoday.scitation.org/doi/10.1063/1.3035086
• The computations for the hydrogen atom in the Heisenberg representation originally from a paper of Pauli https://arxiv.org/ftp/arxiv/papers/1501/1501.05894.pdf