Heisenberg picture

# Heisenberg picture

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Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the Heisenberg picture is a formulation of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger 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 are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...

in which the operators are constant and the states evolve in time. The two models only differ by a basis change with respect to time-dependency, which is the difference between active and passive transformation
Active and passive transformation
In the physical sciences, an active transformation is one which actually changes the physical position of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is a change in the coordinate description of the physical system . The distinction between...

. The Heisenberg picture is the formulation of matrix mechanics
Matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal.

## Mathematical details

In the Heisenberg picture of quantum mechanics the state vector, , does not change with time, and an observable A satisfies

where H is the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

and [·,·] is the commutator
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:...

of A and H. In some sense, the Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

picture is more natural and fundamental than the Schrödinger picture, especially for relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

theories. Lorentz invariance is manifest in the Heisenberg picture.

This approach has a similarity to classical physics
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

: by replacing the commutator above by the Poisson bracket
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...

, the Heisenberg equation becomes an equation in Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

.

By the Stone-von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent.

## Deriving Heisenberg's equation

The expectation value of an observable A, which is a Hermitian
Hermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...

linear operator, for a given state is given by:

In general where is the time evolution operator. For an elementary derivation, we will take Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

to commute with itself at different times, and further, be independent of time, in which case it simplifies to
where H is the Hamiltonian and ħ is Planck's constant divided by . It follows that

With the definition,

it follows:

(differentiating according to the product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

) noting that is the time derivative of A(t), the transformed operator, not the one we started with.

The last passage is valid since : commutes with H. From this results the Heisenberg equation of motion:
,
where [XY] is the commutator
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:...

of two operators and defined as [XY] := XY − YX.

Now, using the operator identity

one obtains for an observable A:

Due to the relationship between Poisson Bracket and Commutators this relation also holds for classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

.

note: the relationship between Poisson Bracket and Commutators is

in classical mechanics

so you can convince yourself that A(t) equation is the Taylor expansion on t=0

## Commutator relations

Obviously, commutator relations are quite different than in the Schrödinger picture because of the time dependency of operators. For example, consider the operators and . The time evolution of those operators depends on the Hamiltonian of the system. For the one-dimensional harmonic oscillator

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

By differentiating both equations one more time and solving them with proper initial conditions