Heisenberg picture

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The Heisenberg picture of quantum mechanics is a model for dealing with time-dependent problems. The following assumptions apply in the Heisenberg picture:

To indicate that you are in the Heisenberg picture, states and operators are occasionally given the index "H": or

Due to the prominent role of the operators in Heisenberg's formulation of quantum mechanics, this was historically also referred to as matrix mechanics . Two other models are the Schrödinger picture and the interaction picture . All models lead to the same expected values .

In the Heisenberg picture, the entire time dependency is in the operators, the states are independent of time:

In the Schrödinger picture, however, the unitary time evolution operator mediates the time evolution of the states:

The operator to be adjoint is , and because of the unitarity we have .

The expected value of the operator must be the same in all images:

The operator in the Heisenberg picture is thus given by the operator in the Schrödinger picture:

In general, the operator can be time-dependent both in the Heisenberg picture and in the Schrödinger picture, an example of this is a Hamilton operator with a time-dependent potential .

The Schrödinger equation for time-dependent wave functions is replaced in the Heisenberg picture by Heisenberg's equation of motion :

in which

  • the commutator from the Hamilton operator and is and
  • as an abbreviation for read.

If the Hamilton operator in the Schrödinger picture does not depend on time, then:

The observable is called the conserved quantity if


If this condition applies, it is also time-independent.