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 :
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.