Heisenberg picture

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

• States are not time-dependent:
• Operators are time-dependent:
• The dynamics of the system is described by the Heisenberg equation of motion.

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

Due to the highlighted role of the operators in the Heisenberg formulation of quantum mechanics, this has historically been referred to as matrix mechanics. Two other models are the Schrödinger picture and the interaction picture. All models give the same expectation values.

In the Schrödinger picture the unitary time evolution operator provides the time evolution of the states:

Is the adjoint operator and is considered by the unitarity.

In the Heisenberg picture, however, the entire time dependence is in the operators and the states are time- independent:

The expectation value a of the operator must be the same in all pictures:

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

It should be noted that in general the operator both in the Heisenberg picture, as well as in the Schrödinger picture can be time-dependent, one example is a Hamiltonian with a time-dependent potential.

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

The commutator of the Hamiltonian and is and is to be read as an abbreviation for.

Depends on the Hamiltonian in the Schrödinger picture is not on the time, then:

The observable is called conserved quantity when

Does this condition, then it is also time-independent.