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

States are not time-dependent :${\ displaystyle | \ psi \ rangle = {\ rm {const}}}$

Operators are time-dependent :${\ displaystyle {\ hat {A}} = {\ hat {A}} (t)}$

To indicate that you are in the Heisenberg picture, states and operators are occasionally given the index "H": or${\ displaystyle | \ psi _ {\ rm {H}} \ rangle}$${\ displaystyle {\ hat {A}} _ {\ rm {H}} (t)}$

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:
${\ displaystyle {\ hat {U}} (t)}$

${\ displaystyle | \ psi _ {\ text {S}} (t) \ rangle = {\ hat {U}} (t) | \ psi _ {\ text {S}} (0) \ rangle}$

The operator to be adjoint is , and because of the unitarity we have .
${\ displaystyle {\ hat {U}} (t)}$${\ displaystyle {\ hat {U}} ^ {\ dagger} (t)}$${\ displaystyle {\ hat {U}} ^ {\ dagger} (t) = {\ hat {U}} (t) ^ {- 1}}$

The expected value of the operator must be the same in all images:
${\ displaystyle a}$${\ displaystyle {\ hat {A}}}$

${\ displaystyle {\ begin {aligned} a & = \ langle \ psi _ {\ text {S}} (t) | {\ hat {A}} _ {\ text {S}} (t) | \ psi _ { \ text {S}} (t) \ rangle = \ langle \ psi _ {\ text {S}} (t) | \ underbrace {{\ hat {U}} (t) {\ hat {U}} ^ { \ dagger} (t)} _ {1} \, {\ hat {A}} _ {\ text {S}} (t) \, \ underbrace {{\ hat {U}} (t) {\ hat { U}} ^ {\ dagger} (t)} _ {1} \, | \ psi _ {\ text {S}} (t) \ rangle \\ & = \ langle {\ hat {U}} ^ {\ dagger} (t) \ psi _ {\ text {S}} (t) | {\ hat {U}} ^ {\ dagger} (t) \, {\ hat {A}} _ {\ text {S} } (t) \, {\ hat {U}} (t) \, | {\ hat {U}} ^ {\ dagger} (t) \ psi _ {\ text {S}} (t) \ rangle = \ langle \ psi _ {\ text {S}} (0) | {\ hat {U}} ^ {\ dagger} (t) \, {\ hat {A}} _ {\ text {S}} (t ) \, {\ hat {U}} (t) | \ psi _ {\ text {S}} (0) \ rangle \\\\ & = \ langle \ psi _ {\ text {H}} | {\ hat {A}} _ {\ text {H}} (t) | \ psi _ {\ text {H}} \ rangle \ end {aligned}}}$

The operator in the Heisenberg picture is thus given by the operator in the Schrödinger picture:
${\ displaystyle {\ hat {A}} _ {\ rm {H}} (t)}$${\ displaystyle {\ hat {A}} _ {\ rm {S}} (t)}$

${\ displaystyle {\ hat {A}} _ {\ rm {H}} (t) = {\ hat {U}} ^ {\ dagger} (t) \, {\ hat {A}} _ {\ rm {S}} (t) \, {\ hat {U}} (t)}$

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 .
${\ displaystyle {\ hat {A}}}$

${\ displaystyle \ left [{\ hat {H}} _ {\ rm {H}} (t), {\ hat {A}} _ {\ rm {H}} (t) \ right]}$the commutator from the Hamilton operator and is and${\ displaystyle {\ hat {H}} _ {\ rm {H}} (t)}$${\ displaystyle {\ hat {A}} _ {\ rm {H}} (t)}$

${\ displaystyle {\ partial \ over \ partial t} {\ hat {A}} _ {\ rm {H}} (t)}$as an abbreviation for read.${\ displaystyle U ^ {\ dagger} (t) \ left ({\ partial \ over \ partial t} {\ hat {A}} _ {\ rm {S}} (t) \ right) U (t)}$

If the Hamilton operator in the Schrödinger picture does not depend on time, then:
${\ displaystyle {\ hat {H}} _ {\ rm {S}}}$

${\ displaystyle {\ hat {H}} _ {\ rm {H}} (t) = {\ hat {H}} _ {\ rm {S}}}$