# Heisenberg picture

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 :${\ displaystyle {\ hat {A}} = {\ hat {A}} (t)}$ • The dynamics of the system are described by Heisenberg's equation of motion .

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:

${\ displaystyle | \ psi _ {\ text {H}} \ rangle = | \ psi _ {\ text {S}} (0) \ rangle}$ 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 . ${\ displaystyle {\ hat {A}}}$ The Schrödinger equation for time-dependent wave functions is replaced in the Heisenberg picture by Heisenberg's equation of motion :

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ hat {A}} _ {\ rm {H}} (t) = {\ partial \ over \ partial t} {\ hat {A}} _ {\ rm {H}} (t) + {i \ over \ hbar} [{\ hat {H}} _ {\ rm {H}} (t) {\ mbox {, }} {\ hat {A}} _ {\ rm {H}} (t)]}$ in which

• ${\ displaystyle \ left [{\ hat {H}} _ {\ rm {H}} (t), {\ hat {A}} _ {\ rm {H}} (t) \ right]}$ 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: ${\ displaystyle {\ hat {H}} _ {\ rm {S}}}$ ${\ displaystyle {\ hat {H}} _ {\ rm {H}} (t) = {\ hat {H}} _ {\ rm {S}}}$ The observable is called the conserved quantity if ${\ displaystyle {\ hat {A}}}$ ${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ hat {A}} _ {\ rm {H}} (t) = 0}$ .

If this condition applies, it is also time-independent. ${\ displaystyle \ langle A \ rangle}$ 