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. ${\displaystyle \langle A\rangle }$