# Heisenberg's equation of motion

The Heisenberg equation of motion , also Heisenberg equation of motion or Heisenberg equation , determines the temporal development of a quantum mechanical system in the matrix representation . It was developed by Werner Heisenberg in the 1920s and is part of the Heisenberg picture of quantum mechanics. The main difference to the formulation of quantum mechanics using the Schrödinger equation is that in this case the states carry the temporal dynamics and the operators are constant, whereas in the Heisenberg representation the operators carry the temporal dynamics, while the state vector on which the operators act , is constant over time. Heisenberg's formulation is therefore closer to classical mechanics, which is also shown by the formal similarity of the classical equations of motion , expressed with the help of Poisson brackets .

The Heisenberg equation of motion replaces the Schrödinger equation of the Schrödinger picture in the Heisenberg picture of quantum mechanics .

The equation of motion itself is:

${\ displaystyle {\ frac {\ mathrm {d} A _ {\ rm {H}}} {\ mathrm {d} t}} = {\ frac {\ mathrm {i}} {\ hbar}} \ left [H_ {\ rm {H}}, A _ {\ rm {H}} \ right] + \ left (\ partial _ {t} A _ {\ rm {S}} \ right) _ {\ rm {H}}}$

where is the Hamilton operator of the system in the Heisenberg picture and is a commutator . The index "H" for the Heisenberg picture and "S" for the Schrödinger picture are inserted to identify the picture. ${\ displaystyle H _ {\ rm {H}}}$${\ displaystyle \ left [H _ {\ rm {H}}, A _ {\ rm {H}} \ right] \ equiv H _ {\ rm {H}} A _ {\ rm {H}} - A _ {\ rm { H}} H _ {\ rm {H}}}$

If an observable in the Schrödinger picture is not explicitly time-dependent and also interchanged with the Hamilton operator, the eigenvalues ​​of the operator are a conserved quantity . ${\ displaystyle A}$${\ displaystyle {\ tfrac {\ partial A _ {\ rm {S}}} {\ partial t}} = 0}$

## Equation of motion for expected values

Since in the Heisenberg picture the states are independent of time ${\ displaystyle {\ tfrac {\ mathrm {d}} {\ mathrm {d} t}} | \ psi _ {\ rm {H}} \ rangle = 0}$

${\ displaystyle \ left \ langle {\ frac {\ mathrm {d} A _ {\ rm {H}}} {\ mathrm {d} t}} \ right \ rangle = \ langle \ psi _ {\ rm {H} } | {\ frac {\ mathrm {d} A _ {\ rm {H}}} {\ mathrm {d} t}} | \ psi _ {\ rm {H}} \ rangle = {\ frac {\ mathrm { d}} {\ mathrm {d} t}} \ langle \ psi _ {\ rm {H}} | A _ {\ rm {H}} | \ psi _ {\ rm {H}} \ rangle = {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ langle A _ {\ rm {H}} \ rangle}$

one can immediately state the Heisenberg equation of the expected values:

${\ displaystyle {\ frac {\ mathrm {d} \ langle A _ {\ rm {H}} \ rangle} {\ mathrm {d} t}} = {\ frac {\ mathrm {i}} {\ hbar}} \ left \ langle \ left [H _ {\ rm {H}}, A _ {\ rm {H}} \ right] \ right \ rangle + \ left \ langle \ left (\ partial _ {t} A _ {\ rm { S}} \ right) _ {\ rm {H}} \ right \ rangle}$

Due to the invariance of the scalar product under image change (the expected values ​​of an operator are the same in all images), the equation can be written independently of the image:

${\ displaystyle {\ frac {\ mathrm {d} \ langle A \ rangle} {\ mathrm {d} t}} = {\ frac {\ mathrm {i}} {\ hbar}} \ left \ langle \ left [ H, A \ right] \ right \ rangle + \ left \ langle \ partial _ {t} A \ right \ rangle}$

This equation is known as the Ehrenfest theorem .

## Equivalence between Schrödinger and Heisenberg equations

In the following, Heisenberg's equation of motion is derived based on the Schrödinger equation . The opposite direction is also possible.

An operator's representation change from the Schrödinger to the Heisenberg image takes place via

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

where is the time evolution operator and its adjoint operator . ${\ displaystyle U (t)}$${\ displaystyle U ^ {\ dagger} (t)}$

By applying the time evolution operator to a state vector in the Schrödinger picture at the point in time , the state vector at the point in time is obtained . In the following, the abbreviated notation is always used: ${\ displaystyle U (t)}$${\ displaystyle t_ {0} = 0}$${\ displaystyle t}$${\ displaystyle U (t, 0) = U (t)}$

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

Inserting the time-dependent wave function into the Schrödinger equation yields: ${\ displaystyle \ mathrm {i} \ hbar {\ tfrac {\ partial} {\ partial t}} | \ psi _ {\ rm {S}} (t) \ rangle = H _ {\ rm {S}} (t ) | \ psi _ {\ rm {S}} (t) \ rangle}$

${\ displaystyle \ mathrm {i} \ hbar {\ frac {\ partial} {\ partial t}} U (t) | \ psi _ {\ rm {S}} (0) \ rangle = H _ {\ rm {S }} (t) U (t) | \ psi _ {\ rm {S}} (0) \ rangle}$

One obtains an operator equation equivalent to the Schrödinger equation :

${\ displaystyle \ mathrm {i} \ hbar {\ frac {\ partial} {\ partial t}} U (t) = H _ {\ rm {S}} (t) U (t)}$

From the operator ${\ displaystyle A _ {\ rm {H}} (t)}$

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

the time derivative is formed using the product rule:

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} A _ {\ rm {H}} = \ left ({\ frac {\ partial} {\ partial t}} U ^ { \ dagger} \ right) A _ {\ rm {S}} \, U + U ^ {\ dagger} \, A _ {\ rm {S}} \ left ({\ frac {\ partial} {\ partial t}} U \ right) + U ^ {\ dagger} \ left ({\ frac {\ partial} {\ partial t}} A _ {\ rm {S}} \ right) U}$

Now the above operator equations and their adjoint

${\ displaystyle {\ frac {\ partial} {\ partial t}} U = - {\ frac {\ mathrm {i}} {\ hbar}} H _ {\ rm {S}} U}$   and   ${\ displaystyle {\ frac {\ partial} {\ partial t}} U ^ {\ dagger} = {\ frac {\ mathrm {i}} {\ hbar}} U ^ {\ dagger} H _ {\ rm {p }}}$

used:

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} A _ {\ rm {H}} = \ left ({\ frac {\ mathrm {i}} {\ hbar}} U ^ {\ dagger} H _ {\ rm {S}} \ right) A _ {\ rm {S}} \, U + U ^ {\ dagger} \, A _ {\ rm {S}} \ left (- {\ frac {\ mathrm {i}} {\ hbar}} H _ {\ rm {S}} U \ right) + U ^ {\ dagger} \ left ({\ frac {\ partial A _ {\ rm {S}}} {\ partial t}} \ right) U}$

Sum up:

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} A _ {\ rm {H}} = {\ frac {\ mathrm {i}} {\ hbar}} \ left (U ^ {\ dagger} H_ {S} A _ {\ rm {S}} UU ^ {\ dagger} A _ {\ rm {S}} H_ {S} U \ right) + U ^ {\ dagger} \ left ({ \ frac {\ partial A _ {\ rm {S}}} {\ partial t}} \ right) U}$

Now you cleverly insert a between and between : ${\ displaystyle 1 = UU ^ {\ dagger}}$${\ displaystyle H _ {\ rm {S}} A _ {\ rm {S}}}$${\ displaystyle A _ {\ rm {S}} H _ {\ rm {S}}}$

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}} A _ {\ rm {H}} = {\ frac {\ mathrm {i}} {\ hbar}} (\ underbrace { U ^ {\ dagger} H _ {\ rm {S}} U} _ {H _ {\ rm {H}}} \ underbrace {U ^ {\ dagger} A _ {\ rm {S}} {\ hat {U} }} _ {A _ {\ rm {H}}} - \ underbrace {U ^ {\ dagger} A _ {\ rm {S}} U} _ {A _ {\ rm {H}}} \ underbrace {U ^ { \ dagger} H _ {\ rm {S}} U} _ {H _ {\ rm {H}}}) + \ underbrace {U ^ {\ dagger} \ left ({\ frac {\ partial A _ {\ rm {p }}} {\ partial t}} \ right) U} _ {\ left (\ partial _ {t} A _ {\ rm {S}} \ right) _ {\ rm {H}}} = {\ frac { \ mathrm {i}} {\ hbar}} (H _ {\ rm {H}} A _ {\ rm {H}} - A _ {\ rm {H}} H _ {\ rm {H}}) + \ left ( \ partial _ {t} A _ {\ rm {S}} \ right) _ {\ rm {H}}}$

With the commutator, Heisenberg's equation of motion can be written compactly:

${\ displaystyle {\ frac {\ mathrm {d} A _ {\ rm {H}}} {\ mathrm {d} t}} = {\ frac {\ mathrm {i}} {\ hbar}} \ left [H_ {\ rm {H}}, A _ {\ rm {H}} \ right] + \ left (\ partial _ {t} A _ {\ rm {S}} \ right) _ {\ rm {H}}}$