Heisenberg Equation of Motion

Suppose that a general state ket is subject to the transformation

This is a time-dependent transformation, because the operator obviously depends on time. The subscript is used to remind us that the transformation is time-dependent. The time evolution of the transformed state ket is given by

(3.17) |

where use has been made of Equations (3.3), (3.5), and the fact that . Clearly, the transformed state ket does not evolve in time. Thus, the transformation (3.16) has the effect of bringing all kets representing states of undisturbed motion of the system to rest.

The transformation must also be applied to bras. The dual of Equation (3.16) yields

(3.18) |

The transformation rule for a general observable is obtained from the requirement that the expectation value should remain invariant. It is easily seen that

Thus, a dynamical variable, which corresponds to a fixed linear operator in the Schrödinger scheme, corresponds to a moving linear operator in this new scheme. It is clear that the transformation (3.16) leads us to a scenario in which the state of the system is represented by a fixed ket, and the dynamical variables are represented by moving linear operators. This is termed the

Consider a dynamical variable corresponding to a fixed linear operator in the Schrödinger picture. According to Equation (3.19), we can write

(3.20) |

Differentiation with respect to time yields

(3.21) |

With the help of Equation (3.14), this reduces to

(3.22) |

or

where

(3.24) |

Equation (3.23) can be written

Equation (3.25) shows how the dynamical variables of the system evolve in the
Heisenberg picture. It is denoted the *Heisenberg equation of motion* [12].
The time-varying dynamical variables in the Heisenberg picture
are usually called *Heisenberg dynamical variables* to distinguish them
from *Schrödinger dynamical variables* (i.e., the corresponding variables in
the Schrödinger picture), which do not evolve in time.

According to Equation (2.22), the Heisenberg equation of motion can be written

where denotes the quantum mechanical Poisson bracket. Let us compare this equation with the classical time evolution equation for a general dynamical variable , which can be written in the form

[See Equation (2.7).] Here, is the classical Poisson bracket, and denotes the classical Hamiltonian. The strong resemblance between Equations (3.26) and (3.27) provides us with further justification for our identification of the linear operator with the energy of the system in quantum mechanics.

Note that if the Hamiltonian does not explicitly depend on time (i.e., the system is not subject to some time-dependent external force) then Equation (3.15) yields

This operator manifestly commutes with , so

(3.29) |

(See Exercise 3.) Furthermore, Equation (3.25) gives

(3.30) |

Thus, if the energy of the system has no explicit time dependence then it is represented by the same non-time-varying operator, , in both the Schrödinger and Heisenberg pictures.

Suppose that is an observable that commutes with the Hamiltonian (and, hence, with the time evolution operator ). It follows from Equation (3.19) that . Heisenberg's equation of motion yields

(3.31) |

Thus, any observable that commutes with the Hamiltonian is a constant of the motion (hence, it is represented by the same fixed operator in both the Schrödinger and Heisenberg pictures). Only those observables that do not commute with the Hamiltonian evolve in time in the Heisenberg picture.