Heisenberg's equations of motion
We have seen that in Schrödinger's scheme the dynamical variables of the system remain fixed during a period of undisturbed motion, whereas the state kets evolve according to Eq. (228). However, this is not the only way in which to represent the time evolution of the system. Suppose that a general state ket is subject to the transformation
(235) |
The transformation must also be applied to bras. The dual of Eq. (234) yields
(236) |
Thus, a dynamical variable, which corresponds to a fixed linear operator in Schrödinger's scheme, corresponds to a moving linear operator in this new scheme. It is clear that the transformation (234) leads us to a scenario in which the state of the system is represented by a fixed vector, and the dynamical variables are represented by moving linear operators. This is termed the Heisenberg picture, as opposed to the Schrödinger picture, which is outlined in Sect. 4.1.
Consider a dynamical variable corresponding to a fixed linear operator in the Schrödinger picture. According to Eq. (237), we can write
(238) |
(239) |
(240) |
where
(242) |
Equation (243) shows how the dynamical variables of the system evolve in the Heisenberg picture. It is denoted Heisenberg's equation of motion. Note that 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 Eq. (112), the Heisenberg equation of motion can be written
Here, is the classical Poisson bracket, and denotes the classical Hamiltonian. The strong resemblance between Eqs. (244) and (245) 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 Eq. (233) yields
(247) |
(248) |
Suppose that is an observable which commutes with the Hamiltonian (and, hence, with the time evolution operator ). It follows from Eq. (237) that . Heisenberg's equation of motion yields
(249) |
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