It reproduces the Schr�dinger equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity.
2.
It is most apparent in the Heisenberg picture of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion.
3.
In the Heisenberg picture it is the other way round, } } is constant while evolves with time according to the Heisenberg equation of motion.
4.
Once the system Hamiltonian is known, one can use the Heisenberg equation of motion to generate the dynamics of a given N-particle operator.
5.
Moreover, this constant energy in the Hamiltonian obviously commutes with and and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.
6.
This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts.
7.
He called the equation for the time evolution of a quantum-mechanical operator, which he was the first to write down, the " Heisenberg equation of motion ".
8.
The zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy and by stating that it has no effect on the Heisenberg equations of motion:
9.
This is Ehrenfest's theorem, which is an obvious corollary of the Heisenberg equations of motion, but is less trivial in the Schr�dinger picture, where Ehrenfest discovered it.
10.
Since the states obey the Schr�dinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of and variables, but it is instructive to see this directly.
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