Thus the zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion.
12.
Note that this equation only holds when the density operator is taken to be in the Schr�dinger picture, even though this equation seems at first look to emulate the Heisenberg equation of motion in the Heisenberg picture, with a crucial sign difference:
13.
The Runge & ndash; Gross proof first shows that there is a one-to-one mapping between external potentials and current densities by invoking the Heisenberg equation of motion for the current density so as to relate time-derivatives of the current density to spatial derivatives of the external potential.
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