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Series in atomic and molecular physics/Series in atomic molecular physics

The Schrodinger picture

Published Dec 1, 2003• 0 citations
Publisher
DOI
10.1201/9781420033649.ch3

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In quantum electrodynamics one can obtain a Hamiltonian which gives reasonable field equations in the Heisenberg picture, but which does not allow of solutions of the wave equation to represent physical states in the Schr\"odinger picture. The inference is that the Heisenberg picture is a good picture, the Schr\"odinger picture is a bad picture, and the two pictures are not equivalent. The usual proof of the equivalence of the two pictures fails because the state vector of the Schr\"odinger picture does not remain in Hilbert space. One can set up quantum electrodynamics entirely in the Heisenberg picture and thereby avoid the worst difficulties encountered in the Schr\"odinger picture. The theory takes a logical form and is not merely an assembly of working rules. It can be applied to the calculation of the anomalous magnetic moment and the Lamb shift, and is then similar to the usual calculations of these effects, with a good deal of dead wood cut away. There is a problem concerned with the general interpretation of quantum mechanics when one cannot use the Schr\"odinger picture and some postulates are proposed for dealing with it.
Although spatial locality is explicit in the Heisenberg picture of quantum dynamics, spatial locality is not explicit in the Schrödinger picture equations of motion. The gauge picture is a modification of Schrödinger’s picture such that locality is explicit in the equations of motion. In order to achieve this explicit locality, the gauge picture utilizes (1) a distinct wavefunction associated with each patch of space, and (2) time-dependent unitary connections to relate the Hilbert spaces associated with nearby patches. In this work, we show that by adding an additional spatially-local term to the gauge picture equations of motion, we can effectively interpolate between the gauge and Schrödinger pictures, such that when this additional term has a large coefficient, all of the gauge picture wavefunctions approach the Schrödginer picture wavefunction (and the connections approach the identity).