Quantum canonical transformations and exact solution of the Schr{umlt o}dinger equation
- Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, T6G 2J1 (Canada)
Time-dependent unitary transformations are used to study the Schr{umlt o}dinger equation for explicitly time-dependent Hamiltonians of the form H(t)={bold R}(t){bold {center_dot}J}, where {bold R} is an arbitrary real vector-valued function of time and {bold J} is the angular momentum operator. The solution of the Schr{umlt o}dinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case {bold R}=(1,0,{nu}(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of {bold J}. It is also shown that by requiring the magnitude of {bold R} to depend on its direction in a particular way, one can solve the Schr{umlt o}dinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)={bold R}(t){bold {center_dot}J} there is another Hamiltonian with the same eigenstates for which the Schr{umlt o}dinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang{endash}Mills gauge theory) is also pointed out. {copyright} {ital 1997 American Institute of Physics.}
- OSTI ID:
- 530075
- Journal Information:
- Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 7 Vol. 38; ISSN JMAPAQ; ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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