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Institute of Physics eBooks

Solving the time-dependent Schrödinger equation and the theory of representations

Colin John Lambert

Published Mar 1, 2021• 0 citations
Publisher
DOI
10.1088/978-0-7503-3639-0ch16

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Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger 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–Mills gauge theory) is also pointed out.
The derivation of a time-dependent Schrödinger equation (TDSE) from a time-independent Schrödinger equation (TISE) in the coherent state representation is considered for the special case of a simple coupled atom-field system described by the soluble Jaynes-Cummings model. The derivation shows why, from the outset, a linear combination of energy eigenstates, instead of a single state, must be used in order to obtain a TDSE for general states. Moreover, this study leads to a method of solving a TDSE by simply solving a TISE.