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Indian Journal of Pure & Applied Physics

Time dependence of Lippmann-Schwinger states

Ritesh Kumar Dubey, V Jayaram Menon, Dharmendra Tripathi

Published Jan 1, 2005• 0 citations
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Abstract

Due to the presence of an infinitesimal parameter e in the resolvent, the Lippmann-Schwinger states cannot be strict eigenstates of the Hamiltonian, and their time evolution is very non-trivial (in agreement with Fermi's golden rule). In sharp contrast, strict Schrodinger eigenstates cannot make transitions into free channels. Artefacts such as adiabatic switching, wave packets, or infinite time span are not needed in our analysis.

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Due to the presence of an infinitesimal parameter e in the resolvent, the Lippmann-Schwinger states cannot be strict eigenstates of the Hamiltonian, and their time evolution is very non-trivial (in agreement with Fermi's golden rule). In sharp contrast, strict Schrodinger eigenstates cannot make transitions into free channels. Artefacts such as adiabatic switching, wave packets, or infinite time span are not needed in our analysis.
Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.
When the Hamiltonian of the system depends on time, the solutions of the Schrödinger equation are non-stationary. Thus the concept of bound states with discrete energy levels and stationary eigenstates should be modified. There are three ways to make these modifications corresponding to the particular kind of approximations, depending on the nature of time dependence of the Hamiltonian. They are: the time dependent perturbation, the harmonic approximation, the adiabatic approximation, the sudden approximation.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The Lippmann–Schwinger–Low (LSL) quantum scattering states involve a resolvent operator depending on an infinitesimal adiabatic parameter ε. We reexamine the LSL formalism by taking the ε → + 0 limit at the end of the analysis (rather than at the outset). It is found that the LSL state vector |ψ kL > does not coincide with the Schrödinger eigen vector in Hilbert space as a whole, and the pair |ψ nL >, |ψ kL > is mutually nonorthogonal if the energy En = Ek, n ≠ k. For this purpose we carefully use a new type of projection operator ηk, a novel nonlinear relation among transition amplitudes, and a separable interaction as illustration. PACS Nos.: 0.3.65.Nk, 0.3.80.+r