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International Journal of Quantum Chemistry

“Stable” quasistationary state: New solution of the time‐dependent Schrödinger equation

Akitomo Tachibana

Published Jul 1, 1982• 6 citations
Publisher

Abstract

Abstract The new concept of the stability of the quantum mechanical state is introduced for the study of the time‐dependent process of the quasistationary state. A stable solution of the time‐dependent Schrodinger equation predicts the presence of the new sharply localized long‐living state at the most repulsive part of the effective potential.

DOI
10.1002/qua.560220117

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Abstract The new concept of the stability of the quantum mechanical state is introduced for the study of the time‐dependent process of the quasistationary state. A stable solution of the time‐dependent Schrodinger equation predicts the presence of the new sharply localized long‐living state at the most repulsive part of the effective potential.
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A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.