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Time delay is a common feature of quantum scattering theory

Serge Richard, Rafael Tiedra de Aldecoa

Journal of Mathematical Analysis and ApplicationsSep 15, 201110 citations

Levinson’s theorem, zero-energy resonances, and time delay in one-dimensional scattering systems

Massimiliano Sassoli de Bianchi

The one-dimensional Levinson’s theorem is derived and used to study zero-energy resonances in a double-potential system. The low energy behavior of time delay is also investigated. In particular, it is shown that the quantum mechanical time delay admits a classical lower bound, in the low energy limit, if the potential has no bound-state solutions.

Journal of Mathematical PhysicsJun 1, 199485 citations

Relation between quantum tunneling times for relativistic particles (6 pages)

Herbert G. Winful

Physical Review AJan 1, 20040 citations

Quantum revivals versus classical periodicity in the infinite square well

Daniel F. Styer

A particle of mass M moves in an infinite square well of width L (the “particle in a box”). Classically, the motion has period L2M/E, which depends on the initial condition through the energy E. Quantum mechanically, any wave function repeats exactly with period 4ML2/πℏ, independent of the initial condition. Given this qualitative difference, how can the classical motion possibly be the limit of the quantal time development? The resolution of this paradox involves the difference between the exact revival (recurrence) of the wave function and the approximate periodicity of expectation values such as 〈x(t)〉. (The latter may recur an odd integral number of times before the full wave function recurs.) The period of the expectation values does depend on the initial condition and can possess the expected classical limit. [An Appendix demonstrates that, under suitably quasiclassical conditions, the quantal time evolution of 〈x(t)〉 passes over to the classical result not only in period, but also in its exact functional form. Another Appendix proves four theorems concerning state-dependent exact revival times.]A particle of mass M moves in an infinite square well of width L (the “particle in a box”). Classically, the motion has period L2M/E, which depends on the initial condition through the energy E. Quantum mechanically, any wave function repeats exactly with period 4ML2/πℏ, independent of the initial condition. Given this qualitative difference, how can the classical motion possibly be the limit of the quantal time development? The resolution of this paradox involves the difference between the exact revival (recurrence) of the wave function and the approximate periodicity of expectation values such as 〈x(t)〉. (The latter may recur an odd integral number of times before the full wave function recurs.) The period of the expectation values does depend on the initial condition and can possess the expected classical limit. [An Appendix demonstrates that, under suitably quasiclassical conditions, the quantal time evolution of 〈x(t)〉 passes over to the classical result not only in period, but also in its exact functional form. Another Appendix proves four theorems concerning state-dependent exact revival times.]

American Journal of PhysicsJan 1, 200164 citations

The Resonance Phenomena Associated with the Time Asymmetry in Non-Hermitian Quantum Mechanics

Nimrod Moiseyev, Romanas Narevich

International Journal of Theoretical PhysicsOct 1, 20030 citations

Levinson's theorems in classical scattering

T. A. Osborn, Richard Froese, S. F. Howes

This paper states and proves Levinson's theorems for the classical scattering of two particles. The derivation is carried out by using time-delay theory. In both classical and quantum mechanics Levinson's theorems are shown to be the consequence of a common principle. This principle is the spectral property of time delay. Consider an arbitrary space region v. In quantum and classical mechanics a time-dependent scattering state traverses v in a definite time. The spectral property states that the sum of transit times for all orbits with total energy \ensuremath{\epsilon} is proportional to the state density in region v. In classical mechanics it is established that this property can be derived from simple features of the time-evolution group.This paper states and proves Levinson's theorems for the classical scattering of two particles. The derivation is carried out by using time-delay theory. In both classical and quantum mechanics Levinson's theorems are shown to be the consequence of a common principle. This principle is the spectral property of time delay. Consider an arbitrary space region v. In quantum and classical mechanics a time-dependent scattering state traverses v in a definite time. The spectral property states that the sum of transit times for all orbits with total energy \ensuremath{\epsilon} is proportional to the state density in region v. In classical mechanics it is established that this property can be derived from simple features of the time-evolution group.

Physical Review AJul 1, 198013 citations

Number of quantal resonances

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Number of quantal resonances

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