Semi-classical approximations of Quantum Mechanical problems

Виктор Павлович Маслов, M. V. Fedoriuk

Виктор Павлович Маслов, M. V. Fedoriuk

Dirk-André Deckert, Leopold Kellers, Travis Norsen +1 more

Bohmian mechanics is an alternative to standard quantum mechanics that does not suffer from the measurement problem. While it agrees with standard quantum mechanics concerning its experimental predictions, it offers novel types of approximations not suggested by the latter. Of particular interest are semi-classical approximations, where part of the system is treated classically. Bohmian semi-classical approximations have been explored before for systems without electromagnetic interactions. Here, the Rabi model is considered as a simple model involving light-matter interaction. This model describes a single mode electromagnetic field interacting with a two-level atom. As is well-known, the quantum treatment and the semi-classical treatment (where the field is treated classically rather than quantum mechanically) give qualitatively different results. We analyze the Rabi model using a different semi-classical approximation based on Bohmian mechanics. In this approximation, the back-reaction from the two-level atom onto the classical field is mediated by the Bohmian configuration of the two-level atom. We find that the Bohmian semi-classical approximation gives results comparable to the usual mean field one for the transition between ground and first excited state. Both semi-classical approximations tend to reproduce the collapse of the population inversion, but fail to reproduce the revival, which is characteristic of the full quantum description. Also an example of a higher excited state is presented where the Bohmian approximation does not perform so well.

Shu Hotta

Nandita Rudra

In quantum mechanics as in classical mechanics, there are a relatively few physical problems that can be solved exactly. Thus one has to take recourse to approximate methods. We start with semi-classical approximation in the next section.

Kenichi Konishi, Giampiero Paffuti

Abstract In this chapter we study the limit ħ → 0 of quantum mechanics. The passage from quantum mechanics to classical mechanics involves precisely this limit, and the approximation schemes developed for small ħ are known as the semi-classical approximation. The semi-classical approximation has a predecessor: the so-called old quantum theory. Many of the questions concerning the connection between the quantum and classical mechanics were already raised during the years of foundation of quantum mechanics, 1900–1927, but satisfactory answers have not always been obtained.