Ermakov-Lewis Invariant in Koopman-von Neumann Mechanics
- PDF / 201,955 Bytes
- 4 Pages / 439.642 x 666.49 pts Page_size
- 71 Downloads / 156 Views
Ermakov-Lewis Invariant in Koopman-von Neumann Mechanics Abhijit Sen1 · Zurab Silagadze2 Published online: 10 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In the paper (Ramos-Prieto et al., Sci. Rep. 8, 8401 2018), among other things, the ErmakovLewis invariant was constructed for the time dependent harmonic oscillator in Koopmanvon Neumann mechanics. We point out that there is a simpler method that allows one to find this invariant. Keywords Koopman-von Neumann mechanics · Ermakov-Lewis invariant · Time dependent harmonic oscillator
1 Introduction In the interesting paper [1], the Ermakov-Lewis invariant is used to study the time-dependent harmonic oscillator (TDHO) in the framework of Koopman-von Neumann (KvN) mechanics [2, 3]. To find the invariant, a system of coupled differential equations was obtained, which then was reduced to a single equation related to the Ermakov equation. Although this method is quite simple, our goal in this short note is to show that there is an even simpler way to find the Ermakov-Lewis invariant for TDHO in KvN mechanics. Recent interest in KvN mechanics is motivated by experiments exploring the quantumclassical border. To formulate a consistent framework for a hybrid quantum-classical dynamics is a long-standing problem [4]. Its solution, in addition to practical interest, for example, in quantum chemistry, can clarify deep conceptual issues in quantum mechanics, such as the problem of measurement. An interesting result in this direction was obtained in [5]. It was found that the Wigner function in the nonrelativistic limit turns into the Koopman-von Neuman wave function, which explains why the Wigner function is not positive-definite.
Abhijit Sen
[email protected] Zurab Silagadze [email protected] 1
Novosibirsk State University, Novosibirsk, 630 090, Russia
2
Budker Institute of Nuclear Physics and Novosibirsk State University, Novosibirsk, 630 090, Russia
2188
International Journal of Theoretical Physics (2020) 59:2187–2190
Time-dependent harmonic oscillators arise in many quantum mechanical systems [6, 7]. At the same time, the existence of Ermakov-Lewis invariants in such systems has attracted much attention [7]. In our opinion, extention of these results to the case of KvN mechanics is of considerable interest.
2 KvN Evolution Equation for TDHO in New Variables The KvN evolution equation for TDHO wave-function has the form [1]: i
∂ ψ(x, p; t) = pˆ λˆ x − k(t)xˆ λˆ p ψ(x, p; t), ∂t
where λˆ x and λˆ p operators satisfy the following commutation rules x, ˆ λˆ x = p, ˆ λˆ p = i.
(1)
(2)
Note that m = 1 and = 1 was assumed for simplicity. As Sudarshan remarked [8], any KvN-mechanical system can be considered as a hidden variable quantum system. Correspondingly, we will make a slight change in notations as follows: x = q, λx = P , λp = −Q,
(3)
where Q and P are quantum variables that are hidden for classical observers. Thus (1) takes the following form in new notations i
∂ ˆ ψ(q, p; t). ψ(q,
Data Loading...
