![]() This is of course what one expects: if quantum coherence is averaged out, then classical mechanics becomes a good approximation. As the temperature is increased, however, the coupling between the molecule and the bath degrees of freedom progressively averages out the coherence structure (“decoherence”), and by the time the temperature is up to 300 K (for the assumed coupling strength in the model) one sees that it has been mostly quenched, and the classical Wigner model has then become a very good approximation. 2 shows P t(r) for the bath temperature T = 0, i.e., the bath being frozen out, and the result is very similar to that for the isolated molecule in Fig 1, showing pronounced coherence structure which the linearized IVR/classical Wigner approximation cannot describe. A number of applications have been carried out using this FB-IVR approach and been seen to give a very good description of quantum coherence effects (because the forward and backward trajectories are different, due to the “jump” caused by operator |$\hat B$| B ̂ at time t). The double phase space average thus becomes a single phase space average over initial conditions (p 0, q 0), but for this more complicated FB trajectory. It digresses too far here to describe these in detail, so I only note that the simplest approximation that is able to include quantum coherence effects is a “forward-backward” (FB) IVR that combines the forward and backward time-evolution operators into a single propagator: 15 trajectories begin at time 0 with initial conditions (p 0, q 0) and are propagated to time t via the usual molecular Hamiltonian, arriving at phase point (p t, q t) here they undergo a “jump” to a new phase point (p t ′, q t ′), determined by operator |$\hat B$| B ̂, and are then propagated back to time 0, arriving at the final phase point (p 0 ′, q 0 ′). It is thus necessary to go beyond the linearized approximation in order to describe quantum coherence effects, and several approaches have been developed to evaluate the double phase space average of Eq. (If there were an infinite number of “slits,” e.g., as for a crystal, rather than just two, then the peaks in the oscillatory pattern would narrow up to be delta functions at the Bragg diffraction angles.) If one neglects this cross term, then one obtains the result of classical mechanics, i.e., the probability distribution at the screen is the sum of probability distributions for the particle going through hole 1 or hole 2, and there is no oscillatory diffraction pattern. There is an oscillatory structure due to the cross term when squaring the sum of the two amplitudes. (The Schrodinger equation is, after all, a wave equation.) The classic example of this is the “2-slit problem,” 1 as is usually discussed in the introductory lecture of a quantum mechanics course: a particle has two possible paths from source to detector (e.g., a screen) by going through hole 1 or hole 2 in a barrier there is a (complex) amplitude associated with each path, one adds these amplitudes, and the square (modulus) of the net amplitude gives the probability distribution at the screen. One of the hallmarks of quantum mechanics, compared to classical mechanics, is the existence of coherence in particle mechanics, caused by interference of probability amplitudes.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |