Abstract
Some coherence effects in chemical dynamics are described correctly by classical mechanics, while others only appear in a quantum treatment--and when these are observed experimentally it is not always immediately obvious whether their origin is classical or quantum. Semiclassical theory provides a systematic way of adding quantum coherence to classical molecular dynamics and thus provides a useful way to distinguish between classical and quantum coherence. Several examples are discussed which illustrate both cases. Particularly interesting is the situation with electronically non-adiabatic processes, where sometimes whether the coherence effects are classical or quantum depends on what specific aspects of the process are observed.
Highlights
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. (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 by going through hole 1 or hole 2 in a barrier; there is a amplitude associated with each path, one adds these amplitudes, and the square of the net amplitude gives the probability distribution at the screen
There is an oscillatory structure due to the cross term when squaring the sum of the two amplitudes. (If there were an infinite number of “slits,” e.g., as for a crystal, rather than just two, the peaks in the oscillatory pattern would narrow up to be delta functions at the Bragg diffraction angles.) If one neglects this cross term, 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
For example, the position-position time correlation function, x(0) x(t), for a classical harmonic oscillator; an elementary calculation gives a result that is proportional to cos(ωt), where ω is the vibrational frequency, i.e., one has “coherent” vibrational motion. [If this degree of freedom is coupled to a number of other (“bath”) degrees of freedom, this oscillatory behavior is damped as time evolves (“de-cohered;” see below) and the correlation function decays to 0.] my opinion is that most “quantum coherence” effects that are seen, for example, in pumpprobe spectroscopic experiments—the oscillatory structure in which is usually identified with one of the vibrational modes in the molecule—are this type of classical vibrational coherence
Summary
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. (The Schrodinger equation is, after all, a wave equation.) The classic example of this is the “2-slit problem,” 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. (The Schrodinger equation is, after all, a wave equation.) The classic example of this is the “2-slit problem,” 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. There is an oscillatory structure due to the cross term when squaring the sum of the two amplitudes. (If there were an infinite number of “slits,” e.g., as for a crystal, rather than just two, the peaks in the oscillatory pattern would narrow up to be delta functions at the Bragg diffraction angles.) If one neglects this cross term, 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
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