![]() The most drastic such approximation is to assume that the only part of the integrand that contributes significantly is when the two phase points, (p 0, q 0) and (p 0 ′, q 0 ′), are infinitesimally close to each other, and then linearizing all quantities in the integrand in their difference. There are thus a variety of approximations and algorithms that have been developed for dealing with this. (1) thus leads to a space average over two sets of initial conditions for the correlation function C A B ( t ) = ( 2 π ℏ ) − F ∫ d p 0 ∫ d q 0 ( 2 π ℏ ) − F ∫ d p 0 ′ × ∫ d q 0 ′ C t ( p 0, q 0 ) C t ( p 0 ′, q 0 ′ ) * e i / ℏ ⟨ p 0, q 0 | A ̂ | p 0 ′, q 0 ′ ⟩ ⟨ p t ′, q t ′ | B ̂ | p t, q t ⟩Īnd the essential difficulty in evaluating this double phase space average (other than of course computing the classical trajectories) is the oscillatory phase factor of the integrand that arises from the difference in the action integrals of the two trajectories this complicates the application of Monte Carlo integration methods (which are the only feasible approach because of the high dimensionality of the phase space for large molecular systems). (2) for the two time evolution operators in Eq. (84)80039-7) IVR that has been most widely used, though there are other versions that are sometimes more useful see Ref. (2) is the coherent state (Herman-Kluk 7 7. Where (p t, q t) are the momenta and coordinates that have evolved to time t from the initial values (p 0, q 0), S t is the classical action integral along this trajectory, and the pre-exponential factor C t is given in terms of the derivatives of p t and q t with respect to p 0 and q 0 (the monodromy matrix). which has proved to be the most useful version of SC theory for application to large molecular systems, expresses the time evolution operator as an average (which must typically be evaluated by Monte Carlo methods) over the phase space of initial conditions of classical trajectories, e − i H ̂ t / ℏ = ( 2 π ℏ ) − F ∫ d q 0 ∫ d p 0 C t ( q 0, p 0 ) e i S t ( p 0, q 0 ) / ℏ × | p t, q t ⟩ ⟨ p 0, q 0 | , The initial value representation (IVR), 4–9 4. In terms of which dynamical quantities in large, complex molecular systems are usually expressed (for various operators A ̂ and B ̂ ). (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.įor reference in the discussions below, it is useful to give a brief summary of the semiclassical description of a typical time correlation function, 3 3. ![]() There is an oscillatory structure due to the cross term when squaring the sum of the two amplitudes. 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. Hibbs, Quantum Mechanics and Path Integrals ( McGraw-Hill, 1965), pp. (The Schrodinger equation is, after all, a wave equation.) The classic example of this is the “2-slit problem,” 1 1. 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.
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