TRANSPORT-EQUATION FOR CALCULATING POWER SPECTRA OF SCHRODINGER WAVES- APPLICATION TO EXCHANGE-NARROWING AND ENVIRONMENTAL ISOMERS

The ''random frequency modulation'' model (introduced, for the sake of explaining exchange narrowing of spectral lines, by Anderson and cons olidated by Kubo) is generalized with an eye toward providing an analy tical basis for some recent computer simulations by Grunwald and Steel on environmental isomerism. The principal problem is to consider a mu ltistate model and calculate the power spectrum of the signal: mu(t) = mu(0) exp[iota integral(0)(t)v(t')dt'] = mu(0) exp[iota x(t)] (-infin ity less than or equal to v less than or equal to +infinity), where \m u(0)\ = 1 and v is a random function of time. The generalized model is specified by probability distributions, g(i)(v), and a transition mat rix, K = (k(ij)), which are defined as follows: When the system is in state i, 1 less than or equal to i less than or equal to s, it generat es a signal mu(t) with a constant angular frequency, so that x = vt; t he probability that the system jumps to state j during a time interval Delta t is k(ij)Delta t + o(Delta t); after the jump, it continues to generate the signal but with frequency v', which is a random variable with density g(j)(v) dv; the possibility that transitions can occur t o the same state is admitted by taking k(ii) > 0. The situations exami ned by Anderson and Kubo correspond to two special cases: (a) a single -state model, i.e. s = 1, and (b) a multistate model with a ''line spe ctrum'', i.e. g(i)(v) = delta(v - v(i)). Divergence from previous trea tments, which deal directly with mu (which lies on the unit circle), s tems from the recognition that since x lies between +/-infinity, the t ask of determining f(x,v;t), the time-dependent probability density in the two-dimensional (x,v)-space, is completely equivalent to finding the probability density in the phase space of a point particle which i s constrained to move (along a straight line) with a velocity that suf fers random changes, Computation of the power spectrum may thus be vie wed as a problem in linear transport theory; indeed, the integrodiffer ential equation which governs the evolution off(x,v;t) for the one-sta te model (case a) turns out to be formally identical with the single r elaxation time approximation to the linear Boltzmann equation. The new approach, which recovers previously published results after minimal m anipulation of the underlying transport equation, is brought to bear o n the issue, raised by Grunwald and Steel, of the distinguishability o f environmental isomers; the conclusions reached by them are corrobora ted and extended.