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Scattering

Published Apr 13, 2006• 0 citations
Publisher

Abstract

Abstract Besides the bounded, periodic classical motion of particles in potentials as shown in Fig. 5.2 (for energy E1 ), there is also the possibility of unbound states which are not localized in space and not repetitive in time; these correspond to particles incident on the potential and which subsequently “bounce “ off (energy E2 in Fig. 5.2) or temporarily change their speeds as they go over the potential (energy E3 ). Classical mechanics, which solves for the exact trajectories in either case, makes little distinction between the two types of motion aside from some tech- nical details,

DOI
10.1093/oso/9780198530978.003.0011

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Abstract Besides the bounded, periodic classical motion of particles in potentials as shown in Fig. 5.2 (for energy E1 ), there is also the possibility of unbound states which are not localized in space and not repetitive in time; these correspond to particles incident on the potential and which subsequently “bounce “ off (energy E2 in Fig. 5.2) or temporarily change their speeds as they go over the potential (energy E3 ). Classical mechanics, which solves for the exact trajectories in either case, makes little distinction between the two types of motion aside from some tech- nical details,
Information theory, which applies to compressing messages, suggests that ln(probability) is information. Shannon’s entropy is -1 times the average information i.e. -Sum over i P(i)ln(P(i)). For a classical gas in the canonical approximation, this is of the form of an average conserved quantity e.g Eave= Sum over i ei P(ei). In deterministic classical mechanics one may characterize an elastic two body collision by e1+e2 = e3 + e4. In a Maxwell-Boltzmann (MB) gas these same deterministic reactions occur, but there is also a static probability P(ei). The question seems to be whether P(ei) brings any new “physical” information into a scattering reaction. One already knows that a particle contributes energy ei. Is there something else brought in? If the answer is no, then using the product of probabilities P(e1)P(e2) and taking ln leads to ln(P(e1)) + ln(P(e2)) suggests that ln(P(e1))= A (e1-B) where A and B are constant. Thus information considerations link probability P(ei) to a deterministic quantity ei. One does not have to consider reaction balance or the maximization of entropy subject to a constraint. One simply finds the appropriate probability Pappropriate(e1) which describes an overall change to the state e1. For the MB case, P(appropriate)(ei) = P(ei) (MB) which gives a static description of the number of particles with ei. In other cases, however, there may be bias. Consider quantum scattering of fermions or bosons. If the collisions are elastic, one again has the classical deterministic e1+e2=e3+e4 expression. There is also a static P(ei) (Fermi-Dirac) for fermions and P(ei) Bose-Einstein for bosons. One is, however, interested in collisions which create and maintain the equilibrium situation. If there is bias in these collisions, then Pappropriate(ei) may differ from P(ei). The information e1 is the full contribution of particle 1 to the collision. Thus one also needs a full probability describing probability movement out of state e1. P(e1) is proportional only to the probability flow out of e1 due to scattering, but there is a “bias” factor namely 1-P(e1) which prevents another particle from entering state e1. This biases a reaction. As a result Pappropriate(e1) = P(e1)/ (1-P(e1)) for a fermion. This may be seen from a reaction balance equation:P(e1)(1-P(e3)) P(e2)(1-P(e4)) = P(e3)(1-P(e1))P(e4)(1-P(e2)) for fermions. For bosons there is an enhancement. One may obtain product probabilities by forming P(ei)/(1-P(ei)). This is the full contribution in terms of probabilities of particle i to a collision and so its ln should equal A(e-B)i which is its full information contribution to the reaction. If the two are not equal, it means there is more information (excluding parameters such as T temperature and u the chemical potential) than ei in the problem.
Abstract A general method is developed of calculation of response signals of N‐particle systems with an unequidistant spectrum to pulse action of coherent physical fields resonant for any transition of the system. The case of a particle system where each particle has three energy levels E1 < E2 < E3 is analysed. It is shown that if such a system is subjected to pulse action of two generators exciting transitions E1 ⟷ E2 and E2 ⟷ E3 at initial time and at time τ to pulse action of a generator with frequency ω31 = h−1 (E3 — E1), the system generates induction and echo signals with power proportional to N2 at all possible transitions. Echo signals arise at times (1 + ϵ) τ, 2 τ, and (1 + 1/ϵ) τ after the first pulse, where ϵ = E2 — E1/E3 — E2. The observed responses may have a physical nature which is both identical with the excitation and different from it. Supplements are given by which the responses of particle systems may be calculated to a series of pulses different from those considered here. When ϵ →1 the results transform to the case of equidistant spectrum.
A Maxwell-Boltzmann gas consists of particles which undergo physical collisions which are associated with the idea of causality, but we argue that reaction balance: e1+e2=e3+e4 and n(e1)n(e2)= n(e3)n(e4) represent steady stream energy-probability conservation and so one need not actually think in terms of direct collisions and causality i.e. one may just as well write: n(e1)n(e2)=n(e1+e2). In the case of quantum mechanics, we argue, as we have done in previous notes, that the free particle form follows from the Lorentz invariant A= -Et+px. In a rest frame, there is energy, but no net momentum. We thus argue for a physical separation of the -Et and px terms. We further suggest that energy is associated with a resonance in time and p momentum with one in space i.e exp(-iEt) and exp(ipx). Given that A=-Et+px is also the relativistic/nonrelativistic classical action, we suggest that these represent probability resonances associated with constant p motion. As a result, given that p is associated with impulse, we suggest that an interaction means forming a new resonance i.e. p1->p2 implies exp(ip1 x)--> exp(ip2 x). As a result, interactions take place in length regions of hbar/p. If this value is very small, as it is in the classical world, then it appears that the resonances occur at points. In the quantum world, it is the opposite. If one wishes to treat quantum mechanics in a statistical manner, one needs to establish an energy-probability balance. Even in classical mechanics, impulses occur that change p, but now one has resonances which occur over a wavelength long piece of space. Furthermore a given p may be anywhere, so one thinks in terms of a(p)exp(ipx) probabilities. We try to show that a probability-energy balance is equivalent to the Schrodinger equation. Thus we argue that quantum mechanics uses a p-probability-energy balance for the time-independent Schrodinger equation and so does not necessarily address causality and physical dynamics.