Probability oscillations

Abstract For the case of small matter effects V perturbation theory using e = 2V E/ Am2 as the expansion parameter. We derive simple and physically transparent formulas for the oscillation probabilities in the lowest order in e which are valid for an arbitrary density profile. They can be applied for the solar and supernova neutrinos propagating in matter of the Earth. Using these formulas we study features of averaging of the oscillation effects over the neutrino energy. Sensitivity of these effects to remote (from a detector), d > PE/AE, structures of the density profile is suppressed. 

We have derived expressions for the oscillation probabilities in matter with arbitrary density profile for the vacuum dominated case V [Pg.411]

Well, one prominent FET manufacturer earlier was quite sure that this doomsday scenario could really happen, and had even stated as much in a certain Application Note (though this section was later removed). On further inquiries (from me in particular) they backed off, and in fact provided fresh data to actually disprove their own earlier assertion. So ultimately, privately they blamed it on one lone engineer of theirs, who didn t quite follow the book when he reported he saw oscillations. Probably a bad scope probe. We will never know. 

The final distribution was obtained by weighting each of the forced oscillator probabilities Pt v by the corresponding Franck- ndon factor to produce a set of classical transition probabilities... 

Finally, some papers which carry the theory of vibrational averaging to higher levels of approximation should be mentioned. Bartell, in his 1955 paper " on the Morse oscillator probability distribution, considers the effect of an increase of temperature on r. Bartell s theory is extended by Bartell and Kuchitsu and by Kuchitsu these papers show in particular that the effective mean amplitude obtained by refinement on the molecular intensity curve is not quite equal to the harmonic mean amplitude calculated from the harmonic force field. Bonham and co-workers have calculated the effect of temperature on both a Morse oscillator and an oscillator in an RKR potential energy curve. In their final paper an informative series of diagrams shows how the quantum-mechanical average passes into the classical average at high temperature. 

Figure Al.1.2. Probability density (v[/ vt/) for the n = 29 state of the hamionic oscillator. The vertical state is chosen as in figure A1.1.1. so that the locations of the turning points comcide with the superimposed potential fiinction.

When looking at the snapshots in Figure A3.13.6 we see that the position of maximal probability oscillates back and forth along the stretching coordinate between the walls at = -20 and +25 pm, with an approximate period of 12 fs, which corresponds to the classical oscillation period r = 1 / v of a pendulum with... 

Before presenting the quantum mechanical description of a hannonic oscillator and selection rules, it is worthwhile presenting the energy level expressions that the reader is probably already familiar with. A vibrational mode v, witii an equilibrium frequency of (in wavenumbers) has energy levels (also in... 

Figure Bl.2.4. Lowest five hannonic oscillator wavefimctions / and probability densities i if.

Application of an oscillating magnetic field at the resonance frequency induces transitions in both directions between the two levels of the spin system. The rate of the induced transitions depends on the MW power which is proportional to the square of oi = (the amplitude of the oscillating magnetic field) (see equation (bl.15.7)) and also depends on the number of spins in each level. Since the probabilities of upward ( P) a)) and downward ( a) p)) transitions are equal, resonance absorption can only be detected when there is a population difference between the two spin levels. This is the case at thennal equilibrium where there is a slight excess of spins in the energetically lower p)-state. The relative population of the two-level system in thennal equilibrium is given by the Boltzmaim distribution... 

Stueckelberg derived a similar fomiula, but assumed that the energy gap is quadratic. As a result, electronic coherence effects enter the picture, and the transition probability oscillates (known as Stueckelberg oscillations) as the particle passes through the non-adiabatic region (see [204] for details). 

Xvf (Ra - Ra,e) Xvi> will be non-zero and probably quite substantial (because, for harmonic oscillator functions these "fundamental" transition integrals are dominant- see earlier) ... 

Finally, the combustion zone does not always proceed at a uniform rate, but oscillates in time, slowing down and dren advancing rapidly. This effect is probably due to die non-uniform packing and distribution of die reactants in the compact. Also visual observation shows that the zone is not uniformly distributed around the periphety of the compact, and the observed movement of a hot spot around the periphery is usually refeiTed to as spin combustion. 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

As we have argued above, this probability is to be averaged over the equilibrium distribution for the -oscillator... 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

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