Quantum probability

At low temperatures, in a sample of very small dimensions, it may happen that the phase-coherence length in Eq.(3) becomes larger than the dimensions of the sample. In a perfect crystal, the electrons will propagate ballistically from one end of the sample and we are in a ballistic regime where the laws of conductivity discussed above no more apply. The propagation of an electron is then directly related to the quantum probability of transmission across the global potential of the sample. 

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

The corresponding quantum mechanical expression of s op in Equation (4.19) is similar except for the quantity Nj, which is replaced by Nfj. However, the physical meaning of some terms are quite different coj represents the frequency corresponding to a transition between two electronic states of the atom separated by an energy Ticoj, and fj is a dimensionless quantity (called the oscillator strength and formally defined in the next chapter, in Section 5.3) related to the quantum probability for this transition, satisfying Jfj fj = l- At this point, it is important to mention that the multiple resonant frequencies coj could be related to multiple valence band to conduction band singularities (transitions), or to transitions due to optical centers. This model does not differentiate between these possible processes it only relates the multiple resonances to different resonance frequencies. 

Quantum mechanics, however, is different from the other applications in that the states themselves require a concept of probability for their physical interpretation. I shall call this intrinsic probability or quantum probability to distinguish it from the above classical or statistical probability. Intrinsic probability is not covered by the definition in 1.1 and cannot be regarded as an ensemble.510... 

Remark. The density matrix is a convenient tool but it conceals the distinction between quantum probability and classical probability. The former is intrinsic to the system, the latter describes our knowledge about it. Certain authors call p the state of the system rather than the state of the ensemble. What we called a state i / is then represented by a density matrix of the special form... 

No instance is known in which the rate of reaction is proportional to a power of the light intensity greater than the first. It is, indeed, very improbable that a molecule would remain in a condition where it possessed so large an excess of energy as a quantum of visible light long enough to acquire a second quantum, since all the time it would be subject to collisions which would rapidly deprive it of its first quantum. Probably all photochemical transformations are one-quantum processes. 

The central point within our consciousness, our "spirit" in the hermetic sense, can now be seen as an entity that can work to control quantum probabilities. To our "spirits" our brain is a quantum sea providing a rich realm in which it can incarnate and manifest patterns down into the electrical/chemical impulses of the nervous system. (It has been calculated that the number of interconnections existing in our brains far exceeds the number of atoms in the whole universe - so in this sense the microcosm truly mirrors the macrocosm ). Our "spirit" can through quantum borrowing momentarily press a certain order into this sea and this manifests as a thought, emotion, etc. Such an ordered state can only exist momentarily, before our spirit or point of consciousness is forced to jump and move to other regions of the brain, where at that moment the pattern of probability waves for the particles in these nerve cells, can reflect the form that our spirit is trying to work with. 

At around 10 to the power -43 of a second, time itself becomes quantised, that is it appears as discontinuous particles of time, for there is no way in which time can manifest in quantities less than 10 to the power -43 (the so called Planck time). For here the borrowed quantum energies distort the fabric of space turning it back upon itself. There time must have a stop. At such short intervals the energies available are enormous enough to create virtual black holes and wormholes in space-time, and at this level we have only a sea of quantum probabilities - the so called Quantum Foam. Contemporary physics suggests that through these virtual wormholes in space-time there are links with all time past and future, and through the virtual black holes even with parallel universes. 

Relative to silicon, the total elemental lithium is even 140 times less abundantin the solar photosphere than on Earth or inmeteorites, as recorded by the solar spectrum. Since the Li/Si element abundance ratio in the meteorites should also have entered the Sun when it formed, one concludes that the Sun must be destroying its initial lithium supply as it ages. This occurs at the base of the surface convection zone of the Sun. The bottom of that zone lies at a depth that is about 1/4 of the Sun s radius. Here the high temperatures (a few million degrees kelvin, MK) that solar-surface nuclei experience is hot enough to destroy lithium, especially 6Li, by nuclear interactions with protons (6Li + p —3He + 4He). Those proton-induced nuclear reactions destroy 6Li much more readily than they do 7Li because the quantum probabilities of the reaction are greater than for 7Li. As a result, to deplete elemental lithium by a factor of 140 in the Sun... 

When He fuses in the hydrogen-exhausted core at the center of a star, itliberates more nuclear power by making l60 than by making 12C, as the smaller atomic mass excess for l60 reveals. The relative amounts of 12C and l60 made during He burning depend on the rate ofthe nuclear reaction12 C + 4He l60. The quantum probability for this reaction to occur has been very hard to pin down. Its precise value is very... 

Figure 1. Snapshots of quantum probabilities, (a) Weak perturbation regime e = 0.05. b) Strong perturbation regime e = 0.2. The other parameters are set as follows E = 0.75, to = 0.3, Ti = 1000/(3ti X 2 ) 0.1036, and cot2 = 0(mod27t).

The numerical results of the pure quantum calculation are shown in Fig. A. 1, which gives the quantum probability in the phase-represented scattering eigenstate that is defined by... 

It was indeed possible in these molecules to partition total space into regions so that the missing information function was simultaneously minimized for each region. The quantum probabilities were in each case dominated by P2(P.) and the most probable partitioning was found to be the one in which pairs of electrons were localized in well-defined regions of space. 

Clearly, the fluctuation in the average population of a region 2 (denoted here by J ( 2)) also approaches zero when a single event dominates the distribution. A difficulty with using quantum probabilities to determine the extent of spatial localization is that their evaluation requires the use of the lull Mh-order density matrix. Because of this, the calculation of P ( 2) rapidly becomes prohibitive with increasing N. The fluctuation A(N, 2) however, can be expressed entirely in terms of the diagonal elements of just the second-order density matrix (Bader and Stephens 1974). 

We wish to compare the quantum probability distributions with those obtained from the classical treatment of the harmonic oscillator at the same energies. The classical probability density P y) as a function of the reduced distance y(—l y l)is given by equation (4.10) and is shown in Figure... 

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