Quantum physical meaning

So far as rule 2 is concerned, since AJ is conventionally taken to refer to J -J", where J is the quantum number of the upper state and J" that of the lower state of the transition, AJ = — 1 has no physical meaning (although it emerges from the quantum mechanics). It is commonly, but incorrectly, thought that AJ = +1 and AJ = — 1 refer to absorption and emission, respectively in fact AJ = +1 applies to both. Transition wavenumbers or frequencies are given by... 

The quantum states of Schrodinger s theory constitute an example of Hilbert space, and their scalar product has a direct physical meaning. Any such example of Hilbert space, where we can actually evaluate the scalar products numerically, is called a representation of Hilbert space. We shall continue discussing the properties of abstract Hilbert space, so that all our conclusions will apply to any and every representation. 

If quantum mechanics is really the fundamental theory of our world, then an effectively classical description of macroscopic systems must emerge from it - the so-called quantum-classical transition (QCT). It turns out that this issue is inextricably connected with the question of the physical meaning of dynamical nonlinearity discussed in the Introduction. The central thesis is that real experimental systems are by definition not isolated, hence the QCT must be viewed in the relevant physical context. 

The previous result is an important one. It indicates that there can be yet another fruitful route to describe lipid bilayers. The idea is to consider the conformational properties of a probe molecule, and then replace all the other molecules by an external potential field (see Figure 11). This external potential may be called the mean-field or self-consistent potential, as it represents the mean behaviour of all molecules self-consistently. There are mean-field theories in many branches of science, for example (quantum) physics, physical chemistry, etc. Very often mean-field theories simplify the system to such an extent that structural as well as thermodynamic properties can be found analytically. This means that there is no need to use a computer. However, the lipid membrane problem is so complicated that the help of the computer is still needed. The method has been refined over the years to a detailed and complex framework, whose results correspond closely with those of MD simulations. The computer time needed for these calculations is however an order of 105 times less (this estimate is certainly too small when SCF calculations are compared with massive MD simulations in which up to 1000 lipids are considered). Indeed, the calculations can be done on a desktop PC with typical... 

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. 

The preparation and study of nanoparticles has attracted a remarkable academic and industrial research effort because of their potential applications, ranging from fundamental studies in quantum physics, fabrication of composite materials, information storage/optoelectronics, immunoassays, to catalysts. The precise control of size and chemical behavior (stabihty and reactivity) by means of the synthesis itself is still one of the main targets because the direct correlation of the new intriguing properties with the particle size which is just between a molecule and a bulk material [140]. 

But more than this, thinking matter has endowed itself with a cosmic affiliation. It has bestowed meaning upon its past, composed of inert, stellar and cloudy matter, and before that, the vacuum (although the latter is now considered to be seething with activity in quantum physics). 

One of the purposes of this work is to make contact between relativistic corrections in quantum mechanics and the weakly relativistic limit of QED for this problem. In particular, we will check how performing plane-wave expectation values of the Breit hamiltonian in the Pauli approximation (only terms depending on c in atomic units) we obtain the proper semi-relativistic functional consistent in order ppl mc ), with the possibility of analyzing the separate contributions of terms with different physical meaning. Also the role of these terms compared to next order ones will be studied. 

These results indicate that the presence of the wave theta is revealed by the value of the expected visibility. If the visibility is one, the 0 waves do not exist, meaning that the quantum waves are mere mathematical probability waves devoid of any physical meaning. If the fringe pattern is blurred and the visibility decreased by a factor of, then it would imply that quantum waves, just like any ordinary wave, are real. 

We saw in Section 4.5 that a quantum mechanical system with symmetry determines a unitary representation of the symmetry group. It is natural then to ask about the physical meaning of representation-theoretic concepts. In this section, we consider the meaning of invariant subspaces and irreducible representations. 

It has been assumed, necessarily, that the reader has some prior familiarity with the basic notions of quantum theory. He is expected to know in a general way what the wave equation is, the significance of the Hamiltonian operator, the physical meaning of a wave function, and so forth, but no detailed knowledge of mathematical intricacies is presumed. Even the contents of a rather qualitative book such as Coulson s Valence should be sufficient, although, of course, further background knowledge will not be amiss. 

The concept of radiationless transitions, namely internal conversion and intersystem crossing1 is one which is widely used in photochemistry today. However, the precise nature of the processes involved is elusive since direct measurement of the yields of radiationless transitions is impossible with the exception of those intersystem crossings between first excited singlet states and lower-lying triplets where the triplet state can be quantitatively estimated by chemical or physical means. In all other cases, the accepted practice is to sum the quantum yields of processes which can be estimated directly, such as decomposition and emission, and attribute those excited molecules not accounted for by such processes to radiationless transitions. 

These contributions were followed by an extensive presentation of the Law of the Black Radiation by Max Planck, who discussed, among other aspects, the physical nature of the constant h. Does this quantum of action, he said, possess a physical meaning for the propagation of electromagnetic radiation in vacuum, or does it intervene only in the emission and absorption processes of radiation by matter ... 

A classification of reactivity indices according to the particular quantum chemical method used for calculation has the advantage that it does not refer to mostly unproven assumptions on the relation between physical meaning of the index and mechanism of reactions, although a pictorial interpretation of the physical meaning... 

On the contrary, the semiclassical approach in the problem of the optical absorption is restricted to a great extent and the adequate description of the phonon-assisted optical bands with a complicated structure caused by the dynamic JTE cannot be done in the framework of this approach [13]. An expressive example is represented by the two-humped absorption band of A —> E <8> e transition. The dip of absorption curve for A —> E <8> e transition to zero has no physical meaning because of the invalidity of the semiclassical approximation for this spectral range due to essentially quantum nature of the density of the vibronic states in the conical intersection of the adiabatic surface. This result is peculiar for the resonance (optical) phenomena in JT systems full discussion of the condition of the applicability of the adiabatic approximation is given in Ref. [13]. 

The physical meaning of the (j, k) element (Ajkn) of the A(r) matrix is the amplitude of the i> if//1 single quantum transition. Substituting Equation (47) into Equation (46) results in ... 

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

In the present book we have used the cogredient expansion form (2.14), where, as distinct from the standard form, an additional normalizing factor has been introduced, namely (—l)< v/(2K + l)/4n. Our expansion of the classical probability density p(0, differs from the standard one in exactly the same way as the expansion of the quantum mechanical density matrix p over 2Tq differs from the expansion over lTg. In Section 5.3 we present a comparison between the physical meaning of the classical polarization moments pg, as used in the present book, and the quantum mechanical polarization moments fg, as determined by the cogredient method using normalization (D.ll). 

---