Observables in quantum mechanics

Primes, H. Classical Observables in Quantum Mechanics (preprint - to be published)... 

Postulate 2. Every observable in quantum mechanics is represented by a linear, hermitian operator. 

Resonances in reactive collisions were first observed in quantum mechanical scattering calculations for the colllnear H + H2 reaction (1-9 for a review of early calculations on this system see reference 22. recent review of the quantum mechanical... 

The simplest quantity we can observe in quantum mechanics is the spin of a particle, around an axis imposed by applying a strong magnetic field. The experiment was first performed by the physicists Stem and Gerlach in 1921 its essentials are indicated in Figure 1. 

Some texts state that (5.14) is derived from (5.12) by taking fft d/dt as the energy operator and multiplication by t as the time operator. However, the energy operator is the Hamiltonian H, and not ih d/dt. Moreover, time is not an observable but is a parameter in quantum mechanics. Hence there is no quantum-mechanical time operator. (The noun observable in quantum mechanics means a physically measurable property of a system.) Equation (5.14) must be derived by a special treatment, which we omit. (See Ballentine, Section 12-3.) The derivation of (5.14) shows that At is to be interpreted as... 

Two basic observables are position (usually—and arbitrarily—in the x direction) and the corresponding linear momentum. In classical mechanics, they are designated X and p. Many other observables are various combinations of these two basic observables. In quantum mechanics, the position operator x is defined by multiplying the function by the variable x ... 

In the preceding section, we constructed second-quantization operators for one- and two-electron operators in such a way that the same matrix elements and hence the same expectation values are obtained in the first and second quantizations. Since the expectation values are the only observables in quantum mechanics, we have arrived at a new representation of many-electron systems with the same physical contents as the standard first-quantization representation. In the present section, we examine this new tool in greater detail by con aring the first- and second-quantization representations of operators. In particular, we show that, for operator products — P, the second-quantization representation of may differ from the product of the second-quantization representations of and unless a complete basis is used. 

The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. 

To nnderstand the internal molecnlar motions, we have placed great store in classical mechanics to obtain a picture of the dynamics of the molecnle and to predict associated patterns that can be observed in quantum spectra. Of course, the classical picture is at best an imprecise image, becanse the molecnlar dynamics are intrinsically quantum mechanical. Nonetheless, the classical metaphor mnst surely possess a large kernel of truth. The classical stnichire brought out by the bifiircation analysis has accounted for real patterns seen in wavefimctions and also for patterns observed in spectra, snch as the existence of local mode doublets, and the... 

Before concluding this sketch of optical phases and passing on to our next topic, the status of the phase in the representation of observables as quantum mechanical operators, we wish to call attention to the theoretical demonstration, provided in [129], that any (discrete, finite dimensional) operator can be constructed through use of optical devices only. 

The concept of two-state systems occupies a central role in quantum mechanics [16,26]. As discussed extensively by Feynmann et al. [16], benzene and ammonia are examples of simple two-state systems Their properties are best described by assuming that the wave function that represents them is a combination of two base states. In the cases of ammonia and benzene, the two base states are equivalent. The two base states necessarily give rise to two independent states, which we named twin states [27,28]. One of them is the ground state, the other an excited states. The twin states are the ones observed experimentally. 

In quantum mechanics, the eigenvalues of an operator represent the only numerical values that can be observed if the physical property corresponding to that operator is measured. Operators for which the eigenvalue spectrum (i.e., the list of eigenvalues) is discrete thus possess discrete spectra when probed experimentally. 

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

III. Experimental observation of Quantum Mechanics. Only this final section should address the rules that govern interpretations of experiments measuring properties of QM systems with macroscopic devices. This includes probability interpretation, uncertainty relations, complementarity and correspondence. Then experiments can be discussed to show how the wave functions manipulated in section I can be used to predict the probabilistic outcome of experiments. 

To achieve their objective the members of group II are compelled to complement the quantities with a correct formal status in quantum mechanics (e.g the observables) with... 

The second postulate states that a physical quantity or observable is represented in quantum mechanics by a hermitian operator. To every classically defined function A(r, p) of position and momentum there corresponds a quantum-mechanical linear hermitian operator A(r, (h/i)V). Thus, to obtain the quantum-mechanical operator, the momentum p in the classical function is replaced by the operator p... 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

In simple English, this implies that if you were in a 4-D universe and launched planets toward a sun, the planets would either fly away to infinity or spiral into the sun. (This is in contrast to a (3 + 1) universe that obviously can, for example, have stable orbits of moons around planets.) A similar problem occurs in quantum mechanics, in which a study of the Schrodinger equation shows that the hydrogen atom has no bound states for n > 5. This seems to suggest that it is difficult for higher universes to be stable over time and contain creatures that can make observations about the universe. 

The Heisenberg space defines the available uncertainty space where, in quantum mechanics, it is possible to perform, direct or indirect, measurements. Outside this space, in the forbidden region, according to the orthodox quantum paradigm, it is impossible to make any measurement prediction. We shall insist that this impossibility does not result from the fact that measuring devices are inherently imperfect and therefore modify, due to the interaction, in an unpredictable way what is supposed to be measured. This results from the fact that, prior to the measurement process, the system does not really possess this property. In this model for describing nature, it is the measurement process itself that, out of a large number of possibilities, creates the physical observable properties of a quantum system. 

P. W. Brumer As we know, in quantum mechanics, time evolution and coherence are synonymous. Thus, if I see time evolution, then coherences underlie the observation. Hence, in moving my arm I have created a molecular coherence. We should all be asking why this is so easy to create compared to the complex experiments described in these talks Is it due to the closely lying energy levels in large systems If so, then it suggests that experiments on larger molecules would be easier. 

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