Quantum mechanics defined

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

An alternative strategy is to synthesize a molecular wave function, on chemical intuition, and progressively modify this function until it solves the molecular wave equation. However, chemical intuition fails to generate molecular wave functions of the required spherical symmetry, as molecules are assumed to have non-spherical three-dimensional structures. The impasse is broken by invoking the Born-Oppenheimer assumption that separates the motion of electrons and nuclei. At this point the strategy ceases to be ab initio and reduces to semi-empirical quantum-mechanical simulation. The assumed three-dimensional nuclear framework is no longer quantum-mechanically defined. The advantage of this model over molecular mechanics is that the electron distribution is defined quantum-mechanically. It has been used to simulate the H2 molecule. 

As discussed subsequently, introduction of the standard p, q representation in classical mechanics, and of the Wigner-Weyl representation in quantum mechanics, defines densities p(p,q) = (p,q p) that both lie in the same Hilbert space. Thus, the essential difference between quantum and classical mechanics... 

Molecular electrostatic potential The molecular electrostatic potential (MEP) associated with a molecule arises from the distribution of electrical charges of the nuclei and electrons of a molecule. The MEP is quantum mechanically defined in terms of the spatial coordinates of the charges on the nuclei and the electronic density function p(r) of the molecule. As the MEP is the net result of the opposing effects of the nuclei and the electrons, electrophiles will be guided to the regions of a molecule where the MEP is most negative. The MEP is a useful quantity in the study of molecular recognition processes. 

One of the postulates of quantum mechanical theory is that for every mechanical quantity there is a mathematical operator. The theory of quantum mechanics defines how these operators are constructed, and they contain derivative operators and multiplication operators. The eigenfunctions and eigenvalues of these operators play a central rede in the theory. For example, the operator that corresponds to the mechanical energy is the Hamiltonian operator, and the time-independent Schrodinger equation is die eigenvalue equation for this operator. For motion in... 

The effect of neutron polarization on diffusion was recognized ten years ago by S. Borowitz and M. Hamermesh. See [3]. Actually, the situation is more complicated than represented in this article, or in the text. The reason is that quantum mechanics defines amplitudes rather than intensities for the two helicities and that these amplitudes are complex rather than real. However, one can define, from the two amplitudes, four real quantities, the so-called statistical matrix. One of the real quantities, Oo(JiP, E, 2, t) is the total flux of both polarizations the three other quantities O, Oy, Og, (which depend on the same variables) describe the state of polarization of the neutrons with energy E, velocity-direction SI, and position x. If these neutrons are impolarized = 0 if the state of polarization is complete,... 

Remember, however, that wavefunctions have symmetry, and so do operators. The light that causes the system to go from one state to another (either by absorption or emission) can be assigned an irreducible representation from the point group of the system of interest. Quantum mechanics defines a specific expression, called a transition moment, to which the irreducible representations can be applied. For an absorption or emission of a photon, the transition moment M is defined as... 

Transition Change from one quantum mechanically defined energy state to another. 

Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at any time t suffices to predict with absolute certainty the properties of the system at any other time t. The situation is quite different in quantum mechanics, however, as it is not possible to know everything about the system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well defined way drat depends on the Hamiltonian operator and the wavefiinction T" according to the last postulate... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

The appropriate quantum mechanical operator fomi of the phase has been the subject of numerous efforts. At present, one can only speak of the best approximate operator, and this also is the subject of debate. A personal historical account by Nieto of various operator definitions for the phase (and of its probability distribution) is in [27] and in companion articles, for example, [130-132] and others, that have appeared in Volume 48 of Physica Scripta T (1993), which is devoted to this subject. (For an introduction to the unitarity requirements placed on a phase operator, one can refer to [133]). In 1927, Dirac proposed a quantum mechanical operator tf), defined in terms of the creation and destruction operators [134], but London [135] showed that this is not Hermitean. (A further source is [136].) Another candidate, e is not unitary. 

The time-dependent Schrddinger equation governs the evolution of a quantum mechanical system from an initial wavepacket. In the case of a semiclassical simulation, this wavepacket must be translated into a set of initial positions and momenta for the pseudoparticles. What the initial wavepacket is depends on the process being studied. This may either be a physically defined situation, such as a molecular beam experiment in which the paiticles are defined in particular quantum states moving relative to one another, or a theoretically defined situation suitable for a mechanistic study of the type what would happen if. .. 

In contrast to the point charge model, which needs atom-centered charges from an external source (because of the geometry dependence of the charge distribution they cannot be parameterized and are often pre-calculated by quantum mechanics), the relatively few different bond dipoles are parameterized. An elegant way to calculate charges is by the use of so-called bond increments (Eq. (26)), which are defined as the charge contribution of each atom j bound to atom i. 

The traditional way to provide the nuclear coordinates to a quantum mechanical program is via a Z-matrix, in which the positions of the nuclei are defined in terms of a set of intei ii.il coordinates (see Section 1.2). Some programs also accept coordinates in Cartesian formal, which can be more convenient for large systems. It can sometimes be important to choow an appropriate set of internal coordinates, especially when locating rninima or transitinn points or when following reaction pathways. This is discussed in more detail in Section 5.7. 

Structure-property relationships are qualitative or quantitative empirically defined relationships between molecular structure and observed properties. In some cases, this may seem to duplicate statistical mechanical or quantum mechanical results. However, structure-property relationships need not be based on any rigorous theoretical principles. 

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