Quantum mechanics, classical expectation values

B. A. Hess The reason that macroscopic motions display coherence is that they are in most cases at the classical limit of quantum dynamics. In this case, a suitable occupation of quantum states ensures that quantum mechanical expectation values equal the classical value of an observable. In particular, the classical state of an electromagnetic field (the coherent state) is one in which the expectation value of the operator of the electromagnetic field equals the classical field strengths. 

The term T is the quantum-mechanical expectation value of the kinetic energy, and is the only term requiring knowledge of P(r, rl) for ri r, while the others have a purely classical interpretation in terms of the distribution functions for a particle and for a pair of particles respectively. These results are valid for all kinds of wavefunctions, or approximate wavefunctions, for any state of any system and because they involve the electron distribution directly it is often possible to get a useful interpretation of molecular properties in terms of the main features of the electron density, without detailed reference to the intricacies of the many-electron wavefunction. A chemical bond, for instance, arises from a concentration of electron density in the bond... 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

It can be concluded, if the DF role must be preserved, that the statistical formalism of expectation values, represented by equation (1), has to be used in classical quantum mechanics for stationary states, in every circumstance. Furthermore, the following conditions must hold ... 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

The quantum mechanical expression for the charge-weighted current density is obtained from Eq. (26) when we replace the classical velocity r (f) by the Dirac velocity operator caL and evaluate its expectation value (21),... 

Molecular electronic dipole moments, pi, and dipole polarizabilities, a, are important in determining the energy, geometry, and intermolecular forces of molecules, and are often related to biological activity. Classically, the pKa electric dipole moment pic can be expressed as a sum of discrete charges multiplied by the position vector r from the origin to the ith charge. Quantum mechanically, the permanent electric dipole moment of a molecule in electronic state Wei is defined simply as an expectation value ... 

In this paper we have derived expressions for the environment-induced correction to the Berry phase, for a spin coupled to an environment. On one hand, we presented a simple quantum-mechanical derivation for the case when the environment is treated as a separate quantum system. On the other hand, we analyzed the case of a spin subject to a random classical field. The quantum-mechanical derivation provides a result which is insensitive to the antisymmetric part of the random-field correlations. In other words, the results for the Lamb shift and the Berry phase are insensitive to whether the different-time values of the random-field operator commute with each other or not. This observation gives rise to the expectation that for a random classical field, with the same noise power, one should obtain the same result. For the quantities at hand, our analysis outlined above involving classical randomly fluctuating fields has confirmed this expectation. 

Thus far we have focussed on the dynamics of quantum-classical systems. In practice, we are primarily interested instead in computing observables that can be compared eventually to experimentally obtainable quantities. To this end, consider the general quantum mechanical expression for the expectation value of an observable,... 

We will in this section consider the mathematical structure for computational procedures when calculating molecular properties of a quantum mechanical subsystem coupled to a classical subsystem. Molecular properties of the quantum subsystem are obtained when considering the interactions between the externally applied time-dependent electromagnetic field and the molecular subsystem in contact with a structured environment such as an aerosol particle. Therefore, we need to study the time evolution of the expectation value of an operator A and we express that as... 

The quantum-mechanical equations for a many-particle system (for more details, see e.g. t 2)) are deduced from the equations of classical mechanics by replacing the physical quantities appearing in them (position, momentum etc...) by appropriate operators the latter operate on certain functions, called wave functions, which describe the possible states of the system. The values of physical observables are the expectation values of the corresponding operators. For instance, the expression... 

Statistical mechanics is the branch of physical science that studies properties of macroscopic systems from the microscopic starting point. For definiteness we focus on the dynamics ofan A-particle system as our underlying microscopic description. In classical mechanics the set of coordinates and momenta, (r, p ) represents a state of the system, and the microscopic representation of observables is provided by the dynamical variables, v4(r, p, Z). The equivalent quantum mechanical objects are the quantum state [/ ofthe system and the associated expectation value Aj = of the operator that corresponds to the classical variable A. The corresponding observables can be thought of as time averages... 

Here we would like to add some comment of a general character. Dirac (1958) argued that the transformation from classical to quantum mechanics should be made, first by constructing the classical Hamiltonian in the Cartesian coordinate system and then by replacing the positions and momenta by their quantum-mechanical operator equivalents, which are determined by the particular representation chosen. The important point is that this transformation should be performed in the Cartesian coordinate system, for it is only in this system that the Heisenberg uncertainty principle for the positions and momenta is usually enunciated. In this connection, notice that some momentum wave functions such as those obtained by Podolsky and Pauling (1929) are correct wave functions that are useful in calculations of the expectation value of any observable, but at the same time they have a drawback in that the momentum variables used there are not conjugate to any relevant position variables (see also, Lombardi, 1980). 

This is the quantum mechanical analog of the classical expression Eq. (11), with the canonical momentum replaced by the expectation value of the momentum operator (P)t. It is then clear that the choice... 

Figure 11. The LCT field inducing selective photoassociation of HI is shown in the upper panel. The bound-state population (5(f)) and the population in the target state (Z v(f)) with quantum number v = 19 are shown in the middle panel of the figure, as indicated. The lower panel compares the quantum mechanical coordinate expectation value ((R)t) with a classical trajectory R(t)).

Let us briefly comment on the relation between the quantum mechanical field-assisted scattering process and its treatment within the classical limit. Therefore, a classical trajectory (R(t)) is determined in the presence of the LCT field-derived quantum mechanically, where the initial condition is defined by the average position and momentum of the initial wavepacket (Eq. (25)). The time evolution of this trajectory is compared to the coordinate expectation value in the lower panel of Fig. 11. It is seen that the trajectory is trapped by the field interaction, leading to a classical vibration at a smaller total energy (0.102 eV as compared to E g = 0.113 eV). Deviations in the two curves are to be expected and arise from the spatial extent of the wavepacket. Here, we encounter a first example for the qualitative relation between quantum and classical dynamics in the case of local control. 

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