Ci 2x, + IC2I X2- However, the individual probabilities do not add in this way for system in a quantum mechanical superposition state. Instead, the expectation value is C P - -C2 2 Ci +C2 P 2)- The classical and quantum mechanical predictions can be very different, because in addition to CipXi + C2pX2, the latter contains an interference term, C C2 Wi x P2) + C Ci( P2 x Pi). [Pg.45]

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. [Pg.25]

Figure 6.4 NMR report of GHZ correlations by Nelson and co-workers in 2000. Quantum mechanics predicts the expectation value of the product of spin components of three spins in a GHZ state is always positive, except for ii GHz x x x I GHz) which is negative and equal to —1. This correlation, which is shown in the experiment, cannot be explained by classical means. Adapted with permission from [17]. |

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. [Pg.94]

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 [Pg.369]

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 [Pg.43]

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 [Pg.9]

Variational Monte Carlo (or VMC, as it is now commonly called) is a method that allows one to calculate quantum expectation values given a trial wavefunction [1,2]. The actual Monte Carlo methodology used for this is almost identical to the usual classical Monte Carlo methods, particularly those of statistical mechanics. Nevertheless, quantum behavior can be studied with this technique. The key idea, as in classical statistical mechanics, is the ability to write the desired property <0> of a system as an average over an ensemble [Pg.38]

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. [Pg.43]

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 [Pg.29]

Despite the demands presented by such a calculation, a number of researchers have used ab initio models to treat the electronic and nuclear degrees of freedom for the quantum motif in molecular mechanics, energy minimization studies. Examples of this include the self-consistant reaction field methods developed by Tapia and coworkers [42-44], which represent only the quantum motif explicitly and use continuum models for the environmental effects (classical and boundary regions), and the methods implemented by Kollman and coworkers [45] in their studies of condensed phase (chemical and biochemical) reaction mechanisms. In both of these implementations the expectation value of the quantum motif Hamiltonian, defined in Eqs. (11) and (14) above, is treated at the Hartree Fock level with relatively small basis sets. [Pg.61]

Hence, if we measure momentum and position in the same direction, the result depends on what has been measured first. These conditions on the commutators are required in order to fulfill the Heisenberg uncertainty relation. The founders of quantum mechanics noted that the only guiding principle for the new quantum theory must be the requirement that results of observations must be reproduced by the theory even if this then collides with classical concepts. The uncertainty relation may be deduced after a couple of steps have been taken starting with the definition of the dispersion of a measurement as the square of the deviation of the actual measurement (expressed by the operator) and the expectation value. In this derivation, which can be found, for instance, in Ref. [45], the nonvanishing commutator of conjugate variables plays a decisive role. [Pg.131]

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