Quantum-mechanical average value

Suppose we wish to measure the position of a particle whose wave function is W(jc, i). The Bom interpretation of F(x, as the probability density for finding the associated particle at position x at time t implies that such a measurement will not yield a unique result. If we have a large number of particles, each of which is in state /) and we measure the position of each of these particles in separate experiments all at some time t, then we will obtain a multitude of different results. We may then calculate the average or mean value x) of these measurements. In quantum mechanics, average values of dynamical quantities are called expectation values. This name is somewhat misleading, because in an experimental measurement one does not expect to obtain the expectation value. 

The angle brackets remind us that these energy terms are quantum-mechanical average values or expectation values each is a functional of the ground-state electron density. Focussing first on the middle term, the one most easily dealt with the nucleus-electron potential energy is the sum over all 2n electrons (as with our treatment of ab initio theory, we will work with a closed-shell molecule which perforce has an even number of electrons) of the potential corresponding to attraction of an electron for all the nuclei A ... 

The ground state electronic energy of our real molecule is the sum of the electron kinetic energies, the nucleus-electron attraction potential energies, and the electron-electron repulsion potential energies (more precisely, the sum of the quantum-mechanical average values or expectation values, each denoted (value)) and each is a functional of the ground-state electron density ... 

The definition of the quantum-mechanical average value is given in Section 3.7 and should not be confused with the time average used in classical mechanics. 

Section 3-5 Quantum-Mechanical Average Value of the Potential Energy... 

The formula for the quantum-mechanical average value [Eq. (6-9)] is equivalent to the arithmetic average of all the possible measured values of a property times their frequency of occurrence [Eq. (6-33)]. This means that it is impossible to devise a function that satisfies the general conditions on xjr and leads to an average energy lower than the lowest eigenvalue of H. 

Perturbation theory also provides the natural mathematical framework for developing chemical concepts and explanations. Because the model H(0) corresponds to a simpler physical system that is presumably well understood, we can determine how the properties of the more complex system H evolve term by term from the perturbative corrections in Eq. (1.5a), and thereby elucidate how these properties originate from the terms contained in //(pertJ. For example, Eq. (1.5c) shows that the first-order correction E11 is merely the average (quantum-mechanical expectation value) of the perturbation H(pert) in the unperturbed eigenstate 0), a highly intuitive result. Most physical explanations in quantum mechanics can be traced back to this kind of perturbative reasoning, wherein the connection is drawn from what is well understood to the specific phenomenon of interest. 

Evidently a double averaging is involved, the quantum mechanical averaging implicit in the expectation value and the ensemble averaging associated with our inability to specify more completely the condition of an individual system. The time-honoured axiom of statistical mechanics (that of equal a priori probabilities and... 

It is thus evident that the qnantnm Brownian motion of a particle in a spin-bath may be calculated, in principle, as a stochastic process by solving the Langevin Equation 9.34 for qnantnm mechanical mean values simultaneously with quantum correction equations, which describe the quantum mechanical fluctuation or dispersion around them. Before closing this section, we mention that an essential element of this approach is to express the quantum statistical average as a sum of statistical averages over a set of functions of the quantum mechanical mean values and... 

The integration of Equations 9.49 and 9.51 is carried out using the second-order Heun s algorithm, with a very small time step of 0.001. These equations differ from the corresponding classical equations in two ways First, the noise correlation of c-number spin-bath variables r t) are quantum mechanical in nature, as evident from the correlation function in Equation 9.42, which is numerically fitted by the superposition of exponential functions with D, and X . Second, the knowledge of Q requires the quantum correction equations that yield quantum dispersion around the quantum mechanical mean values q and p for the system. Statistical averaging over noise is... 

The same idea is used to compute a quantum-mechanical average. For the average value of r we take each possible value of r times its frequency (given by ir dv) and sum... 

Here < J A J > is a reduced matrix element. In effect, it is simply a number depending on J which can be found in tabulations by Elliott and Stevens (1953). The factor is the quantum mechanical average of 1/r using the wave functions of the open-shell electrons. It is commonly given in units of (uo is the Bohr radius) and denoted as atomic units (au). In that case, the value of the hyperfine coupling constant is = 12.51 kT/au. 

This provides a recipe for calculating the average value of the system property associated with the quantum-mechanical operator A, for a specific but arbitrary choice of the wavefiinction T, notably those choices which are not eigenfunctions of A. 

Suppose that the system property A is of interest, and that it corresponds to the quantum-mechanical operator A. The average value of A obtained m a series of measurements can be calculated by exploiting the corollary to the fifth postulate... 

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. 

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