Quantum-mechanical average value potential energy

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 ... 

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

In addition to- predicting a total energy, quantum mechanics enables one to calculate the average values of its kinetic and potential energy contributions, the latter consisting of the nuclear-electron attractive potential energy (V, ),... 

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. 

There are a few other analytically solvable systems, but most are variations on the themes presented here and in the last chapter. For now, we will halt our treatment of model systems and move on to a system that is more obviously relevant chemically. But before we do, it is important to reemphasize a few conclusions about the systems we have treated so far. (1) In all of our model systems, the total energy (kinetic -I- potential) is quantized. This is a result of the postulates of quantum mechanics. (2) In some of the systems, other observables are also quantized and have analytic expressions for their quantized values (like momentum). Whether other observables have analytic expressions for their quantized values depends on the system. Average values, rather than quantized values, may be all that can be determined. (3) All of these model systems have approximate analogs in reality, so that the conclusions obtained from the analysis of these systems can be applied approximately to known chemical systems (much in the same way ideal gas laws are applied to the behavior of real gases). (4) Classical mechanics was unable to rationalize these observations of atomic and molecular systems. It is this last point that makes quantum mechanics worth understanding in order to understand chemistry. 

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