Expectation value, quantum mechanics

Expectation value Quantum mechanics only gives a probability of a quantity having a certain value - the expectation value. 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

Many physical properties of a molecule can be calculated as expectation values of a corresponding quantum mechanical operator. The evaluation of other properties can be formulated in terms of the "response" (i.e., derivative) of the electronic energy with respect to the application of an external field perturbation. 

For a quantum mechanical calculation, the single point calculation leads to a wave function for the molecular system and considerably more information than just the energy and gradient are available. In principle, any expectation value might be computed. You can get plots of the individual orbitals, the total (or spin) electron density and the electrostatic field around the molecule. You can see the orbital energies in the status line when you plot an orbital. Finally, the log file contains additional information including the dipole moment of the molecule. The level of detail may be controlled by the PrintLevel entry in the chem.ini file. 

This two general Cl function expressions, along with the results obtained in the section 5.1 above, permit to compute the expected value form of any quantum mechanical operator in a most complete general way. 

In his pioneering work Baetzold used the Hartree-Fock (HF) method for quantum mechanical calculations for the cluster structure (the details are summarized in Reference 33). The value of the HF procedure is that it yields the best possible single-determinant wave function, which in turn should give correct values for expectation values of single-particle operators such as electric moments and... 

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. 

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 appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. 

The position, momentum, and energy are all dynamical quantities and consequently possess quantum-mechanical operators from which expectation values at any given time may be determined. Time, on the other hand, has a unique role in non-relativistic quantum theory as an independent variable dynamical quantities are functions of time. Thus, the uncertainty in time cannot be related to a range of expectation values. 

Show explicitly for a hydrogen atom in the Is state that the total energy is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum-mechanical virial theorem. 

Nevertheless, the situation is not completely hopeless. There is a recipe for systematically approaching the wave function of the ground state P0> i- c., the state which delivers the lowest energy E0. This is the variational principle, which holds a very prominent place in all quantum-chemical applications. We recall from standard quantum mechanics that the expectation value of a particular observable represented by the appropriate operator O using any, possibly complex, wave function Etrial that is normalized according to equation (1-10) is given by... 

The motion of the neutron polarization P(t) - the quantum mechanical expectancy value of neutron spin - is described by the Bloch equation... 

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