Kinetic energy quantum mechanics

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, ),... 

In addition to the usual terms of kinetic energy and potential energy, quantum mechanical particles possess another property called spin, which has the dimensions of angular momentum. The values of spin are quantized to half integer or integer multiples ofB. In the following we omit the factor of ft when we discuss spin values, for brevity. If the total spin of a particle is s then there are 2s -k 1 states associated with it, because the projection of the spin onto a particular axis can have that many possible values, ranging from +s to -s in increments of 1. The axis of spin projection is usually labeled the z axis, so a spin of s = 1/2 can have projections on the z axis = +1 /2 and = -1/2 sl spin of s = 1 can have = -1,0, +1, and so on. 

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

When we wish to replace the quantum mechanical operators with the corresponding classical variables, the well-known expression for the kinetic energy in hyperspherical coordinates [73] is... 

Traditionally, for molecular systems, one proceeds by considering the electronic Hamiltonian which consists of the quantum mechanical operators for the kinetic energy of the electrons, their mutual Coulombic repulsions, and... 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

The theory of chemical reactions has many facets including elaborate quantum mechanical scattering approaches that treat the kinetic energy of atoms by proper wave mechanical methods. These approaches to chemical reaction theory go far beyond the capabilities of a product like HyperChem as many of the ideas are yet to have wide-spread practical implementations. 

Schrodinger postulated that the form of the hamiltonian in quantum mechanics is obtained by replacing the kinetic energy in Equation (1.20), giving... 

Just as for an atom, the hamiltonian H for a diatomic or polyatomic molecule is the sum of the kinetic energy T, or its quantum mechanical equivalent, and the potential energy V, as in Equation (1.20). In a molecule the kinetic energy T consists of contributions and from the motions of the electrons and nuclei, respectively. The potential energy comprises two terms, and F , due to coulombic repulsions between the electrons and between the nuclei, respectively, and a third term Fg , due to attractive forces between the electrons and nuclei, giving... 

Freeman, A. J., and Lowdin, P.-O., On a quantum mechanical kinetic energy transformation."... 

---