Time-independent wave equation description

The objective of WFT is the exact solution of the time-independent Schrodinger equation (TISE), H V = ET, for a system of interest. (We recall that in quantum mechanics, associated with each measurable parameter in a physical system is an operator, and the operator associated with the energy of a system is called the Hamiltonian H. The Hamiltonian contains the operations associated with the kinetic and potential energies of aU particles that comprise a system. We further note that the terms function, operator, and functional are to be understood such that a function is a prescription which maps one or more numbers to another number, an operator is a prescription which maps one function to another function, and a functional takes a function and provides a number.) The solution to the TISE yields the wave function T as weU as the energy E for the system of interest. In a systematic, variational search one looks for the wave function that produces the lowest energy, and arrives at a description for the system in its ground state. 

Shock-compressed solids and shock-compression processes have been described in this book from a perspective of solid state physics and solid state chemistry. This viewpoint has been developed independently from the traditional emphasis on mechanical deformation as determined from measurements of shock and particle velocities, or from time-resolved wave profiles. The physical and chemical studies show that the mechanical descriptions provide an overly restrictive basis for identifying and quantifying shock processes in solids. These equations of state or strength investigations are certainly necessary to the description of shock-compressed matter, and are of great value, but they are not sufficient to develop a fundamental understanding of the processes. 

In many applications of quantum mechanics in physics and chemistry, interest is primarily in the description of the stationary, or time-independent, states of a system. Thus, it is sufficient to determine the energies and wave-functions with the use of the Schitidinger equation in the form... 

Various difficulties of classical physics, including inadequate description of atoms and molecules, led to new ways of visualizing physical realities, ways which are embodied in the methods of quantum mechanics. Quantum mechanics is based on the description of particle motion by a wave function, satisfying the Schrodinger equation, which in its time-independent form is ... 

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

Inherent to the optical matrix description of the reflectivity of neutron beams is the assumption that the neutron beam is a wave. Alternatively and entirely equivalently, the neutron could be viewed as a quantum mechnical particle with the reflection process the result of the interaction of the neutron with a potential energy barrier, V. The reflectivity is then the modulus squared of the reflected wave amplitude, the latter quantity being related to the potential energy barrier by the time-independent Schrodinger wave equation. 

The use of wave groups or wave packets in physics, and certainly in chemistry, was limited to a few theoretical examples in the applications of quantum mechanics. The solution of the time-dependent Schrodinger equation for a particle in a box, or for a harmonic oscillator, and the elucidation of the uncertainty principle by superposition of waves are two of these examples. However, essentially all theoretical problems are presented as solutions in the time-independent frame picture. In part, this practice is due to the desire to start from a quantum-state description. But, more importantly, it was due to the lack of experimental ability to synthesize wave packets. 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

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