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Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science by Mark F. Horstemeyer

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6.3 THEORETICAL BACKGROUND

In quantum mechanics, a particle is characterized by a wave function ψ(r, t) that contains the information about the spatial state of the particle at time t and position r. The wave function ψ(r, t) is a complex function of the three coordinates x, y, and z, and of the time t. The interpretation of the wave function is as follows: the probability dP(r, t) of the particle being at the time t in a volume element d3r = dxdydz located at the point r is

c06e001 (6.1)

where C is the normalization constant. The probability of finding the particle within the entire volume is of course 1.0, such that the

c06e002 (6.2)

c06e003 (6.3)

Such that the normalization constant C = [∫|ψ(r, t)|2d3r]−1. A point to remember is that the ψ(r, t) must be defined and be continuous everywhere.

For the Schrödinger Equation 6.4, let us consider a particle of mass subjected to the potential V(r,t). The time evolution of the wave function is governed by the Schrödinger equation:

c06e004 (6.4)

where ∇2 is the Laplacian operator. There are several mathematical and physical properties associated with this above-mentioned ...

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