68111087-2-1-Schrodinger-Equation - Chapter 2: The...

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Chapter 2: The Schrödinger equation and its applications* Outline 2.1 Wave Functions of a Single Particle. 2.2 The Schrodinger Equation. 2.3 Particle in a Time-Independent Potential. 2.4 Probability Density and Probability Current. 2.5 Scalar Product of Wave Functions; Operators. In this chapter we will discuss one of the central equation of the quantum physics. The Schrodinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. We have seen that electrons and photons behave in a very similar fashion—both exhibit diffraction effects, as in the double slit experiment, both have particle like or quantum behavior. We can in fact give a complete analysis of photon behavior—we can figure out how the electromagnetic wave propagates, using Maxwell’s equations, then find the probability that a photon is in a given small volume of space dxdydz , is proportional to | E | 2 dxdydz , the energy density. On the other hand, our analysis of the electron’s behavior is incomplete—we know that it must also be described by a wave function analogous to E , such that gives the probability of finding the electron in a small volume dxdydz around the point ( x , y , z ) at the time t . However, we do not yet have the analog of Maxwell’s equations to tell us how ψ varies in time and space . The purpose of this section is to give a plausible derivation of such an equation by examining how the Maxwell wave equation works for a single-particle (photon) wave, and constructing parallel equations for particles which, unlike photons, have nonzero rest mass. Maxwell’s Wave Equation Let us examine what Maxwell’s equations tell us about the motion of the simplest type of electromagnetic wave—a monochromatic wave in empty space, with no currents or charges present. First, we briefly review the derivation of the wave equation from Maxwell’s equations in empty space: Where , is speed of light. The continuity equation can either be regarded as an empirical law expressing (local) charge conservation, or can be derived as a consequence of two of Maxwell's equations.
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To derive the wave equation, we take the curl of the fourth equation, together with the vector operator identity, gives that Meanwhile we note that, this equations shows that time dependent electric field produce magnetic field and time dependent magnetic field produce electric field. The fields are perpendicular to each others and they produce electromagnetic wave. This propety has found many technological applications. (antenna, accelerators etc). For a plane wave moving in the x -direction this reduces to The solution to this wave equation has the form (Another possible solution is proportional to cos( kx - ωt ) or cos( kx - ωt ). In order to find relation between parameters we applying the wave equation differential operator to our plane wave solution and it leads to This is just the familiar statement that the wave must travel at c .
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This note was uploaded on 01/12/2012 for the course CHEM 133 taught by Professor Staff during the Spring '08 term at UCSD.

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68111087-2-1-Schrodinger-Equation - Chapter 2: The...

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