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qmwaveequation03152001

# qmwaveequation03152001 - Waves and the Schroedinger...

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Waves and the Schroedinger Equation 5 april 2010 1 The Wave Equation We have seen from previous discussions that the wave-particle duality of matter requires we describe entities through some wave-form based represe- nation. The most natural consideration are classical waves, and arriving at a way to describe their spatial and temporal evolution. In the following discussion, pursuing this description for quantum entities will lead us to the Schroedinger equation, our starting point for treating atomic and molecular systems. The motion of classical, non-dispersive waves requires some definitions: frequency = 1 T = ν wavelength = λ velocity = v = λν A general expression for a wave moving in the +x direction: ψ ( x, t ) = A sin 2 π x λ - t T ψ ( x, t ) = A sin 2 πx λ - 2 πt T ψ ( x, t ) = A sin ( kx - ωt ) wavevector, k k = | k | = 2 π λ wave vector units of inverse length ( 1 length ) Recall: λ = h p (1) 1

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Angular frequency (radians/second) ω = 2 πν Note that a phase-shift can be introduced in order to change the origin of the waveform: ψ ( x, t ) = A sin ( kx - ωt + φ ) Now, let’s consider stationary, or standing, waves ; for such entities, the nodes remain fixed in time and space (though the wave is moving with velocity, v). Standing waves are generated from interference of waves of equal frequency and amplitude traveling in opposite directions . ψ ( x, t ) = A sin ( kx - ωt ) + A sin ( kx + ωt ) ψ ( x, t ) = 2 Asin ( kx ) cos ( ωt ) ψ ( x, t ) = ψ ( x ) cos ( ωt ) (2) ψ ( x, t ) = ψ ( x ) | {z } time - independent cos ( ωt ) | {z } time - dependent Thus, from the last expression, we see that stationary waves have fixed nodal points; zero amplitude versus time at fixed points). Now, we have gone about things in a reverse manner, but we can consider the following. We have written a representation of a wave-particle entity as a sinusoidal function.
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