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lecture07_umn - Lecture 7 Free Particle Wave Function The...

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Lecture 7 Free Particle Wave Function The Particle in a Box Representing Wave Functions September 23 2009 Chapter 3 McQuarrie and Simon pages 73-105
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Free Particle Time-independent Schrödinger equation in one dimension (7-1) For a free particle (particle being free over all space), the potential energy V is everywhere zero . This leads to a simplified equation (7-2) Eq. 7-2 has a simpler looking form if we select a constant (7-3) " h 2 2 m d 2 dx 2 + V ( x ) # $ % ( ) x ( ) = E ) x ( ) " h 2 2 m d 2 dx 2 " E # $ % ( ) x ( ) = 0 k = 2 mE h
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Free Particle Wave Function We may write eq. 7-2 as (7-4) This is one of the simplest possible differential equations. It has solutions of the form (7-5) A and B are arbitrary constants for the general case of eq. 7-4. If we wish to ensure normalization, there are some rather tricky details that arise, but we will ignore them for this particular system as they do not really matter. " x ( ) = Ae ikx + Be # ikx d 2 dx 2 + k 2 " # $ % ( x ( ) = 0
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Solutions of the differential equation Let us focus on the important details of this wave function. First, there is no quantization . From eq. 7-3, it is apparent that all positive values of E are allowed (positive because the particle has only kinetic energy, and kinetic energy is bounded from below by zero). (7-3) " x ( ) = Ae ikx + Be # ikx k = 2 mE h
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Series Expansion Let us use the series expansion of the function e ix (7-6) If we were to take A = B in eq. 7-5, we would have (7-7) e ix = 1 + ix " 1 2! x 2 " 1 3! ix 3 + 1 4! x 4 + 1 5! ix 5 + L = 1 " 1 2! x 2 + 1 4! x 4 " L # $ % ( + i x " 1 3! x 3 + 1 5! x 5 + L # $ % ( = cos x + i sin x " x ( ) = N cos kx e " ix = cos x " i sin x
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Free Particle Wave Function And if we were to take A = B in eq. 7-5, we would have (7-8) where N is some arbitrary normalization constant. So, a free particle has a wave function that can be represented as either a sine or cosine function, which is to say, it is highly delocalized. The wavelength of the wave function decreases with higher energy and with greater mass (recall the definition of k ). " x ( ) = N sin kx k = 2 mE h
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Because the free particle is not quantized , we sometimes say it is ‘in the continuum’ , meaning that there is a continuous range of energies available to it. The free particle wave function is readily generalized
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

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lecture07_umn - Lecture 7 Free Particle Wave Function The...

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