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This 7 page Class Notes was uploaded by Citlalli Sauer on Tuesday October 20, 2015. The Class Notes belongs to PHYS410 at San Diego State University taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/225321/phys410-san-diego-state-university in Physics 2 at San Diego State University.
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Date Created: 10/20/15
Lecture 3 Outline Probability Density 0 Probability theory continuous variables 132 o Normalisation and complex numbers 14 continuous vs discrete variables 0 Probability of measurement between a and LE dx ie Px padr Where prob density 0 thus probability of measurement between a and b b Pab d9 Discrete Continuous 1 Silo PO 1 fl p93d95 ltjgt 2390ij ltxgt f1 aspltxgtdx ltfjgt 2390 fjPj ltf93gt fl f56p95d93 02 Aj2gt ltj2gt W2 02 A2gt 99gt 56gt2 Rock falling off cliff Grif ths Example 11 cliff height h Rock accelerates 9 position 9315 9152 Velocity ill f gt ie dt 3 1 Total ight time T ie 93T h Probability camera ashes in interval dt is dt dx 9 dx 9 g 1 dx T gtlZh glZxVZh Zxhx Probability density px 2 O S a S h See Grif ths Figure 16 probability density px Rock still falling Off cliff Grif ths Example 11 contcl 0 Probability density px O S a S h hltgtd 1 h id 1Mill 1 9 92 9 92 A NEo NE 0 h 1 h 1 1 2 3 h 9 a xdx midx x ltgt pltgt Mf0 Mg 0 Wavefunction Normalisation Wavefunction x 15 probability density IJU t2 00 IJ9IS2 dx 1 But how does this work with the Schr dinger Eqn 8IJU t E2 32IJU t 75 7 V 15 Il 15 Note if11xt is a soln then so is A x t nonnormalisable trivial soln x t O nonnormalisable solns Where f IJU t2 00 squareintegrable solns f11xt2 1 Wavefunction Evolution 0 Assuming f Ilxt O2 dx 1 What about t later 0 ie does the Schrodinger Eqn keep it normalised d 00 00 E m Ilxt2 dx OO q 2 6w mu da Using the product rule a 2 a myav Given the Schrodinger Eqn and its conjugate 8amp1 23971 6211 i 6amp1quot 23971 WWquot 239 11 11 3t 2m 3x2 EV and 3t 2m 3x2 EV Wavefunction Evolution contd w g a 1 2m 3 a IJ 2 3t x7tl And the Change in the normalisation is given by d 00 00 8 iii 8Q 8Q IJ 2 IJ 11 d dt 00 I Sam 00 3m 2m 3x 3x 9 00 iii 8 8V IJ IJ 2m 3x 3x O 0 Assuming lim 9 gt 00 x t gt O
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