Lecture_09

Lecture_09 - PHYS 342 Fall 2010 Lecture Lecture 09:...

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PHYS 342 Fall 2010 ecture 09: Lecture 09: Expectation Values, Operators Ron Reifenberger irck anotechnology Center Birck Nanotechnology Center Purdue University Lecture 09 1
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Measuring the position of an oscillating mass No. of occurrences Measured value ring n 1 x 1 n 2 x 2 n 3 x 3 +Force spring constant k (N/m) .. =0 n m x m x0 x final hat is the average value of x? U(x) E What is the average value of x? 1 ii NN i nx N n   K.E. 11 1 ; i N i i xN N n 0 x o x o 2
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Whenwedonothaveapred ictab leoutcomeforameasurement ,wemustdea l ith robability ensities nd/or robability istributions onsider ome The probability distribution with probability densities and/or probability distributions. Consider some quantity that varies continuously, for example the height of a Purdue student. We could coerce a group of students to report their height and we could make a plot of height in 10-cm intervals as shown below. Nobody will be exactly 180 cm or 190 cm tall, so we just group students into height intervals and systematically round things up at the edges of each bin. If we want finer detail, we could group heights in 1-cm intervals or 1-mm intervals, which will continue to make things discrete. But as the intervals 3 become smaller, the histogram resembles a continuous curve. Thiscurve is known as the probability distribution . http://theochemlab.asu.edu/teaching/phy571/supp02.pdf
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If there is a known probability distribution for height, P(h), then () hP h dh hP h dh hh      1 Phdh  hP h dh if area under P h is normalized to unity  n QM the Probability Density is defined as P= * In QM, the Probability Density is defined as P= Ψ Ψ , so we say the expectation value of a quantity, say a particle’s position x, is given by * ** ) () xd x x x dx IF x x is normalized to unity      * () () x dx   4
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Example: What is the expectation value for the position (x ) and position squared (x 2 ) of an electron in the n=2 state of an infinite square well? U(x) n (x) E Levels n n 1E o 22 8 hm L 2 cos x L L    2 x 2 2 24 E o E 48 L 8 m L sin x LL 23 s x n =2 0 L/2 L/2 3 9E o 4 16E 98 16 8 L cos sin x L 2 <x>=expectation value for x: o 2 /2 *2 () s i n n 0 L x x x x d x x d x   even odd 2 sin n L xx    5
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2 ven ven <x 2 >=expectation value for x 2 : 2 /2 22 * 2 2 () s in n L x x LL x x d x x d x      even even 2 2 2 sin 1 s i n 2 n L L xx x      3 23 2 21 cos 2 2 6 48 x L L   0.0708 0.
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This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue.

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Lecture_09 - PHYS 342 Fall 2010 Lecture Lecture 09:...

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