Chem370_HO_01_11_08

Chem370_HO_01_11_08 - CHM 370 Handout One The Mean value of...

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Unformatted text preview: CHM 370 Handout One The Mean value of a Random Value_ The definition of the average (mean) value may be given as ' I . N v = lim -1— [value of v obtained in the ith measurement] N-e- N 1-! The rationale behind this is that the variable v is a random variable. This may result from variations in experimental procedure or. as will be discovered later, from the very nature of the system. In either case. the number of measurements N is extended to a large number in order to define best the average of the observable measured. with the number extended to infinity providing the true mean value of the system. (That is. the larger the statistical sample, the more accurate the measure.) In quantum mechanics. one will find that because of the nature of the system, data points will occur several times. We now define the function N(v,.) to be the number of times data point v; occurs in the series of measurements. Let M be the number of different data points in the statistical sample. Using the function N(v,.). the summation becomes ' N v ) five] 1-! N N M 2 [value of v obtained in the ith measurement] = 2v, a: I'Il ii] I"! Since the summation in brackets merely counts the number of times v.- shows up, the overall summation simply becomes 7 M Nan is] j-l i-l Upon returning to the definition of the average value. we note that the limit operation on the summation is linear. that is, the sum of several limits equals the limit of the sum Along with the multiplication of l/N , then (hang-fife Ntv.)= igniv.[§§d]=iv[fim&;2] is! in] . The definition of a probability function P(v,) is This is the fraction of all measurements that result in v.1. where extending the number of measurements to infinity yields the most accurate fraction. (Example: flipping a fair coin has a one-half probability of landing on the obverse [heads] side. meaning that as the number of coin flips becomes infinite. heads will show up half of the time.) Substituting, then CHM 370 HandoutL Two Proof that E(T,V) z kT2 6T _ V From equations in the lecture notes, 1 E(T, V) 2 Z(T,V)ZEexp[~ But Em} g] = W [aexpE-TE/kTUV. Comparing equations (1) and (2) gives 5W”): Z(1E,V)ZkT2[a—afexp[%fly allE z 2(r1, V) kT2[aiT§exp[_—gfl]y However, Thus equation (3) becOmes or equivaiently the desired result. (1) (2) (3) (4) ...
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Chem370_HO_01_11_08 - CHM 370 Handout One The Mean value of...

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