intermediate quantum 491: homework 1 solutions g
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Intermediate Quantum 491: Homework 1 Solutions
1-1
Consider the sets of test grades gi = 75.71 71.43 41.43 64.29 77.14 65.7158.57 68.57 24.29 54.29 48.57 68.57 78.57 61.43 58.57
Then we get the mean = 61.1, STD = 14.6.
Figure 1: Histogram of data with 10 bins from 0 to 100: 〈x〉 = 60.3 ∆x = 14.1
Let bin centers be x[i] and number in each bin n[i] with∑
n[i] = Nthe total number of grades.
Then mean
〈x〉 =
∑n[i]x[i]
N
and variance
(∆x)2 =
∑n[i](x[i]− 〈x〉)2
N − 1
Where we divide by N-1 because we used the data to calculate themean.
The probability to have a grade in the ith bin is estimated from thisdata as p[i] = n[i]/N .
Michael Gold
Michael Gold
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Michael Gold
(1.4)
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Michael Gold
(1.6)
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Michael Gold
This P is for inside the range. Outside is 1-P = 1/4
Michael Gold
(1.8)
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Michael Gold
+
Michael Gold
P has is dimensionless probability, so J has dimensions of inverse time.
Michael Gold
(1.9)
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Michael Gold
= -(1/𝛕) P
Michael Gold
I check dimensions as c𝞽 = ħc/(2𝝘) = (energy*distance)/(energy)
Michael Gold
(1.10)
Michael Gold
P(t) = exp(-t/𝞽)
Michael Gold
then,
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Michael Gold
(8) (1.12)