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HW1 solutions

HW1 solutions - Physics 4120 Spring 2008 Homework#1...

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Physics 4120 Spring 2008 Homework #1 Solutions 1) 2)This problem is much easier if you use a spreadsheet rather than do every calculation by hand. J n(j) P(j) j*P(j) Δj Δj^2 Δj^2*P(j) j^2 j^2*P(j) 17 1 0.038462 0.653846 -4.5 20.25 0.778846 289 11.11538 18 3 0.115385 2.076923 -3.5 12.25 1.413462 324 37.38462 20 2 0.076923 1.538462 -1.5 2.25 0.173077 400 30.76923 21 4 0.153846 3.230769 -0.5 0.25 0.038462 441 67.84615 22 8 0.307692 6.769231 0.5 0.25 0.076923 484 148.9231 23 5 0.192308 4.423077 1.5 2.25 0.432692 529 101.7308 24 2 0.076923 1.846154 2.5 6.25 0.480769 576 44.30769 25 1 0.038462 0.961538 3.5 12.25 0.471154 625 24.03846 Sum: 26 1 21.5 3.865385 466.1154 N <j> σ^2 <j^2> a) From our spreadsheet calculation, <j> = 21.5 b) j for each j is given in the spreadsheet above c) The highlighted cell above shows σ 2 calculated using our original equation. To find σ, we just take the square root, so σ = 1.966 d) j 2 for each j is given in the spreadsheet above. e) From our spreadsheet calculation, <j 2 > = 466.115 f) To find σ 2 using the equation we derived in problem 1, we just take ‹j 2 › - ‹j› 2 = 466.115 – 21.5 2 = 466.115 – 462.25 = 3.865 = σ 2 , so σ = 1.966 and the two methods agree as expected. ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 j j j j j j j P j j jP j j P j j P j j P j j j P j j P j j j j j j j j j j j j j j j j j j - = + - = + - = + - = + - = + - = - = σ

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3) a) To find A, we impose the restraint that the integral of our probability function is 1.
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HW1 solutions - Physics 4120 Spring 2008 Homework#1...

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