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Expectation Value

  1. Jun 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the expectation value <x> if:
    from 0 <= x <= a, psi = A x/a
    from a <= x <= b, psi = A(b-x)/(b-a)

    Normalizing gives me that A = sqrt(3/b) (verified correct)


    2. Relevant equations



    3. The attempt at a solution
    [tex]<x> = \int_0^b x \psi^2 dx = \int_0^a \frac{A^2 x^3}{a^2}dx + \int_a^b A^2 \frac{x (b-x)^2 }{(b-a)^2} dx[/tex]


    [tex]\int_0^a A^2 \frac{ x^3 }{ a^2} = A^2 \frac{ a^2 }{4} = 3/4 a^2/b[/tex]

    [tex]A^2 \int_a^b \frac{x(b-x)^2}{(b-a)^2} dx = -A^2 \frac{x (b-x)^3}{(b-a)^2} - A^2 \frac{(b-x)^4}{12(b-a)^2}[/tex] eval from b to a

    [tex]<x> = \frac{3a^2}{4b } + \frac{+ 3(b-a)a}{b} + \frac{3(b-a)^2}{12b} = \frac{-8a^2 + b^2 + 10ab}{4b}[/tex]

    The answer is supposed to be (2a+b)/4... Does anyone see where I've gone wrong? I have double checked the antideriv. for the x(b-x)^2 integral and it seems to work.
     
    Last edited: Jun 27, 2009
  2. jcsd
  3. Jun 27, 2009 #2

    Dick

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    The antiderivative of the x(b-x)^2 does NOT work. Try it again. How did you get that form anyway?
     
  4. Jun 27, 2009 #3
    Thanks for your reply

    I used integration by parts,

    u = x
    du = dx

    dv = (b-x)^2
    v = -(b-x)^3/3

    vu - int(v du)

    [tex]\frac{-(b-x)^3 x}{3} - \frac{(b-x)^4}{12}[/tex]

    so
    [tex]A^2 \int_a^b \frac{x(b-x)^2}{(b-a)^2} dx = \frac{A^2}{(b-a)^2} * (\frac{-x(b-x)^3}{4} - \frac{(b-x)^4}{12})[/tex]
    I ran through it again but I can't seem to see what I am doing incorrectly.
     
    Last edited: Jun 27, 2009
  5. Jun 27, 2009 #4
    The 0 to a integration seems to be correct.

    The a to b integration seems little bit weird. The approach that you did by using the integration by part can lead you to the answer.

    [tex]A^2 \int_a^b \frac{x(b-x)^2}{(b-a)^2} dx = \frac{A^2}{(b-a)^2} * (\frac{-x(b-x)^3}{3} - \frac{(b-x)^4}{12})[/tex]

    [tex] = \frac{3}{b} (\frac{a(b-a)}{3}+\frac{(b-a)^2}{12})[/tex]

    Now you can get <x> value by doing:

    [tex]<x>=\frac{3a^2}{4b}+\frac{4a(b-a)}{4b}+\frac{(b-a)^2}{4b}=\frac{2a+b}{4}[/tex]

    Pretty much a straight forward process.

    Hope it helped =)
     
  6. Jun 27, 2009 #5

    Dick

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    In the original post you were missing the denominator 3 on the (b-x)^3 part. In post 3 you had it except in the last line you replaced it with a 4. You are doing everything right. I think you are just making a bookkeeping error. You could also integrate x*(b-x)^2 by just expanding it into xb^2-2bx^2-x^3. That's what I did.
     
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