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Average value in a one-dimensional well

  • Thread starter kreil
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kreil
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Homework Statement


Show that the average value of x2 in the one-dimensional well is

[tex](x^2)_{av}=L^2(\frac{1}{3}-\frac{1}{2n^2 \pi^2})[/tex]


Homework Equations



wave fuction in 1-dim well:
[tex]\psi_n(x)=\sqrt{\frac{2}{L}}sin(\frac{n \pi x}{L})[/tex]

[tex]x^2_{av}=\int_{0}^{L}|\psi(x)|^2 x^2 dx[/tex]


The Attempt at a Solution



Im having trouble evaluating the integral:

[tex]x^2_{av}=\frac{2}{L} \int_0^Lsin^2(\frac{n \pi x}{L})x^2dx[/tex]

i think this needs to be integrated by parts, but could it be in a table somewhere?
 
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Answers and Replies

  • #2
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There should be a table, when I took Quantum mechanics, usually the professor gave a table with solution of some integral(like this one for example), even in exams he did that.
But try to do it manually, more experience..
 
  • #3
13
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yah, simplest way to solve it is integration by parts.
dont forget to use double angles to get rid of
[tex]sin^2[/tex]
 

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