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W is real valued?

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Homework Statement


The question is in the post below.

Show the equation W posted is real valued.




The Attempt at a Solution


The idea is to show that W conjugate = W but there is a complex exponential in W which makes things tricky?
 
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Answers and Replies

  • #2
HallsofIvy
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You have posted a link to a restricted web-site.
 
  • #3
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Okay. I have put the equation up in the document. Show W is real valued.
 

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  • #4
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Is it because psi conjugated produces a negative in the y value so y has only an imaginary component. mod spi sqaured is real. since y has a factor or i, it will cancel with the i already apparent in the exponential. So everything in the integral is positive hence W is positive.
 
  • #5
HallsofIvy
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Yes. The imaginary part of the integrand is an odd function ([itex]e^{iy}= cos(y)+ i sin(y)[/itex] and sin is odd) so its integral over a region symmetric about 0 ([itex]-\infty[/itex] to [itex]\infty[/itex] is 0.
 
  • #6
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That is very neat. But is my long explanation also correct? Although in my explantion, I said that y is purely imaginary. Is that correct would sin(y) make sense then?

In that integral y is claimed to be the integration variable but how does that make sense? Why do you need an integration variable? why not integrate wrt p or x?

This means y can't be complex valued which raises the question why are the arguments in psi have plus and minus y/2 for non conjugate and conjugate psi.
 
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  • #7
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That is very neat. But is my long explanation also correct? Although in my explantion, I said that y is purely imaginary. Is that correct would sin(y) make sense then?

In that integral y is claimed to be the integration variable but how does that make sense? Why do you need an integration variable? why not integrate wrt p or x?

This means y can't be complex valued which raises the question why are the arguments in psi have plus and minus y/2 for non conjugate and conjugate psi.
[itex] y [/itex] is a dummy variable. Imagine if you had a function:

[tex] f(x) = \sum_{n=1}^x 1 [/tex]

The integral is no different, [itex] y [/itex] is used to "increment" (so to speak), just as [itex] n [/itex] is.
 
  • #8
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[itex] y [/itex] is a dummy variable. Imagine if you had a function:

[tex] f(x) = \sum_{n=1}^x 1 [/tex]

The integral is no different, [itex] y [/itex] is used to "increment" (so to speak), just as [itex] n [/itex] is.
Right, W is a function of x and p and the integral has nothing to do with x or p, it is there to evaluate a number with constants x and p.
 

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