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Orthonormal basis

  • Thread starter asi123
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  • #1
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Homework Statement



Hey guys.

http://img39.imageshack.us/img39/2345/27760913.jpg [Broken]

I need to show that these wave functions are orthonormal.
I'm a bit confuse, what's i and what's j?
I mean, do I need to take both of the functions, put them in the integral and to show that the result is the Kronecker delta?
Can I neglect the exponent for this?

Thanks a lot.


Homework Equations





The Attempt at a Solution

 
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Answers and Replies

  • #2
180
4
i and j are just arbitrary indices. Yes you have to calculate the integrals for different cases. When i=j the exponential cancels (when you take the complex conjugate it changes sign). When i and j aren't equal you'll get some exponential dependence as well but in that case you should get zero anyway.
 
  • #3
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i and j are just arbitrary indices. Yes you have to calculate the integrals for different cases. When i=j the exponential cancels (when you take the complex conjugate it changes sign). When i and j aren't equal you'll get some exponential dependence as well but in that case you should get zero anyway.
Well, where are i and j in my problem?
I mean, this is not a series, it's a function.
 
  • #4
gabbagabbahey
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Well, where are i and j in my problem?
I mean, this is not a series, it's a function.
You have two wave functions, [itex]\psi_1[/itex] and [itex]\psi_2[/itex], so the indices [itex]i[/itex] and [itex]j[/itex] can each take on the values [itex]1[/itex] and [itex]2[/itex].
 
  • #5
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You have two wave functions, [itex]\psi_1[/itex] and [itex]\psi_2[/itex], so the indices [itex]i[/itex] and [itex]j[/itex] can each take on the values [itex]1[/itex] and [itex]2[/itex].
Yeah, but I don't have i and j inside the functions so how can I come up with the kronecker delta?

How can I show that if i=j then it's 1 and if i does not equal to j, it's 0 if I don't have i and j?

Thanks.
 
  • #6
gabbagabbahey
Homework Helper
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Yeah, but I don't have i and j inside the functions so how can I come up with the kronecker delta?

How can I show that if i=j then it's 1 and if i does not equal to j, it's 0 if I don't have i and j?

Thanks.
Showing that

[tex]\int \psi_i \psi_j dx =\delta_{ij}[/tex]

just means that you need to show:

[tex]\int \psi_1 \psi_1 dx =\int \psi_2 \psi_2 dx =1[/tex]

and

[tex]\int \psi_1 \psi_2 dx=\int \psi_2 \psi_1 dx =0[/tex]
 
  • #7
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Showing that

[tex]\int \psi_i \psi_j dx =\delta_{ij}[/tex]

just means that you need to show:

[tex]\int \psi_1 \psi_1 dx =\int \psi_2 \psi_2 dx =1[/tex]

and

[tex]\int \psi_1 \psi_2 dx=\int \psi_2 \psi_1 dx =0[/tex]
Oh, now I get it.

Thanks a lot.
 
  • #8
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Well, here is the second part of the question

http://img207.imageshack.us/img207/879/95899388.jpg [Broken]

I also posted there answer.
I think they have a mistake, I marked it in the red box.
Shouldn't it be A^2=1/2 ?
Am I missing something?

Thanks a lot.
 
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