Quantum Mechanics: k basis

  • Thread starter rsaad
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  • #1
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Homework Statement



write the following in K basis:

A=∫|x><x|dx where the integral limits are from -a to a


Homework Equations





The Attempt at a Solution



I tried solving it by inserting the identity
I=∫|k><k|dk where the integral limits are from -∞ to +∞

but then I do not know how to proceed from there. What to do about the two integrals with varying limits!
 

Answers and Replies

  • #2
CompuChip
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Why is it a problem that the integrals have different limits?

More relevant question to help your forward: what is <k|x> ?
 
  • #3
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<k|x>= exp(-ikx)/(2*pi)^0.5
 
  • #4
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I am getting a very weird answer.
 
  • #5
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I introduced the identity twice and on simplifying, I get 1/2pi ∫∫dk dx ??
 
  • #6
CompuChip
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If you introduce the identity twice, you should use two different integration variables. So I expect a triple integration, e.g. over x, k and k'.
 
  • #7
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Yes, I know that. I simplified things and I got that answer.
 
  • #8
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Could you please solve the solve question and suggest the steps?
 
  • #9
CompuChip
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OK, so I was thinking

[tex]\int |x\rangle \langle x | \, dx =
\iiint |k\rangle \langle k | x\rangle \langle x | |k'\rangle \langle k' | \, dx \, dk \, dk'
\propto \iiint e^{-i(k - k')x}|k\rangle \langle k' | \, dx \, dk \, dk'[/tex]

Is that where you got to as well?

And then you go on to use
[tex]\int e^{i(k - k')x} \, dx \propto \delta(k - k')[/tex]
but I don't see how the |k> <k'| disappeared from your suggested answer... after all, what you should get is similar in form to |x> <x|.
 
Last edited:

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