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Homework Help: Hilbert Space

  1. Oct 9, 2007 #1
    [SOLVED] Hilbert Space

    1. The problem statement, all variables and given/known data
    For What Values of [tex]\psi(x)=\frac{1}{x^{\alpha}}[/tex] belong in a Hilbert Sapce?


    2. Relevant equations
    [tex]\int x^{a}=\frac{1}{a+1} x^{a+1} [/tex]


    3. The attempt at a solution
    I tried to use the condition that function in Hilbert space should satisfy:
    [tex]\int\psi^{2}=A[/tex] but it seems always infinite exist in x=0 or x=infinite
     
    Last edited: Oct 9, 2007
  2. jcsd
  3. Oct 9, 2007 #2

    Gokul43201

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    Why are you writing the integral of x^a, when you want to examine the integral of 1/x^{2a}?

    Also, have you written down the question completely? What is the domain on which [itex]\psi(x)[/itex] is defined?
     
  4. Oct 9, 2007 #3
    one is alpha and another is a. I just write a integral equation, a = 2*alpha. I think it is not the key point. The question is complete. I asked my professor if there is some constrain of x, she said x can be any value. that is to say the integral will from minus infinite to infinite
     
  5. Oct 9, 2007 #4

    Gokul43201

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    Okay, they both looked the same to me.
    It can not be. There's at least a couple of missing words. Here's one way to write a somewhat more complete question:

    For what values of [itex]\alpha [/itex] does [itex]\psi(x)=1/{x^{\alpha}}[/itex] belong in a Hilbert space?

    This needs to be specified in the question. You have not completely specified a function unless you describe its domain.
     
  6. Oct 9, 2007 #5
    Thanks, I explained it in the interval of minus infinite to minus zero and zero to infinite. I guess a wave function with singularity is not a good one in physic....
     
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