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Structure of the wave function space F

  1. Nov 21, 2008 #1
    hello all
  2. jcsd
  3. Nov 21, 2008 #2
    Re: testing

    I have some concerns about Structure of the wave function space F Im refering to chapter II of QUANTUM MECHANICS OF Cohen-Tannoudji
    The item A-1.a of this chapter say:

    It can easily be shown that F satisfies all the criteria of a vector space. As an example, we demostrate that if [tex]\psi[/tex]1(r) and
    [tex]\psi[/tex]2(r) [tex]\in[/tex] F. then*

    [tex]\psi[/tex](r) = [tex]\lambda[/tex]1[tex]\psi[/tex]1(r) + [tex]\lambda[/tex]2[tex]\psi[/tex]2(r) [tex]\in[/tex] F

    where [tex]\lambda[/tex]1 and [tex]\lambda[/tex]2 are two arbitrary complex numbers

    In order to show that [tex]\psi[/tex](r) is square integrable
    expand [tex]\left|[/tex] [tex]\psi[/tex](r)|2 :


    |[tex]\psi[/tex](r)|2 = |[tex]\lambda[/tex]1|2|[tex]\psi[/tex]1(r)|2 + |[tex]\lambda[/tex]2|2|[tex]\psi[/tex]2(r)|2 + [tex]\lambda[/tex]1*[tex]\lambda[/tex]2[tex]\psi[/tex]1[tex]^{}*[/tex](r)[tex]\psi[/tex]2(r)+[tex]\lambda[/tex]1[tex]\lambda[/tex]2[tex]\psi[/tex]1(r)[tex]\psi[/tex]2*(r)

    |[tex]\psi[/tex](r)|2 is therefore smaller than a function whose
    integral converges, since [tex]\psi[/tex]1
    and [tex]\psi[/tex]2 are aquare-integrable
    Last edited: Nov 22, 2008
  4. Nov 22, 2008 #3


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    Re: testing

    Yes, and more?
  5. Nov 22, 2008 #4
    Re: testing

    On my last comment referred to the space functions F we have the |[tex]\psi[/tex](r)|2 expanded expression given by (A-3)

    The last two terms of (A-3) have the same modulus, wich has as an upper limit:

    |[tex]\lambda[/tex]1||[tex]\lambda[/tex]2|[|[tex]\psi[/tex]1(r)|2 + [tex]\psi[/tex]2(r)|2]

    Its OK, tha last two terms have the same modulus.

    The question is:
    Why the last two terms of (A-3) have the above expression??
    What does mean "upper limit"??
    What is the relation of this question with "triangular inequality" referred to complex-variable?
    see Churchil -----"Analysis of complex-variable"-----

    The Author´s comment is not clear for me
    Can someone explain me this a little better?

    thank you
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