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Reviewing a possible wrong result of QUANTUM MECHANICS text by Cohen-Tannoudji

  1. Dec 13, 2008 #1
    I have some concerns about Structure of the wave function space F Im refering to QUANTUM MECHANICS chapter II of Cohen-Tannoudji text
    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]
    [/tex]1(r)[tex]\psi[/tex]2*(r) (A-3)

    (*) simbol means complex-conjugate

    |[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 square-integrable

    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]

    It is OK, the last two terms of (A-3) have the same modulus.

    The questions are:

    Why has the author used the above expression as "upper limit" term?
    What is the relation of this question with "triangular inequality" referred to complex-variable?

    bearing in mind that the last two terms of (A-3) have the same modules
    I have expressed some ideas about this problem in the attached file

    see Churchil text -----"Analysis of complex-variable"-----

    The Author´s comment is not clear for me
    Can someone explain me this a little better?
    all suggestions will be welcome

    thank you all


    Attached Files:

  2. jcsd
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