I have some concerns about Structure of the wave function space(adsbygoogle = window.adsbygoogle || []).push({}); FIm refering to QUANTUM MECHANICS chapter II of Cohen-Tannoudji text

The item A-1.a of this chapter say:

it can easily be shown thatFsatisfies 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) (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

NORBERT

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

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