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

1. Dec 13, 2008

### norbert

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
$$\psi$$1(r) and $$\psi$$2(r) $$\in$$ F. then

$$\psi$$(r) = $$\lambda$$1$$\psi$$1(r)+$$\lambda$$2$$\psi$$2(r) $$\in$$ F

where $$\lambda$$1 and $$\lambda$$2 are two arbitrary complex numbers

In order to show that $$\psi$$(r) is square integrable
expand |$$\left|$$$$\psi$$(r)|2 :

|$$\psi$$(r)|2 = |$$\lambda$$1|2|$$\psi$$1(r)|2+|$$\lambda$$2|2|$$\psi$$2(r)|2+$$\lambda$$1*$$\lambda$$2$$\psi$$1$$^{}*$$(r)$$\psi$$2(r)+$$\lambda$$1$$\lambda$$2*$$\psi$$1(r)$$\psi$$2*(r) (A-3)

(*) simbol means complex-conjugate

|$$\psi$$(r)|2 is therefore smaller than a function whose
integral converges, since $$\psi$$1 and $$\psi$$2 are square-integrable

On my last comment referred to the space
functions F we have the |$$\psi$$(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:

|$$\lambda$$1||$$\lambda$$2|[|$$\psi$$1(r)|2 + |$$\psi$$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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