testing

[norbert]

hello all

[norbert]

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 \psi1(r) and
\psi2(r) \in F. then*

\psi(r) = \lambda1\psi1(r) + \lambda2\psi2(r) \in F

where \lambda1 and \lambda2 are two arbitrary complex numbers

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

\psi(r)


|\psi(r)|2 = |\lambda1|2|\psi1(r)|2 + |\lambda2|2|\psi2(r)|2 + \lambda1*\lambda2\psi1^{}*(r)\psi2(r)+\lambda1\lam bda2\psi1(r)\psi2*(r)


|\psi(r)|2 is therefore smaller than a function whose
integral converges, since \psi1
and \psi2 are aquare-integrable

[malawi_glenn]

Yes, and more?

[norbert]

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:

|\lambda1||\lambda2|[|\psi1(r)|2 + \psi2(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