Structure of the wave function space F

[norbert]

hello all

 

[norbert]

I have some concerns about Structure of the wave function space F I am referring 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\lambda₂\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, which 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