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 $$\psi$$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)


|$$\psi$$(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)


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

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