# Structure of the wave function space F

1. Nov 21, 2008

### norbert

hello all

2. Nov 21, 2008

### norbert

Re: testing

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 $$\psi$$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

Last edited: Nov 22, 2008
3. Nov 22, 2008

### malawi_glenn

Re: testing

Yes, and more?

4. Nov 22, 2008

### norbert

Re: testing

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]

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 AuthorÂ´s comment is not clear for me
Can someone explain me this a little better?

thank you