# Help with a extract of Schaum

1. Jun 15, 2011

### atomqwerty

Hello,
In http://img96.imageshack.us/img96/2825/sinttulobdd.png" [Broken] from Schaum's Quantum Mechanics, I don't get why they say that A2 (with ' ) must be zero; I think in that case $\Phi_{II}(x)$ will diverge if x=+infinite, and we don't want that function to diverge, do we?

Thanks!

Last edited by a moderator: May 5, 2017
2. Jun 15, 2011

### vanhees71

With this step you simply fulfill the boundary condition that there should be incoming particles only from the left and not from the right.

3. Jun 15, 2011

### atomqwerty

Yes, but it you put the condition that when x->infinite, the wave function must be zero, the it doesn't match!
Maybe this condition that I say is just when E<V...

4. Jun 16, 2011

### vanhees71

If you are in the continuous part of the spectrum your "eigenstates" are not Hilbert-space vectors but generalized eigenstates. They belong to the dual of the nuclear space in the sense of the G'elfand triple (="rigged Hilbert space").

The most simple example are the "eigenstates" of the momentum operator. These are (in position representation) just the plane waves (here for simplicity written for the one-dimensional motion of a particle)

$$u_p(x)=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} x p).$$

This is not a square-integrable function. It can not be normalized to 1 in the usual sense but only to a $\delta$ distribution, according to

$$\langle p_1|p_2 \rangle = \int_{-\infty}^{\infty} \mathrm{d} x \; u_{p_1}^*(x) u_{p_2}(x)=\delta(p_1-p_2).$$