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1) Meaning of Bound and Scattering States.

The bound states I have studied thus far are limited to the infinite square well and the quantum harmonic oscillator. In the case of a harmonic oscillator, a particle is able to "leak" out of its potential well.

a) Would I be right in saying that even bound states can experience this "leaking" into classically forbidden zones (finite potentials only), but is limited to the vicinity of its "well"?

b) On the other hand, is it right to say that a scattering state can not only be found in classically forbidden zones, but is itself a also travelling wave (unlike the bound state)? I.e it is not bound by any sort of "well" and is free to move about

2) Delta-function potential.

If we were to work out the (bound-state) wave-function for a potential ##-\alpha \delta (x)##, this would yield a wave-function peaked at x = 0. However, there is a non-zero probability that we can find said particle outside the delta-function well.

a) Is this infinitely deep well not analogous to an infinite square well, except that its depth is from 0 to ##-\infty## rather than 0 to ##\infty##? Why would we still expect that the particle can be found outside of the vicinity of x = 0?

b) Is the Delta-function potential considered to be an even, or odd function? Scouring the internet seems to bring up conflicting opinions.

3) Even potentials and their solutions.

a) It was stated in the book that an even potential gives rise to either even or odd solutions to the Schrodinger equation. Say I obtain the even and odd solutions seperately, in some domain of x. Do I then sum the even and odd solutions in their respective domains to get the actual wavefunction? I.e is the actual wavefunction for an even potential a sum of even and odd solutions?

Thanks very much in advance!

PS: Would it be possible to discuss these in a not-so-technical manner for now?