Read about:
1. energy eigenstates (and momentum eigenstates) for a constant potential (free particle, V=0...).
gaussian wave-packets:
a) creating them (b(k)) and
b) time dependence (spreading and propagation).
2. Scattering in 1 dimension. what happens when a plan wave encounters a bump, e.g., a delta function?
For next month, read about angular momentum and the hydrogen atom. Chapters 9 and 10 of Liboff look pretty good for that to me. That might be a good notation to follow, i think, except maybe we could leave the 0 off of the a for the Bohr radius...
Thought question: What determines the magnitude of the Bohr radius? (That is, what sets the scale for the size of atoms?)
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I think ultimately the uncertainty principle is responsible for the Bohr radius. The position of the electron cannot be determined accurately (since its momentum would shoot to infinity), so given a certain probability that the electron will a distance "r" from the nucleus, we know that p ~ h/r due to the uncertainty relation. The K.E. is then ~p^2/2m = h^2/2mr^2.
ReplyDeleteThe P.E. depends on the electron's charge "e". Some unit analysis hints at P.E. = -e^2/r(4pi e_o). The denominator is [4 pi epsilon naught multiplied by our likely position for the electron].
So, E = h^2/2mr^2 - e^2/r(4pi e_o).
Taking the derivative with respect to "r" and noting that the Energy wants to be a minimum: dE/dr = -h^2/mr^3 + e^2/r^2(4pi e_0) = 0.
Setting equal to zero in order to minimize E.
SO; r = h^2(4pi e_0)/me^2. (these are all constants) r = .528 angstroms
The plane wave encountering a bump - is this like a repulsive delta function? That is, instead of:
ReplyDeleteV = -(alpha)delta(x - x_0)
we now have:
V = (alpha)delta(x - x_0)
?
Jon,
ReplyDeleteI don't see why you can do that. I see you get the right value but why should that be? The heisenberg uncertainty principle relates delx and delp. You have used the relationship between the standard deviation of momentum with the standard deviation of position, but why can you just say this IS (or even close to) the relationship between its expected momentum and position.