I know you probably are getting to hate me for posting too much HW, not giving you enough warning, etc, etc., but here is another HW that will help you be ready for Tuesday's class, in which we will solve the H-atom problem. In fact just being aware of these questions will help you be prepared: [Comments and questions below would be much appreciated (and will get you credit for this assignment). How about: if you comment, then you don't have to hand it in. How does that seem? PS. These problems are probably more important, right now, than 7 and 8 from problem set #7. ]
** The quarter will be over soon. I think we have only 2 more weeks --4 more chances to meet and discuss quantum mechanics. I hope you don't miss this rare and fleeting opportunity to learn about quantum physics.
1. For a particle of mass m in the potential, V(r) = -e^2/r (in units, as Liboff uses, in which e^2/r has units of energy):
a) What is a characteristic length scale that you can construct using e^2, hbar and m?
b) How do the units of e^2 compare with the units of alpha, where alpha is the strength of a 1D delta function? (comments welcome)
[notice the similarity between the form of the characteristic lengths for this potential, -e^2/r, and the 1D delta function, -alpha delta(x).] (Please comment below)
c) What is a characteristic kinetic energy scale associated with that length scale?
d) For this potential function, what is a characteristic potential energy scale that you can make with that length scale?
2. check later please.
reading: On Tuesday we will go at a brisk pace through the process of presenting the Schrodinger eqn in 3D in spherical coordinates (Del^2) (bring your 3D glasses if you have them), separation of variables, solving for L^2 eigenstates, Ylm, solving the radial equation with the substitution u(r)=rR(r). This is pretty standard and you can read about it in any quantum book if you want to be prepared in that way.
[in Liboff, it is a little scattered: pages 367-380, more or less, treat the theta-phi part and the angular momentum eigenstates; then page 413 shows the Laplacian in spherical coordinates and pages 446 to 449, more of less, cover the hydrogen atom energies, eigenstates, degeneracies, wave function shapes and hybridization*,...
The most interested part will be the way in which we discover the quantization of energy when we solve the radial eqn. We will use the characteristic length scale in that and show how both length scale and energy are quantized together...
* hybridization refers to combining degenerate eigenstates to produce new bases of degeneracy manifolds. This provide a choice of ways to look at and use the eigenstates.
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Thanks for the page references
ReplyDeleteFor 1b, since Int[delta[x],{x,-1,1}] = 1 is unitless, and dx has units of length, then delta[x] has units of 1/length. However, if V = -alpha*delta[x], then alpha must have units of lenght*energy. Does that reasoning seem correct?
ReplyDeletethat reasoning sounds correct to me, MikeK.
ReplyDeleteme too
ReplyDeleteI imagine you may have gone over this on Tuesday in class...
ReplyDeleteFor 1a, if the potential is in terms of energy, then I'm getting e^2 must be of the form energy*length. So I started with (e^2)/r, decomposed it into basic units (Kg, meters, seconds), and got length goes like :
e^2 s^2
-----------
m^2 kg
Which is like:
e^2 kg
-----------
p^2
Does that seem reasonable? There should probably be an h-bar in there somewhere...
Ahh, I see it was indeed gone over in class in the form of a quiz.
ReplyDelete(my face) <------ (egg)
For 1c), If we take the KE operator from Schrodinger's (-hbar^2/2m)Del^2, and we know the characteristic length is a=hbar^2/me^2, are we simply able to substitute to get:
ReplyDeleteKE ~ [-ae^2/2]Del^2 ?
The units work out to be (Jm^2)Del^2, but I think the Del^2 operator works to get rid of the m^2 term.
1(c)
ReplyDeleteKE for delta is m*alpha^2/(2*hbar^2)
e^2 ~ alpha as seen from (b)
so KE ~ m*e^4/(2*hbar^2) for V(r)=e^2/r
1(d)
ReplyDeletePlug answer from (a) into r for V(r)=-e^2/r?
Then V(r)~ -m*e^4/hbar^2
compared to KE, KE=V/2... reminds me of the virial theorem.