Issued Wed, Oct 2 Due Fri, Oct 11

(Note: Exam 1 is Th, Oct 3, 7:30 PM in Muen E064)

Required reading for this week: Gas. Ch. 5.

There are HINTS for this homework!

1) Gas #5-1

2) Gas 5-7. The very last part of Gas' question ("and that it will yield...") should be restated as follows:

Show that if you're looking for a bound state, then you should replace . Then, clearly state an argument that for a bound state, the denominators in the matrix should vanish. Finally, show that this gives the correct result for the bound state energy of the delta function (if lambda is negative)

(If you have any problems or questions with either problem 1 or 2, you should first look at Griffith's "Introduction to Q.M."! (there's a copy on reserve in the library) Section 2.7 (p.66-67))

3) Gas 5-4

(I completely ignored Gas' hints on this problem, and suggest you do too)

4) Gas 5-11

5) Consider a "double square well" potential, as shown.

Let V0 and a be fixed,

(and be large enough that

several bound states occur).

This problem is meant to be qualitative, no calculations required!

a) Sketch the ground-state wave function u1(x), and the first excited state u2(x) for the cases i) b=0, ii) b = a and iii) b>>a. (Briefly, discuss your sketches.)

b) Qualitatively, how do the corresponding energies (E1 and E2) vary, as b goes from 0 to infinity? Sketch E1(b) and E2(b) on one graph. (Briefly, discuss your sketch.)

c) The double square well (like the double delta function well from class) is a primitive 1-D model for the potential an electron feels in a diatomic molecule. (The wells represent the electrical attraction of the nuclei) If the nuclei are free to move, they will find the configuration of lowest energy. In view of your conclusions in b) above, does the electron tend to draw the nuclei together, or push them apart? (Never mind the internuclear repulsion, which is something separate!)

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