Issued Wed, Oct 30 Due Wed, Nov 6

Note: Exam 2 will be Thur, Nov 7, 7:30 PM in Muen E064

Exam 2 will cover Gas. through Ch 7, with emphasis on material in Chapters 5 and 6, plus the extra lecture material I gave to complement Ch. 6. (Ch. 7 is really just an application of what we learned in Ch. 6, so in that sense is good practise and fair game for the exam)

Required reading for this week: Get started on Ch. 8

Note: The object in these problems is to start thinking in an "operatory" way, rather than a "wave functiony" way. E.g. you should not do (or even write down) any integrals for problems 1 and 2. Think about operator properties!

1) Gas. 7.3

2) Gas. 7.10

3) Gas 7.11a only.

Don't just use the results from H directly, but rather SHOW that N has eigenvalues n=0,1,2... from scratch, using the same ideas and principles as Gas did (and we did in class) for showing that H had discrete eigenvalues.

4) Consider the Hamiltonian describing a one-dimensional particle in a gravitational field, .

a) Calculate the equation of motion for the operators x(t) and p(t),

using (7-63), and the commutation relation .

b) Show that the equation of motion is just the classical equation of motion.

c) Solve for p(t) and x(t) in terms of p(0) and x(0).

(The only other constants appearing in your answer should be m and g).

d) Show that .

(Operators that commute at the same time need not commute at different times)

Extra Credit: Gas 7.12

Note: Here is an interactive Java applet which helps demonstrate the results of the previous problem!