Physics 312 – Problem Set 10
1. Shankar Exercise 21.1.3 (pg 589)
2. Shankar Exercise 21.1.7 (pg 597) (submit)
3. Shankar Exercise 21.1.8 (pg 597) (submit)
4. Shankar Exercise 21.1.9 (pg 598)
5. Shankar Exercise 21.1.11 (pg 600)
6. (submit) Find the propagator K(xb , tb ; xa , ta ) for the Lagrangian
L=
1
1
mẋ2 − mω 2 x2 − e(t)x .
2
2
Your result will be in terms of time integrals over the function e(t), along with sine functions. Show that you
recapture the expected results for
• e(t) = 0.
• e(t) = constant, ω = 0.
• e(t)=0, ω=0.
7. (submit) Consider the function
Z
∞
K(h̄) =
dtexp(iS(t)/h̄) .
−∞
Using the method of stationary phases, show that regions where S 0 (t) 6= 0 can be neglected in the limit where
h̄ → 0. Do this by considering a region a ≤ t ≤ b where S 0 (t) 6= 0, change variables, and show that this term
goes to zero faster than those with S 0 (t) = 0. (For a “well-behaved” S(t).)
8. Shankar Exercise 21.1.16 (pg 609) (submit)
1