Phys 7211: Homework Set 11
Problem 1 (Barton #8.1 p. 197)
Solve J t D
2
2
x
expIx 2 4 D tM
N y = 0 using yH x, tL = dx K0 H x  x , t  t0 L yH x , t0 L with K0 Hx, tL =
4pDt

(i) Clearly y H x, tL = x is the solution with yH x, 0L = x. The magic rule
Phys 7211: Homework Set 10
Problem 1 (Barton #6.1 p. 154)
Given that rHrL = 0 for 0 8 x, y< p and that  y H x, 0L = 1, x H0, yL = 1, and x Hp, yL = y H x, pL = 0, the magic rule gives
p
p
yHrL = dS Hn yL H Hr r L up to the irrelevant additive constant
Phys 7211: Homework Set 9
Again, the veritable obscenity 0 will be taken as 0 = 1.
Problem 1 (Barton #5.1 p. 138)
(i) By Gauss' Law and the spherical symmetry of the problem, E =
potential then is ymax =
qmax
4pr
q
4 p r2
`
r and Emax = 3 106 V/m, thus qm
Phys 7211: Homework Set 8
Throughout this homework set, the veritable obscenity 0 will be taken as 0 = 1. Theorists will rejoice. Experimenalists and
engineers will kvetch. The mathematicians will ask me to disprove this by counterexample. You know I like
Phys 7211: Homework Set 7
Problem 1 (A&W #8.3.1 p. 514)
This is almost trivial. in fact, I'm not really sure what A&W want here. So, I guess, "by math" we have
I2 +k 2 M Ha1 y1 + a2 y2 L = I2 +k 2 M a1 y1 + I2 +k 2 M a2 y2 = a1 I2 +k 2 M y1 + a2 I2 +k 2 M
Phys 7211: Homework Set 6
Problem 1 (Barton #2.8 p. 69) Note: This is an IVP, not a BVP
.
(i) Working in analogy to Barton #2.7 (Problem 5 from HW 5), we have Equation of motion for t 0 is x  w 2 x = a sin HW tL
x2 Ht L x1 HtL x1 Ht L x2 HtL
W Ht L
with
Phys 7211: Homework Set 5
Problem 1 (Barton #2.1, p. 67)
(i) If y1 H xL and y2 H xL are two linearly independent solutions to y + qH xL y + rH xL y 0, then the Wronskian
x
W = y1 y2  y1 y2 = 0 and, by Abel's formula, ln W H xL =  dx qH x L. Considering
Phys 7211: Homework Set 4
Problem 1 (Barton #1.8, p. 39)
Given the Schrdinger equation 
d2 y
dx2
+ l dH xL y = k 2 y, we solve for the wave function of a particle that is incident, from the
left, on the delta function potential. We can assume a solution
Phys 7211: Homework Set 2
Problem 1 (AW p. 395398)
Ex . 6.1.1:
(a) z1 =
1
x+i y
=
xi y
1
x+i y xi y
=
xi y
x2 + y2
(b) In polar form, z = r cosHfL + i y sinHfL = r expHi fL where r =
x2 + y2 and f = AtanI x M = atanI x M + sgnH yL QH xL p. Note:
y
y
Phys 7211: Homework Set 1
Problem 1 (AW p. 241)
Ex . 4.1.3: The special linear group SL(2) consists of all 2x2 matrices with determinant +1. Show
that all such matrices form a group.
Answer: Consider A,B, and CSL(2).
1) Since A and B are 2x2 matrices, the