2.1 Proofs.
2.2 Proof. Pick a point and look at the behavior of y for both increasing and decreasing x.
2.4 <x> = a/2
<x2>
= a2[(1/3)
- 1/(2n2p2)]
<p> = 0
<p2>
= (n p hbar / a )2
sx
= [a/2][(1/2)
- 2/(n2p2)]1/2
sp
= n p hbar / a
sx
sp
= (hbar/2)[(np)2/3 - 2]1/2,
closest at n=1.
2.5 (a) A = 1/Ö2
(b) Y(x, t)
= (1/Öa)
e -iwt [
sin(px/a)
+ sin(2px/a)
e -3iwt
]
|Y(x, t)|2
= (1/a) [
sin2(px/a)
+ sin2(2px/a)
+ 2 sin(px/a)
sin(2px/a)
cos(3wt)]
(c) <x> = (a/2)
[ 1 - 32/(9 p2)
cos(3wt)]
(d) Use <p> = m d<x>/dt to get
<p> = [8 hbbar /(3a)]sin(3wt)
(d) <H> = (1/2)(E1+E2)
(f) classical frequency
= Ö(10/9) the quantum frequency.
2.6 [almost no algebra] Take the answers for 2.6 and replace 3wt by 3wt -f wherever w occurs.
2.36 Answer given in problem. Use the solution in the text as a template, making modifications as necessary to accomodate the allowed values of x.
| Last revised 2005/02/22 |