Physics 501, Problem Set 5 Solutions
1) In class we obtained the general formula for atomic transition rates involving the emission of a single photon:
e2 k 3
Rif (k)d2 =
8 2 0
d2 | < n | (k) r|m > |2 .
Here we must sum over polarization vectors and
Physics 501, Midterm Exam with Solutions
1) Consider a particle moving in one dimensional with a delta-function potential and Hamiltonian:
where may be positive or negative.
a) For what values of do one or more bound states exi
Physics 501 Problem Set 1 Solutions
1) Probability current conservation implies:
1 |f |2 = |1 + f+ |2 .
|f |2 + |f+ |2 = 2Ref+ .
1 + f = 1 + f+ .
This can be rearranged to give:
2) Continuity at the origin gives:
The derivative as x approaches
Physics 501 Problem Set 3 Solutions
1a) The Hamiltonian in the region r < R is:
Vhf VS + B
where I is the unit matrix. The matrix of constants has eigenvalues:
(VT VS B)2 /4 + Vhf
= (1/2)(VT + VS + B)
VT VS B thes
Physics 501, Problem Set 4 Solutions
(1)n q 2n
2 n=0 (2n + 1)!
drV (r)r2n+1 .
sin2 (/2) = [1 cos ]/2 = (1 u)/2.
q 2 = 2k 2 (1 u).
we see that
Thus the coecient is
an (u) =
2m [2(u 1)]n
2 (2n + 1)!
drV (r)r2n+2 .
Physics 501, Problem Set 6 Solutions
1a) Noting that
x = x , y = y , z = z
cfw_ i , j = 2 ij I
it can be seen that
where I is the 2 2 identity matrix. It then follows that these matrices also obey the Dirac algebra.
REMARK: To show that this set
Physics 501 Problem Set 2 Solutions
1a) Because eikz is a solution of the V = 0 Schroedinger equation it can be expanded in a complete set of solutions of
that equation. We might expect this to involve a sum over all the spherical harmonics but the ones w