Nuclear Physics
Physics 422
Homework VI
Due March 9, 2006
1. (F-H 8.1) We wish to derive the commutation relation between
J
x
and
J
y
.
(a) There are two ways to treat rotations in quantum mechanics. One can rotate the
states or rotate the operators. Suppose is some quantum mechanical state and ^
F
is
some operator. Let ^
U
y
(
) be the unitary opertor which rotates the wavefunction about the
y-axis:
0
(
~r
) = ^
U
y
(
)(
~r
). Show that the matrix element of operator ^
F
between rotated
states is equivalent to the matrix element of the rotated operator, ^
F
R
=
U
y
y
(
)^
FU
y
(
),
between unrotated states.
(b) Assume
~
F
= (
F
x
; F
y
; F
z
) is a vector (i.e. with the standard behavior under rotations.)
Consider an innitesimal rotation of
~
F
by the angle
about the y-axis. Write down the
components of the rotated vector,
~
F
R
in terms of
~
F
and
.
(c) As you learned previously,
J
y
is the generator of rotations about the y-axis, i.e.
U
y
(
) =
e
iJ
y
(
h
=1). Use the innitesimal rotation results from (b) with
~
F
!
~
J
to derive the
commutation relation between
J
x
and
J
y
.
2. Consider the addition of orbital (
~
L
) and spin (
~s
) angular momentum the individual
nucleons of the ourine nucleus. The conguration space states are labeled by the orbital
angular momentum quantum numbers,
l
and
m
l
; and the spin states are labeled by the
spin angular momentum quantum numbers,
s
= 1
=
2 and
m
s
. Your task is to construct
angular momentum states of good TOTAL angular momentum labeled as
j
and
m
j
. (a)
Consider the specic case of
l
= 2 and
s
= 1
=
2. How many angular momentum states are
there altogether? Show that you get same number of states from combining the orbital
and spin angular smomentum states to form states of good total angular momentum.
(b) Using the Clebsch-gordon tables, write down the following states of good total angular
momentum in terms of the individual orbital and spin angular momentum states:
j
l
=
2
; m
l
i
and
j
s
= 1
=
2
; m
s
i
.
(i)
j
j
= 5
=
2
; m
j
= +1
=
2
i
(ii)
j
j
= 3
=
2
; m
j
= +1
=
2
i
(iii) Invert these results to nd the product states:
j
l
= 2
; m
l
= 1
ij
s
= 1
=
2
; m
s
=
1
=
2
i
and
j
l
= 2
; m
l
= 0
ij
s
= 1
=
2
; m
s
= +1
=
2
i
in terms of the states of good total angular
momentum.
3. Consider the vector addition of isospin in the pion-nucleon system. Specically, using
the Clebsch-gordon tables, write down the following states of good total isospin in terms
of the individual isospin states:
j
i
>
=
j
t
= 1
; t
i
i
and
j
N
i
>
=
j
t
= 1
=
2
; t
i
i
.
(a)
j
3
=
2
;
+1
=
2
i
(b)
j
1
=
2
;
+1
=
2
i
(c) Invert the results from (a) and (b) to nd the product states:
j
1
;
+1
ij
1
=
2
;
1
=
2
i
and
j
1
;
0
ij
1
=
2
;
+1
=
2
i
in terms of the states of good total isospin.
4. (F-H 8.6) Justify that the isospin of the deuteron is zero: (a) By using experimental
information (namely, that there is no bound state of two neutrons), and (b) By considering
the generalized Pauli exclusion principle whereby the total wave function, assumed to be
