a μ
þ
ð Þ
¼ e
iδ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ
ð
Þ j þ μ þ 1
ð
Þ
p
δ : an arbitrary real number
ð
Þ ,
ð3:99Þ
where e
iδ is a phase factor. From (3.96) we also get
a μ
À
ð Þ
¼ e
Àiδ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ þ 1
ð
Þ j þ μ
ð
Þ
p
:
ð3:100Þ
In (3.99) and (3.100), we routinely put δ ¼ 0 so that a μ
(+) and a μ
(À) can be positive
numbers. Explicitly rewriting (3.77), we get
J
þ
ð Þ
jζ, μi ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ
ð
Þ j þ μ þ 1
ð
Þ
p
jζ, μ þ 1i,
J
À
ð Þ
jζ, μi ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ þ 1
ð
Þ j þ μ
ð
Þ
p
jζ, μ À 1i,
ð3:101Þ
where j is a fixed given number chosen from among zero, positive integers, and
positive half-integers (or half-odd-integers).
As discussed above, we have derived various properties and relations with respect
to the generalized angular momentum on the basis of (i) the relation (3.30) or (3.69)
and (ii) the fact that J x , J y , and J z are Hermitian operators. This notion is very useful
in dealing with various angular momenta of different origins (e.g., orbital angular
momentum, spin angular momentum) from a unified point of view. In Chap. 20, we
will revisit this issue in more detail.
3.5 Orbital Angular Momentum: Operator Approach
In Sect. 3.4 we have derived various important results on angular momenta on the
basis of the commutation relations (3.69) and the assumption that J x , J y , and J z are
Hermitian. Now, let us return to the discussion on orbital angular momenta we dealt
with in Sects. 3.2 and 3.3. First, we treat the orbital angular momenta via operator
approach. This approach enables us to understand why a quantity j introduced in
Sect. 3.4 takes a value zero or positive integers with the orbital angular momenta. In
the next section (Sect 3.6) we will deal with the related issues by an analytical
method.
In (3.28) we introduced differential operators L
(+) and L
(À) . According to Sect.
3.4, we define following operators to eliminate ħ so that we can deal with dimensionless quantities:
M L=ħ ¼ e 1 e 2 e 3
ð
Þ
M x
M y
M z
0
B
@
1
C
A,
3.5 Orbital Angular Momentum: Operator Approach
77
þ
ð Þ
¼ e
iδ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ
ð
Þ j þ μ þ 1
ð
Þ
p
δ : an arbitrary real number
ð
Þ ,
ð3:99Þ
where e
iδ is a phase factor. From (3.96) we also get
a μ
À
ð Þ
¼ e
Àiδ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ þ 1
ð
Þ j þ μ
ð
Þ
p
:
ð3:100Þ
In (3.99) and (3.100), we routinely put δ ¼ 0 so that a μ
(+) and a μ
(À) can be positive
numbers. Explicitly rewriting (3.77), we get
J
þ
ð Þ
jζ, μi ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ
ð
Þ j þ μ þ 1
ð
Þ
p
jζ, μ þ 1i,
J
À
ð Þ
jζ, μi ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j À μ þ 1
ð
Þ j þ μ
ð
Þ
p
jζ, μ À 1i,
ð3:101Þ
where j is a fixed given number chosen from among zero, positive integers, and
positive half-integers (or half-odd-integers).
As discussed above, we have derived various properties and relations with respect
to the generalized angular momentum on the basis of (i) the relation (3.30) or (3.69)
and (ii) the fact that J x , J y , and J z are Hermitian operators. This notion is very useful
in dealing with various angular momenta of different origins (e.g., orbital angular
momentum, spin angular momentum) from a unified point of view. In Chap. 20, we
will revisit this issue in more detail.
3.5 Orbital Angular Momentum: Operator Approach
In Sect. 3.4 we have derived various important results on angular momenta on the
basis of the commutation relations (3.69) and the assumption that J x , J y , and J z are
Hermitian. Now, let us return to the discussion on orbital angular momenta we dealt
with in Sects. 3.2 and 3.3. First, we treat the orbital angular momenta via operator
approach. This approach enables us to understand why a quantity j introduced in
Sect. 3.4 takes a value zero or positive integers with the orbital angular momenta. In
the next section (Sect 3.6) we will deal with the related issues by an analytical
method.
In (3.28) we introduced differential operators L
(+) and L
(À) . According to Sect.
3.4, we define following operators to eliminate ħ so that we can deal with dimensionless quantities:
M L=ħ ¼ e 1 e 2 e 3
ð
Þ
M x
M y
M z
0
B
@
1
C
A,
3.5 Orbital Angular Momentum: Operator Approach
77