Jan 31, 2002

Summary of angular momentum formalisms Summary of angular momentum formalisms

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Coordinate representation of orbital angular momentum

In spherical polar coordinates, the operator representing the squared angular momentum L2 takes the form:

L2 = -(h/2p)2 ì
í
î 		1
sinq

q sinq
q
 + 1
sin2 q
 2
f2
 ü
ý
þ 		,
(1)
while the operator representing z-component of angular momentum takes the form:
Lz = -i (h/2p)
f
 .
(2)
The spherical harmonic functions Ylm are eigenfunctions of both L2 and Lz with
L2 Ylm = (h/2p)2 l(l+1)
(3)
and
Lz Ylm = (h/2p) m.
(4)
Some of these spherical harmonic functions are:
Y00 = 1
4p
(5)
Y10 =   æ
 ú
Ö
3
4p
 
 cosq
(6)
Y1±1 = -±   æ
 ú
Ö
3
8p
 
 sinqe±i f
(7)

In the process of evaluating the differential eigenvalue equations, we find that the ``quantum numbers'' l must be postive integers (l = 0, 1, 2, ¼), and m is restricted to the integer values between -l £ m £ l.

Operator representation of general angular momentum

The following derivation follows the discussion of Shankar's text ( Principles of Quantum Mechanics, 2nd edition, Chapter 12). It turns out that a very similar eigenvalue structure can be derived in an operator formalism. In this operator formalism, we will see that additional half-integer solutions for the angular momentum quantum numbers are also possible. For this generalization we will use J2 and Jz to represent the square and z- components of the angular momentum, respectively. Furthermore, we will assume that we can find the eigenvalues of these operators which we will denote by a and b for the moment:

J2 |abñ = a |abñ
(8)
Jz |abñ = b |abñ.
(9)
We can now introduce 2 other operators which will prove to be very helpful:
J± º Jx ±Jy.
(10)
We can show that these operators have the effect of incrementing or decrementing the b eigenvalue of |abñ by one.

First we note the following commutation relations:

[Jz,J±] = ±(h/2p) J±
(11)
and
[J2, J±] = 0.
(12)
Later, we will also need to use the result
[J-,J+] = -2 (h/2p) Jz,
(13)
which follows from the identity
[Jx,Jy] = i (h/2p) Jz.
(14)
We can then show that the function (J±|abñ) has eigenvalues a and b ±(h/2p) of J2 and Jz, respectively. Acting on (J±|abñ) with J2:
J2 (J±|abñ) = J± J2 |abñ = J± a |abñ = a (J± |abñ).
(15)
Acting on (J±|abñ) with Jz:
Jz(J±|abñ) = ±(h/2p) |abñ+ J± Jz |abñ = ±(h/2p) |abñ+ J± b |abñ = (±(h/2p) + b) (J±|abñ).
(16)
This mean that we can write the function (J± |abñ) as N |a(b±(h/2p))ñ, where N is a normalization constant determined from:
N2 áa(b±(h/2p))|a(b±(h/2p))ñ = áab|J±f J± |abñ = áab|(J2 - Jz2 -±(h/2p) Jz|abñ = a - b2 -±(h/2p) b,
(17)
assuming that áab||abñ = 1. This result means that
N = Ö
a-b2 -±(h/2p) b
 
 .
(18)
In order to make further progress, we notice that since the normalization cannot be negative, for a given value of a, there are restrictions on the value of b. In particular, we can safely assume that there is a maximum value of b which we will denote by bmax. From the behavior of a maximum value, we know that
J+ |abmaxñ = 0.
(19)
Now multiplying the above equation by J-, we find
J-J+ |abmaxñ = 0 = (Jx2 + Jy2 + i[Jx,Jy])|abmaxñ = (J2 - Jz2 - (h/2p) Jz) |abmaxñ = a -b2max - (h/2p) bmax.
(20)
This defines the eigenvalue a in terms of bmax to be
a = bmax( bmax + (h/2p)).
(21)
We can also use Eq. (18) to argue that b has a minimum value bmin and analyzing the properties of |abminñ using similar steps as above, we can also show that
a = bmin( bmin - (h/2p)).
(22)
Comparing Eqs. (21) and (22), it is apparent that
bmin = - bmax.
(23)
It is now convenient to define bmax º (h/2p) j so that the eigenvalue a can be written
a = (h/2p)2 j(j+1).
(24)
This analysis then suggests that if we define a general value of the eigenvalue b to take the form
b º (h/2p) mj,
(25)
the results tell us that mj can take the values -j £ mj £ j, (2j+1 different values in all for a given j). With these definitions, the normalized increment or decrement operation can be written:
J± |j mj ñ = (h/2p) Ö
j(j+1) - mj(mj ±1)
 
 |j (mj ±1)ñ.
(26)
This structure of the eigenvalues jmj is very similar to the eigenvalues of orbital angular moment lm. There is one new ``wrinkle'', however. The above arguments tell us that we can get from the maximum value of mj = j to the minimum value mj = -j in a number of applications of the operator J-. Suppose that that number of applications is U. This means that the sequence of values of the eigenvalue mj is
j, j-1, j-2, ¼j-U,
(27)
so that
j-U = -j
(28)
or
j = U
2
 .
(29)
Since U must be an integer, j can be an integer if U is even, but can also be a half-integer if U is odd!! This means that we can use this formalism to describe orbital, spin, and total angular momentum.

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On 31 Jan 2002, 13:38.