On angular momentum

92 Pages·1952·5.47 MB·English

by  Julian Schwinger

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NXO-3071
ON ANGULAR MOAfENTUM
' 1. Introduction
One of the methods of treating a general angular mo-
mentum in quantum mechanics is to regard it-as the euper-
popition of a number of elementary "epins", or angular
molrenta w i t h 3 = 1/2. Such a spin assembly, considered as
a Rose-Einateln system, can he usefully discussed by the
method of second quantization. We shall see that this pro-
I
cedure unites the compact symbolism of the group theoretical
approach with the explicit operator techniques of quantum
mechanics
e
We introduce spin creation and annihilation operator8
associated w i t h a given spatial reference system,
+ +
ax+ = (a+,a_) and at = (a+,a-),. which aati-sfy
,
The number of spins and the resultant angular momentum m e
-
then given by
With the conventional matrix representation for p, the com-
-
ponents of J appear as
-
J+ = J1 + iJ2 = a++ a,, J, = ~1 iJ2 = a,+a + 9
.
Of course, this realization of the angular momentum commu-
tation properties i n terms of those of harmonic oscillators v
can be introduced without explicit reference t o 'the compos-
ition of spins,
To evaluate the square of the total angular momentum .
we employ the matrix elements of the spin permutation oper-
ator
Thus
I Y
and
According to the commutation relations (1,1), >
Whence
\
(leg)
I
- .
*' '
<I.
no-3071 P. 3
...
I . j = 81 n = 0, 81 , 1, (1.101
--
e?
We further note that, according to (1.3), a state w i t h
F a fixed number of positive,and negative spins also has a
definite magnetic quantum number,
-
m = -1(n n-) , (1.11)
2 +
Therefore, fmm the eigenvector of a state w i t h prescribed
.
r
occupation numbers,
1 ,
(1.12)
- we obtain the angular momentum eigenvector"
.: 1 5
P
-
Familiar as a symbolic expression of the transformation pro-
\
a++$ ,
perties of angular momentm eigenvectors this form is
here a precise operator construction of the eigenvector.
$
On multiplying (1.13) w i t h an analogous monomial con-
structed from the components of the arbitrary spinor
U
A direct proof is given i n Appendix A
*
See, f o r example, H. leyl, The Theory of %Groupsa nd
5 Quantum Mechanics (E. P., Dutton and Company, Inc.,
New Pork, 1931) p. 189
* ,
NYO -3 071 P. 4
we obtain, after summation with respect to m, and then with y
respect t o j,
.
(1.16)
ih which we have written
(1.17)
To illustrate the utility of (1.16), concieved of as an eigen-
vector generating function, we shall verify the orthogonalitg
.
and normalization of the eigenvectors (1.13) Consid.er, then,
& qo
According t o the comutation relations (l.l)#an d a = 0,
P
we have
(1.19)
I
NYO-3071 P. 5
\
(1.21)
As a second elementary example, we shall obtain the
-
1.
matrix elements of powers of J, - by considering the effect
-
of the operators elJ? on (le16). We have
,&I
and therefore
j'm'
which, on expansion, yields the non-vanishing matrix element
id-
s -
Similarly
&
Npo-3071
,
and
A particular consequence of (1.24) and (1.26) is
rbioh details the constfiction of an arbitrary eigenvector
from those possessing the maximum values of )ml compatible
with a given j.
It is &so possible to exhibit an operator which permits
the construction of an arbitrary eigenvector from that pos-
searing the minimum value of j compatible with a given m.
fisaeed, (1.131, written in the form
staters that
. "
mere K+, and two associated operators are defined by
,