The Hamiltonian of an isotropic harmonic oscillator is invariant under unitary transformations in three dimensions. This well‐known invariance is exploited in a treatment of the Talmi transformation, viz., the transformation of two‐particle oscillator functions to center‐of‐mass and relative coordinates. A simple and transparent form of this transformation in terms of rotation matrices and Wigner coefficients of SU3 is given. The calculation of these Wigner coefficients is described and the problem of degeneracies discussed.

1.
I.
Talmi
,
Helv. Phys. Acta
25
,
185
(
1952
).
2.
R.
Thieberger
,
Nucl. Phys.
2
,
533
(
1956–1957
).
3.
K. W.
Ford
and
E. J.
Konopinski
,
Nucl. Phys.
9
,
218
(
1957–1958
).
4.
M.
Moshinsky
,
Nucl. Phys.
13
,
104
(
1959
).
5.
T. A.
Brody
,
Rev. Mex. Fis.
8
,
139
(
1959
).
6.
R. D.
Lawson
and
M.
Goeppert‐Mayer
,
Phys. Rey.
117
,
174
(
1960
).
7.
V. V.
Balashov
and
V. A.
Eltekov
,
Nucl. Phys.
16
,
423
(
1960
).
8.
A.
Arima
and
T.
Terasawa
,
Prog. Theoret. Phys. (Kyoto)
23
,
115
(
1960
).
9.
T. A. Brody and M. Moshinsky, Tables of Transformation Brackets, (Universidad de Mexico, Mexico City, 1960).
10.
J. M.
Jauch
and
E. L.
Hill
,
Phys. Rev.
57
,
641
(
1940
).
11.
G. A.
Baker
, Jr.
,
Phys. Rev.
103
,
1119
(
1956
).
12.
J. P.
Elliott
,
Proc. Roy. Soc. (London)
245A
,
128
,
562
(
1958
).
13.
V.
Bargmann
and
M.
Moshinsky
,
Nucl. Phys.
18
,
697
(
1960
);
V.
Bargmann
and
M.
Moshinsky
,
23
,
177
(
1961
).,
Nucl. Phys.
14.
M.
Kretzschmar
,
Z. Phys.
157
,
433
(
1960
).
15.
M.
Moshinsky
,
J. Math. Phys.
4
,
1128
(
1963
).
16.
G. Racah, Lectures at the Istanbul Summer School of Theoretical Physics, 1962 (to be published).
17.
R. Sen, “Construction of the Irreducible Representations of SU3,” Ph.D. thesis, Jerusalem, Israel (1963).
18.
G. Racah, Group Theory and Spectroscopy, Lecture Notes, Princeton (1951) (reprint CERN 61‐8).
19.
M. Hamermesh, Group Theory and its Application to Physical Problems (Pergamon Press, New York, 1962), Chap. 10.
20.
H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publications, Inc., New York, 1931), 2nd ed. (revised), Chap. V 12.
21.
M.
Moshinsky
,
Nucl. Phys.
31
,
384
(
1962
).
22.
The operator F2 [Eq. (19)] is closely related to the contraction Q2 of the above‐mentioned tensor of rank 2; one has the operator indentity
Q2 = 2F2+16H2+H12J2
. Furthermore, F2, which is the Casimir operator of U2, is closely related to the Casimir operator G2 of SU3, defined18 as
G216Σi,kAikAki
. It may be verified that the following operator identity holds: 6G212J2+Q2, so that 6G2≡2F2+16H2+H. Hence the eigenvalue of G2 in the representation 12} is given by
g2 = 191222−λ1λ2+3λ1)
in accordance with Eq. (106) of Ref. 18.
23.
The situation is quite different if we restrict SU3 to its subgroup U2, as is done in the applications of SU3 to strong interaction symmetries.24U2 is labeled by three quantum numbers (hypercharge, isospin, and third component of isospin), and consequently in the decomposition of SU3 every representation of U2 occurs at most once, so that the U2 scheme is completely defined.
24.
R. E.
Behrends
,
J.
Dreitlein
,
C.
Fronsdal
, and
W.
Lee
,
Rev. Mod. Phys.
34
,
1
(
1962
).
25.
G.
Racah
,
Rev. Mod. Phys.
21
,
494
(
1949
).
26.
The tensor b is not the Hermitian conjugate of a, but rather b−m = (−)m(am)*.
27.
The matrix elements for all operators under consideration are independent of m; henceforth m will be omitted from the designation of the function.
28.
In establishing the sign of the matrix element, we use the fact that the parity of li, equals the parity of εi, and that furthermore, ε′i = εi±1 in the matrix elements. This allows us to replace (−)l1+l2 by (−)ε22+1 = (−)E+1.
29.
A twofold degeneracy appears for E⩾6, a threefold degeneracy for E⩾12. See Table 2 in Ref. 13.
30.
A. R. Edmonds, Angular Momentum in Quantum Mechanics, (Princeton University Press, Princeton, New Jersey, 1957), Chap. 4.5.
31.
M.
Moshinsky
and
T. A.
Brody
,
Rev. Mex. Fis.
9
,
181
(
1960
).
This content is only available via PDF.
You do not currently have access to this content.