A generalized class of invariants, I (t), for the three‐dimensional, time‐dependent harmonic oscillator is presented in both classical and quantum mechanics. For convenience a simple notation for types of harmonic oscillator is introduced. Two interpretations, one in terms of angular momentum and the other employing a canonical transformation, are offered for I (t). An invariant symmetric tensor, Imn(t), is constructed and shown to reduce to Fradkin’s invariant tensor for time‐independent systems. The usual SU(3) (compact) or SU(2,1) (noncompact) is shown to be a noninvariance group for the time‐dependent oscillator with S{U(2) ⊗U(1) } as the invariance subgroup. Extensions to anisotropic systems and the singular quadratic perturbation problem are discussed.

1.
H. R.
Lewis
, Jr.,
Phys. Rev. Lett.
18
,
510
,
636
(
1967
).
Google Scholar
Crossref
Search ADS
 
2.
H. R.
Lewis
, Jr.,
J. Math. Phys.
9
,
1976
(
1968
).
Google Scholar
Crossref
Search ADS
 
3.
H. R.
Lewis
, Jr. and
W. B.
Riesenfeld
,
J. Math. Phys.
10
,
1458
(
1969
).
Google Scholar
Crossref
Search ADS
 
4.
M.
Kruskal
,
J. Math. Phys.
3
,
806
(
1962
).
Google Scholar
Crossref
Search ADS
 
5.
E.
Pinney
,
Proc. Am. Math. Soc.
1
,
681
(
1950
).
Google Scholar
 
6.
C. J.
Eliezer
 and
A.
Gray
,
S.I.A.M.
30
,
463
(
1976
).
Google Scholar
 
7.
D. M.
Fradkin
,
Am. J. Phys.
33
,
207
(
1965
).
Google Scholar
Crossref
Search ADS
 
8.
D. M.
Fradkin
,
Prog. Theor. Phys.
37
,
798
(
1967
).
Google Scholar
Crossref
Search ADS
 
9.
J. M.
Jauch
 and
E. L.
Hill
,
Phys. Rev.
57
,
641
(
1940
).
Google Scholar
Crossref
Search ADS
 
10.
W. R.
Hamilton
,
Proc. R. Irish Acad.
3
,
344
(
1847
);
Google Scholar
 
C. Runge, Vektoranalysis (Dutton, New York, 1919), Vol. 1;
W.
Lenz
,
Z. Phys.
24
,
197
(
1924
).
Google Scholar
Crossref
Search ADS
 
11.
The Runge‐Lenz vector for the Schrödinger‐Coulomb Hamiltonian, H = 12p2−Ze2/r, 
R = 12(L×p−p×L)+Ze2r̂
, has magnitude R = a(1−ε) and lies along the line 0R from the center of force to the perihelion of the ellipse
1r = 1l(l+1){Rcosθ+1}
, where
e = electron
charge,
a = length
of the semimajor axis,
l = angular
momentum quantum
number = 0
, 1, 2, ⋯,
θ = plane
polar angle measured from 0R, and
ε = eccentricity
of the ellipse. The constancy of R([H,R] = 0), may be viewed as an expression of the constancy of the orientation of the axes of the ellipse. A similar interpretation applies to Hmn. The commutation relations for H≡E<0, 
[Lk,Lm] = iεkmnLn
,
[Rk,Lm] = iεkmnRn
,
Rk,Rm] = iεkmn(−2HZ2e−4)Ln
, are an algebraic statement that the hydrogen atom possesses the dynamical symmetry O(4) = SO(3)⊗SO(3).
12.
P.
Camiz
 et al.,
J. Math. Phys.
12
,
2040
(
1971
) and Refs. cited therein.
Google Scholar
Crossref
Search ADS
 
13.
D. C.
Khandekar
 and
S. V.
Lawande
,
J. Math. Phys.
16
,
384
(
1975
).
Google Scholar
Crossref
Search ADS
 
14.
E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (Wiley, New York, 1974), p. 58.
15.
The identities (46) and (47) differ from their classical counterparts (15) and (12) respectively. The additional terms which appear have coefficients either of ℏ or ℏ2 and so the difference is due to quantum mechanical effects and not to any change in the structure of the tensor Imn.
16.
Non‐Compact Groups in Particle Physics, edited by Y. Chow (Benjamin, New York, 1966).
17.
N.
Mukunda
,
L.
O’Raifeartaigh
, and
E. C. G.
Sudarshan
,
Phys. Rev. Lett.
15
,
1041
(
1965
).
Google Scholar
Crossref
Search ADS
 
18.
E. C. G.
Sudarshan
,
N.
Mukunda
, and
L.
O’Raifeartaigh
,
Phys. Lett.
19
,
332
(
1965
).
Google Scholar
Crossref
Search ADS
 
19.
R. Gilmore, Lie Groups, lie Algebras and Some of Their Applications (Wiley, New York, 1974).
20.
Yu. N.
Demkov
,
Zh. Eksp. Teor. Fiz.
44
,
2007
(
1963
)
Google Scholar
 
[
Yu. N.
Demkov
,
Sov. Phys. JETP
17
,
1349
(
1963
)].
Google Scholar
 
21.
I.
Vendramin
,
Nuovo Cimento A
54
,
190
(
1968
).
Google Scholar
Crossref
Search ADS
 
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1977
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