A set of generalized polar coordinate systems are defined in N‐dimensional space. The total orbital angular‐momentum operator as defined by Louck is found to be a tensor invariant on the (N − 1)‐dimensional unit sphere; hence it is easily explicitly determined in any of the possible generalized polar coordinate systems. Commuting operators can be found and the eigenvalue problem solved in many coordinate systems. Two examples are given: (1) a coordinate system of 3M dimensions where M is an integer, and (2) a coordinate system of 8 dimensions.

1.
J. D.
Louck
,
J. Mol. Spectr.
4
,
298
(
1960
).
2.
H. H.
Nielsen
,
Rev. Mod. Phys.
23
,
90
(
1951
).
3.
W. H.
Shaffer
,
Rev. Mod. Phys.
16
,
245
(
1944
).
4.
In the above equation and in the rest of the paper, the summation over identical upper and lower indices in a term will be assumed, Latin indices being summed from 1 to N, Greek indices being summed from 1 to N−1.
5.
See, for instance, J. L. Synge and A. Schild, Tensor Calculus (University of Toronto Press, Toronto, Ontario, Canada, 1949).
6.
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw‐Hill Book Company, Inc., New York, 1953), p. 1730.
7.
Commuting operators in the form of functions of a single variable and the other operators (as above) can be constructed once a coordinate system is chosen (if they exist) by the well‐known separation‐of‐variables technique. The separation constants become the eigenvalues.
8.
Reference 6, pp. 539, 542.
9.
James D. Louck, “Theory of Angular Momentum in N‐Dimensional Space,” Los Alamos Scientific Laboratory report LA‐2451 (1960) (unpublished).
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