Algebraic Matric Groups.
نویسنده
چکیده
where the diagonal elements of DI were denoted by cos qpj + i sin (pi. As is well known, the number of real parameters in R, if (4) is not fulfilled, can be counted as follows: the first column Rji is an n dimensional vector of length 1, giving n1 parameters; the second column Rp is of length 1 and is orthogonal to the first column giving n 2 parameters; the third column of R must be a unit vector orthogonal to the first and the second column, giving n 3 parameters, etc. Altogether R has n 1 + (n 2) + (n 3) + . .. + 1 = 1/2((n1)n parameters. If (4) is fulfilled, the second column must be orthogonal to the first column and to the vectors with the components R,1 cos .pj and Rjl sin sj. It has, therefore, if these three vectors are linearly independent, only n 4 parameters instead of the n 2 as above. Hence, the condition U12 = 0 fixes two paraimeters and cannot be satisfied, in general, by varying the energy. If only two of the vectors Rj1, R:n cos wp, Rfl sin (pj are independent, Rj2 still has to be orthogonal to one more vector than if the condition U12 = 0 is not imposed. In addition, the linear relation between the vectors Rfl, Rfl cos (pj, Rfl sin pj gives at least one additional condition so that the number of free parameters in U with U12 = 0 is, in this case, at least two smaller than it is in the general U. If both Rij cos (,o and Rji sin .pj are multiples of Rj, all (p, are equal which involves certainly more than two conditions. This shows that the conclusion of the preceding paragraph is valid in general and completes the demonstration of (ii).
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ورودعنوان ژورنال:
- Proceedings of the National Academy of Sciences of the United States of America
دوره 32 12 شماره
صفحات -
تاریخ انتشار 1946