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Pure and applied physics

Representations and Eigenfunctions

Published Jan 1, 1959• 0 citations
Publisher
DOI
10.1016/b978-0-12-750550-3.50016-1

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The paper respectively studies the transformations of spin component operators and their eigenfunctions from to representation. The studies demonstrate that if using the matrix of obtaining from the equation of transforming the eigenfunctions of and in terms of the transformation arising from Ref. [7] for all of the spin component operators from to representation, a negative symbol will occur in the result, which is not concordant with the commutation relation between and . Moreover, this matrix of transformation can not be applied to the transformation of the eigenfunctions of or . It therefore concludes that with respect to the transformation of representation of an operator and its eigenfunctions, the matrix of transformation should be obtained from the equation of transforming the eigenfunctions of the same operator from the old to the new representation.
By discussing the relationship between the coordinate representation and the discretional mechanical parameter representation,this paper analyses the origin of the δ function normalization of eigenfunction with continuous eigenvalue.A conclusion is obtained that the eigenfunction of coordinate in the discretional mechanical parameter representation and the eigenfunction of discretional mechanical parameter in the coordinate representation are conjugated complex number of each other.Using this conclusion we give a new physical explanation of the eigenfunction closure.
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