 By L. Hopf

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Extra resources for Introduction to the Differential Equations of Physics

Sample text

4. Angular momentum 27 After raising to the power n and passing to the limit, the matrix element is the exponential of [vl2 2 lul 2 2 z~ i ~ -~ ~ ( l - e i~sin,~, T)+(fiz-¢v)e i),sin~ T + five2i~ . This looks complicated, and needs some comments. First of all, what we are supposed to have computed is <¢(u),n(z,¢,2;~)¢(v)> where n(z,¢,2A)=e za+-~a-+2iAa° . The language of future chapters will replace coherent vectors ¢ ( u ) ¢ ( v ) by e x p o n e n t i a l v e c t o r s C ( u ) E(v) proportional to the preceding ones, but with a different normalization so that < ~ ( u ) , E(v) > = e < u,v>.

Since S . commutes with Z . it keeps Cp stable, and decomposing the representation amounts to finding the eigenvectors of S in each Cp. Recalling that Z = S - I - 2N = 2J - 2 N , we see that for u even (odd) all components have integer (half-integer) spin. 7) the greatest allowed value of spin is exactly u/2, and corresponds to the chaos Co of order 0. e. 's, conceniently normalized. On the first chaos Ca, the eigenvalue of Z is u/2 - 1, and therefore any ladder built on an eigenvector of J in Ca has at least u rungs.

Weyl operators 3 Weyl suggested the following approach to the quantization problem of associating a q u a n t u m observable F(P, Q) with a function F(p, q) on classical phase space. First define the u n i t a r y operators e - i (rP+sQ) for real r , s. Then take the Fourier transform of the classical observable F ( r , s) = f e -i (~P+sq) r ( p , q) dpdq , and transform it bazkwards into the operator, F(P, Q) = f e i (rP+sQ) F(r, s)drds For a discussion of this method, which is extremely interesting as a version of pseudodifferential calculus, see Folland [Fol].