Group Theory: And its Application to the Quantum Mechanics by Eugene P. Wigner, J. J. Griffin PDF

By Eugene P. Wigner, J. J. Griffin

ISBN-10: 0127505504

ISBN-13: 9780127505503

Workforce conception And Its software To The Quantum Mechanics Of Atomic Spectra

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9) E W e break the sum over I on the left side into t w o parts b y writing the term with I = K separately. 7a), obtaining K KL L^K 2 LK\ L^K^ — J^I K~ &L HL K This yields a new eigenvalue F to terms o f order X 2 K 2 JUL . 10) L The new eigenfunction ÇP is given b y K V

U-i)t (Ut)t = U = (U- )- . 17) The Principal Axis Transformation for Unitary and Hermitian Matrices Every unitary matrix V and every Hermitian matrix Η can be brought into diagonal form by a similarity transformation with a unitary matrix U. For such matrices, the exceptional case mentioned on page 22 cannot occur. First of all, we point out that a unitary (or Hermitian) matrix remains unitary (or Hermitian) after a unitary transformation. Since it is a product of three unitary matrices, U VU is itself unitary.

From U^VU = A , w e (U*)îVU* = Λ*. , whether b y U or U*), the diagonal form A can also be written as Λ*. Thus the numbers Λ Λ ,·'·,Λ are the same as the numbers Λ*, Λ*, * · · , A*. This implies that THE COMPLEX V V Ν EIGENVALUES o f a real orthogonal matrix V OCCUR IN CONJUGATE PAIRS. 2 Η Moreover since VV — 1, they all have absolute value 1 ; the real eigenvalues are therefore ^ 1 . I n an odd-dimensional matrix at least one eigenvalue must be real. 2) ALREADY SHOWS. GROUP T H E O R Y A N D ATOMIC SPECTRA 30 If D is an eigenvector for the eigenvalue λ, then t)* is an eigenvector for the complex conjugate value A*.

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Group Theory: And its Application to the Quantum Mechanics of Atomic Spectra by Eugene P. Wigner, J. J. Griffin

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