Applications of Group Theory in Quantum Mechanics by M. I. Petrashen, J. L. Trifonov

By M. I. Petrashen, J. L. Trifonov

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Publish yr note: First released November fifteenth 1969

Geared towards postgraduate scholars, theoretical physicists, and researchers, this complicated textual content explores the function of recent group-theoretical tools in quantum concept. The authors established their textual content on a physics path they taught at a popular Soviet college. Readers will locate it a lucid advisor to workforce idea and matrix representations that develops suggestions to the extent required for applications.

The text's major concentration rests upon element and area teams, with functions to digital and vibrational states. extra subject matters comprise non-stop rotation teams, permutation teams, and Lorentz teams. a few difficulties contain reviews of the symmetry homes of the Schroedinger wave functionality, in addition to the reason of "additional" degeneracy within the Coulomb box and likely topics in solid-state physics. The textual content concludes with an instructive account of difficulties regarding the stipulations for relativistic invariance in quantum theory.[b][/b]

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The condition that the representation is reducible can therefore be formulated as follows. A representation D is reducible if there exists a non-singular matrix V, such that the matrices V–1DV are quasi-diagonal. 6 Schur’s first lemma We shall now prove an important theorem known as Schur’s first lemma: A matrix which commutes with all the matrices of an irreducible representation is a multiple of the unit matrix. Proof. Let D(g) be the matrices of an irreducible representation of order n of the group G, g ∈ G.

83) where χ(i)(g) and χ(j) (g) are the characters of the irreducible representations D(i) and D(j), respectively. 83) for unitary representations. 85) 62 which was to be proved. The function χ(g) has the same value for all the elements of a given class. 86) where ks is the number of elements in class Cs, and is the value of the character of the representation corresponding to the elements of this class. d. The character of a reducible representation D is equal to the sum of characters of irreducible representations into which it can be decomposed.

If the group elements are linear transformations, the matrices of these transformations themselves form a representation which is isomorphic to the group. These two representations correspond to the trivial invariant sub-groups which were mentioned in Chapter 1. To illustrate other representations of a group, consider the derivation of one of the representations of the group C of matrices of linear transformations of n variables x1, x2, . . 10) Transformation of the variables x1, x2, . . , xn induces a transformation of the coefficients of this form.

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