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Journal of Chemical Education

Compounding Selection Rules in Quantum Mechanics

Kenneth R. Brownstein, K. S. Capelle

Published Jul 1, 1999• 0 citations
Publisher

Abstract

The selection rule for matrix elements, <n j |αβ| n k >, of a product of operators, αβ, can be obtained by compounding the selection rules for the matrix elements <n j |α| n k > and <n s |β| n k > of the individual operators through intermediate states | n s . We point out that a naive application of this technique may lead to misleading results due to nonexistent intermediate "ghost" states.

DOI
10.1021/ed076p992

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The selection rule for matrix elements, <n j |αβ| n k >, of a product of operators, αβ, can be obtained by compounding the selection rules for the matrix elements <n j |α| n k > and <n s |β| n k > of the individual operators through intermediate states | n s . We point out that a naive application of this technique may lead to misleading results due to nonexistent intermediate "ghost" states.
The recognition of antiunitary symmetry in a quantum mechanical system is considered in relation to selection rules. It is shown that the interaction of unitary symmetry and the anti-invariance of matrix elements under antiunitary operators can produce extra selection rules in magnetic systems. The criteria obtained for the occurrence of these additional selection rules are presented in the form of group character tests, and in a special case provide an application for the symmetrized and antisymmetrized squares of group representations.
The phenomenological-velocity radical αlsin β= L+κ L−/κ, with κ= 1−v2/c2, is usually analysed by squaring, which yields a v-free compatibility constraint and the conclusion that the equation says nothing about v. That conclusion holds only for the squared equation. Working on the principal branch, we show that the product of principal roots equals ε L+L− with a branch factor ε∈{±1}that depends on the signs of L± and on whether |v|<cor |v|>c, and that squaring erases it. On the solution locus, ε reduces to sgn(lα) for |v|<c and to +1 for |v|>c, giving a selection rule: solutions with |v|<c require sin β >0, solutions with |v|>c require sgn(lα) = sgn sin β, and for β ∈[0,π] the |v|>cregime is admitted if and only if lα>0. The light cone |v|= c is simultaneously the branch discontinuity, the locus where L−/κ is undefined, and the value returned by symbolic solvers. On the locus, εis a nonseparable function of sgn(lα) and the velocity regime: the Z/2 parity obstruction of a maximally CHSH-violating sign table, produced here by a branch cut with no probability. Its definedness locus coincides pointwise with a setting-dependent acceptance map under which Bell-local responses carry −cos(a−b) exactly on accepted data with unbiased marginals; what the model relinquishes is the settings-independence of the trial set, not locality, and we state the protocol under which the mechanism predicts no violation and is thereby refutable. Transporting the asymmetric decoration into an operator setting yields E=−cos(a−b+ 2ϑ(v)) with S = 2√2 |cos 2ϑ|. Keywords: principal branch; square-root identity; Riemann surface; Lorentz factor; light cone; phenomenological velocity; selection rule; Bell–CHSH; computer algebra.
For orthodox quantum systems, pure states correspond one-to-one to all one-dimensional subspaces and observables correspond one-to-one to all selfadjoint operators in the state space. A relaxation of this one-to-one correspondence will bring about superselection rules. A general formulation of superselection rules is technically complicated. 1 We shall adopt an intuitive approach here and limit our discussion to some special cases to see how superselection rules can be incorporated in a Hilbert space structure of quantum theory.
By a quantum mechanical analysis of the additive rule F α[F β[f]]=F α+β[f], which the fractional Fourier transformation (FrFT) F α[f] should satisfy, we reveal that the position-momentum mutual-transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.