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The Journal of Chemical Physics

Reply to ‘‘Comment on ‘Dirac states for unit position and momentum: Phase consistency of their angular momentum representations’ ’’a)

Robert F. Snider

Published Mar 1, 1983• 0 citations
Publisher

Abstract

It is stressed that the defined states and their relationship deal exclusively with the quantum mechanics of directions and angular momentum. In particular, they have nothing to do with the possibility of factoring three dimensional motion into magnitude and directional components. A magnitude independent derivation of the previously obtained result is given.

DOI
10.1063/1.444998

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Comment on ‘‘Dirac states for unit position and momentum: Phase consistency of their angular momentum representations’’

G. E. Hahne

The argument presented makes two points about Snider’s [J. Chem. Phys. 76, 3543 (1982)] transformation between ‘‘unit position’’ states ‖r̂〉 and ‘‘unit momentum’’ states ‖p〉 for the motion of a structureless particle in three space: Snider’s transformation (1) implicitly defines an operator p̂ that has a classical limit a unit vector that is perpendicular to both r̂ and l=r×p, while on a classical scattering trajectory for r→∞, the physical unit vector p/p≊±r̂, and (2) attempts, inconsistent with the kinematics of states and observables of the system, to pass from r̂ to p̂ amplitudes of a state vector without any reference to the scalar r or p dependence of the wave functions.

The Journal of Chemical Physics•Mar 1, 1983•1 citation

Dirac states for unit position and momentum: Phase consistency of their angular momentum representations

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It is shown that the position and momentum directional representations of angular momentum states must satisfy Σλs〈r̂‖λs〉〈λs‖p̂〉 = Σλs(i)λYλs(r̂)Yλs(p̂)*. This imposes phase constraints on the relation between 〈r̂‖λs〉, 〈p̂‖λs〉, Yλs(r̂), and Yλs(p̂). In the accompanying paper, it is shown that this resolves a problem in the centrifugal sudden approximation of molecular collision theory.

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See also commentary Comment on ’’Spin and uncertainty in the interpretation of quantum mechanics’’

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It is stressed that the defined states and their relationship deal exclusively with the quantum mechanics of directions and angular momentum. In particular, they have nothing to do with the possibility of factoring three dimensional motion into magnitude and directional components. A magnitude independent derivation of the previously obtained result is given.
The argument presented makes two points about Snider’s [J. Chem. Phys. 76, 3543 (1982)] transformation between ‘‘unit position’’ states ‖r̂〉 and ‘‘unit momentum’’ states ‖p〉 for the motion of a structureless particle in three space: Snider’s transformation (1) implicitly defines an operator p̂ that has a classical limit a unit vector that is perpendicular to both r̂ and l=r×p, while on a classical scattering trajectory for r→∞, the physical unit vector p/p≊±r̂, and (2) attempts, inconsistent with the kinematics of states and observables of the system, to pass from r̂ to p̂ amplitudes of a state vector without any reference to the scalar r or p dependence of the wave functions.
The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator L^z=ℏı (x∂∂y−y∂∂x){\hat{L}}_{z} = {\hbar\over{\imath}}\,(x{\partial\over{\partial y}} - y{\partial\over{\partial x}} )L^z​=ℏ​(x∂y∂​−y∂x∂​) is not that it is the operatorfor the zzz-component LzL_{z}Lz​ of the angular momentum L{\mathbf{L}}L, but rather the expression of the operator for the angular momentum L{\mathbf{L}}L when it is aligned with the zzz-axis. Hence what we are used to note (erroneously) as L^z{\hat{L}}_{z}L^z​ is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.