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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Constraints in Wave Mechanics and Induced Rotation in Nuclear Scattering

ERNEST A. SEGLIE, David Rittenhouse Inglis

Published Mar 15, 1973• 0 citations
Publisher

Abstract

As an illustration of the problem of handling classical constraints in wave mechanics, the correct wave equation is obtained for a model consisting of two particles, one of them constrained to have a fixed radial distance from the origin and the other free save for an angular constraint confining the two radius vectors to be coincident. The particle with fixed radius is here equivalent to a thin rigid rod. The wave equation is obtained by starting with a system of two unconstrained particles in which the constraints are replaced by restoring forces, writing the wave equation in appropriate polar coordinates by transforming the Laplacian from Cartesian coordinates, and then going to the limit of strong restoring forces. If, in this problem of a rotating rigid rod collinear with the position of an otherwise free particle, one follows the rather usual procedure of using the derivative with respect to the conjugate angular coordinate as the operator for a canonical angular momentum, one obtains in the radial wave equation in good approximation a "centrifugal potential" term with the familiar coefficient $l(l+1)$. The limiting procedure shows that the correct result instead has a similar term but with coefficient $l(l+1)+1$. Also the relation between the wave function and the solution of the radial equation is not the same as for a single particle. It is thereby shown that an application sometimes made to angular coordinates of a metric transformation in analogy to one from Cartesian to curvilinear coordinates is not strictly correct. The result derived by the limiting procedure is related to recent discussions of $^{16}\mathrm{O}$-$^{16}\mathrm{O}$ scattering.

DOI
10.1103/physrevd.7.1712

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As an illustration of the problem of handling classical constraints in wave mechanics, the correct wave equation is obtained for a model consisting of two particles, one of them constrained to have a fixed radial distance from the origin and the other free save for an angular constraint confining the two radius vectors to be coincident. The particle with fixed radius is here equivalent to a thin rigid rod. The wave equation is obtained by starting with a system of two unconstrained particles in which the constraints are replaced by restoring forces, writing the wave equation in appropriate polar coordinates by transforming the Laplacian from Cartesian coordinates, and then going to the limit of strong restoring forces. If, in this problem of a rotating rigid rod collinear with the position of an otherwise free particle, one follows the rather usual procedure of using the derivative with respect to the conjugate angular coordinate as the operator for a canonical angular momentum, one obtains in the radial wave equation in good approximation a "centrifugal potential" term with the familiar coefficient l(l+1)l(l+1)l(l+1). The limiting procedure shows that the correct result instead has a similar term but with coefficient l(l+1)+1l(l+1)+1l(l+1)+1. Also the relation between the wave function and the solution of the radial equation is not the same as for a single particle. It is thereby shown that an application sometimes made to angular coordinates of a metric transformation in analogy to one from Cartesian to curvilinear coordinates is not strictly correct. The result derived by the limiting procedure is related to recent discussions of 16O^{16}\mathrm{O}16O-16O^{16}\mathrm{O}16O scattering.
We discuss the introduction of constraints in a system of n noninteracting particles which obey the laws of nonrelativistic quantum mechanics. In this paper the particles are first thought of as being unconstrained (described by the 3n3n3n Cartesian coordinates of a flat space RRR), but subject to an external potential VVV which, in a certian suitable limit, forces the system to remain in a curved subspace VVV of RRR. This idea was already employed in a previous work where we have discussed the motion of one constrained particle. It was then shown that in order to obtain a meaningful result the particle wave function should be "uniformly compressed" into a surface (or curve), avoiding, in this way, the tangential forces which correspond to the dissipative constraints of classical mechanics. The resulting Schr\"odinger equation could then be separated in such a way that the part which contained the surface (or curve) variables was independent of the potential VVV employed in the constraining process. In the present paper we show that this procedure cannot be carried out for all subspaces VVV of RRR in the case of a many-particle system, unless VVV satisfies a certain geometrical condition. The many-particle Schr\"odinger equation (as was already the case of the one-particle systems) contains, besides the kinetic energy term, potentials which are not bending invariants, that is, cannot be obtained from the metric tensor gij{g}_{\mathrm{ij}}gij​ of VVV or its derivatives. This gives rise to different Schr\"odinger equations for isometric subspaces, in striking contrast with the usual quantization procedures which start from the classical Lagrangian of the constrained system.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
The Hamiltonian describing a system of particles in a rotating coordinate system is derived and it is shown that the simple classical solution of rigid flow is forbidden in quantum mechanics, even at very low angular velocities. This effect is closely parallel to the Aharonov–Bohm effect, which likewise has its origin in the single-valuedness requirement of the wave function. An analytical approach to perturbation theory is used to include the effects of the Coriolis and centrifugal forces and to derive the current flows for some independent-particle systems, that is, for the Inglis cranking model. It is shown, by explicit construction, that the currents are not rigid even when the moment of inertia assumes the rigid-flow value, as it does for the harmonic oscillator single-particle potential under conditions of self-consistency. Furthermore, it is shown that, for a more general potential, even the moment of inertia is not rigid.
The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result. The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory. The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.