Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Must be the wave function single-valued? Ring vs. periodic boundary conditions, spinors and double rotations

Josep Planelles

Published Jan 5, 2022• 0 citations
Publisher

Abstract

This is a lecture notes for undergraduate students. We try to tackle the single valuedness of spatial and double valuedness of spin functions. Also, we adress the need of spinors to accommodate spin functions with some parallelism to the need of axial vectors (or antisymmetric traceless tensors) to accommodate angular momentum. Finally, we revisit the Dirac and Weyl tricks on the non-equivalence of a 2 pi and a 4 pi rotation related the topology of rotation and unitary groups.

DOI
10.26434/chemrxiv-2022-x5x7d

Related papers

Single Valuedness of Wave Functions

Eugen Merzbacher

The requirement that the quantal wave function be single-valued is examined in the light of two recent developments: The effect of a magnetic vector potential on a particle moving in a multiply connected field-free region (Aharonov-Bohm effect), and the flux quantization inside a superconducting ring. Both of these effects are new applications of conventional elementary quantum mechanics. The single-valuedness problem is considered, with its historical background, for particles with no spin, particles with spin 12, and rigid bodies. It is shown that the single-valuedness condition, if properly adapted to the particular physical model, is deeply rooted in the foundations of quantum mechanics. However, wave functions which superficially appear to be double-valued are useful in an helicity representation of spinors and in the construction of nuclear wave functions.

American Journal of Physics•Apr 1, 1962•117 citations

Critical comments on the quantization of the angular momentum: II. Analysis based on the requirement that the eigenfunction of the third component of the operator of the angular momentum must be a single valued periodic function

G. S. Japaridze, Anzor Khelashvili, K. Sh. Turashvili

We discuss the requirement of single valuedness and periodicity of eigenfunction of the third component of the operator of angular momentum. This condition, imposed on a non observable, is often used to derive that the eigenvalues of angular momentum could be only integer. We reexamine the arguments based on this requirement and alternate condition imposed by Pauli and show that they do not follow from the first principles and therefore these constraints can dropped. Consequently, we arrive to the same conclusion as in [1]: there exist regular, normalizable eigenfunctions with the non-integer eigenvalues thus a non-integer angular momentum is perfectly admissible from the theoretical viewpoint. The issue of the nature of eigenvalues forming the spectrum of the angular momentum remains open. What can be derived from the first principles is that to a fixed value of the angular momentum L corresponds a discrete spectrum of eigenvalues of the third component of the angular momentum, m, defined by the relation |m|=L-k, k=0,1,...,[L], where [L] is an integer part of L. As a mathematical byproduct of our analysis of eigenfunctions, we present an alternate definition of a power of a complex number allowing to retain initial translational invariance of a base.

arXiv:2004.10673•Apr 17, 2020•1 citation

Remark Concerning the Eigenvalues of Orbital Angular Momentum

H. A. Buchdahl

That the eigenvalues of the components of orbital angular momentum are integral is most often inferred from the requirement that the Schrödinger representatives of the angular momentum eigenkets shall be single-valued functions of the spatial coordinates. Physical interpretation, however, requires only that the squared modulus of any wavefunction shall be single-valued, which leads only to the weaker conclusion that the eigenvalues are either all integral or all half-odd integral. It is shown here in an elementary manner that the eigenvalues are in fact integral, any explicit appeal to single-valuedness being avoided.

American Journal of Physics•Nov 1, 1962•11 citations

A note on angular momentum and sums over classical paths

A Norcliffe

The spherical top propagator and spectral operator kernel for all spins are evaluated in terms of new rotational coordinates that are related to the Euler angles in a simple way. Both operator kernels are expressible exactly as sums over classical paths. The angular momentum wavefunctions are single-valued functions of the new rotational coordinates for all values of j.

Journal of Physics A Mathematical Nuclear and General•Sep 1, 1973•1 citation

Some notes on the `true relativistic spin operator'

Daniel R. Terno

Operators that are associated with several important quantities, like angular momentum, play a double role: they are both representations of generators of the symmetry group and `observables'. We show that Dirac spin operator cannot be associated with a `spin observable' in the standard sense.

•Aug 12, 2002•1 citation

The Dirac Equation

Theodore Frankel

Spin is what makes the world go 'round. The Groups SO (3) and SU (2) How does SU(2) act on its Lie algebra? For physical and mathematical motivation for this section (which involves nonrelativistic quantum mechanics) we refer the reader to some remarks of Feynman and of Weyl. Specifically, Feynman, in his section entitled “Degeneracy,” shows that a process involving a specific choice of direction in space requires that the process be described not by a single wave function ψ but rather by a multicomponent column vector of wave functions Ψ = (ψ 1 , …, ψ N ) T . He then indicates, roughly speaking, that since the physics cannot depend on the choice of cartesian coordinates ( x 1 , x 2 , x 3 ) of space, the N -tuples must transform under some representation ρ: SO (3) → U ( N ) of the rotation group SO (3) of space. This is not quite accurate; since e i γ Ψ represents the same wave function (when γ is a constant), ρ is only a “ray” representation, ρ( g )ρ( h ) = e i γ( g,h ) ρ( gh ) for a function γ( g, h ). Weyl shows that this can be made into a genuine representation, except that it is (perhaps) double-valued. We shall show in this section that there is a natural 2 : 1 homomorphism π of the special unitary group SU (2) onto SO (3), thus yielding a (perhaps double valued) representation of SU (2) into U ( N ).

Cambridge University Press eBooks•Nov 24, 2003•0 citations
This is a lecture notes for undergraduate students. We try to tackle the single valuedness of spatial and double valuedness of spin functions. Also, we adress the need of spinors to accommodate spin functions with some parallelism to the need of axial vectors (or antisymmetric traceless tensors) to accommodate angular momentum. Finally, we revisit the Dirac and Weyl tricks on the non-equivalence of a 2 pi and a 4 pi rotation related the topology of rotation and unitary groups.
The requirement that the quantal wave function be single-valued is examined in the light of two recent developments: The effect of a magnetic vector potential on a particle moving in a multiply connected field-free region (Aharonov-Bohm effect), and the flux quantization inside a superconducting ring. Both of these effects are new applications of conventional elementary quantum mechanics. The single-valuedness problem is considered, with its historical background, for particles with no spin, particles with spin 12, and rigid bodies. It is shown that the single-valuedness condition, if properly adapted to the particular physical model, is deeply rooted in the foundations of quantum mechanics. However, wave functions which superficially appear to be double-valued are useful in an helicity representation of spinors and in the construction of nuclear wave functions.
We discuss the requirement of single valuedness and periodicity of eigenfunction of the third component of the operator of angular momentum. This condition, imposed on a non observable, is often used to derive that the eigenvalues of angular momentum could be only integer. We reexamine the arguments based on this requirement and alternate condition imposed by Pauli and show that they do not follow from the first principles and therefore these constraints can dropped. Consequently, we arrive to the same conclusion as in [1]: there exist regular, normalizable eigenfunctions with the non-integer eigenvalues thus a non-integer angular momentum is perfectly admissible from the theoretical viewpoint. The issue of the nature of eigenvalues forming the spectrum of the angular momentum remains open. What can be derived from the first principles is that to a fixed value of the angular momentum L corresponds a discrete spectrum of eigenvalues of the third component of the angular momentum, m, defined by the relation |m|=L-k, k=0,1,...,[L], where [L] is an integer part of L. As a mathematical byproduct of our analysis of eigenfunctions, we present an alternate definition of a power of a complex number allowing to retain initial translational invariance of a base.
That the eigenvalues of the components of orbital angular momentum are integral is most often inferred from the requirement that the Schrödinger representatives of the angular momentum eigenkets shall be single-valued functions of the spatial coordinates. Physical interpretation, however, requires only that the squared modulus of any wavefunction shall be single-valued, which leads only to the weaker conclusion that the eigenvalues are either all integral or all half-odd integral. It is shown here in an elementary manner that the eigenvalues are in fact integral, any explicit appeal to single-valuedness being avoided.
Spin is what makes the world go 'round. The Groups SO (3) and SU (2) How does SU(2) act on its Lie algebra? For physical and mathematical motivation for this section (which involves nonrelativistic quantum mechanics) we refer the reader to some remarks of Feynman and of Weyl. Specifically, Feynman, in his section entitled “Degeneracy,” shows that a process involving a specific choice of direction in space requires that the process be described not by a single wave function ψ but rather by a multicomponent column vector of wave functions Ψ = (ψ 1 , …, ψ N ) T . He then indicates, roughly speaking, that since the physics cannot depend on the choice of cartesian coordinates ( x 1 , x 2 , x 3 ) of space, the N -tuples must transform under some representation ρ: SO (3) → U ( N ) of the rotation group SO (3) of space. This is not quite accurate; since e i γ Ψ represents the same wave function (when γ is a constant), ρ is only a “ray” representation, ρ( g )ρ( h ) = e i γ( g,h ) ρ( gh ) for a function γ( g, h ). Weyl shows that this can be made into a genuine representation, except that it is (perhaps) double-valued. We shall show in this section that there is a natural 2 : 1 homomorphism π of the special unitary group SU (2) onto SO (3), thus yielding a (perhaps double valued) representation of SU (2) into U ( N ).