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arXiv:2206.03282 · Physics

Novel Outlook on the Eigenvalue Problem for the Orbital Angular Momentum Operator

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

Published Jun 1, 2022• 0 citations
arXivHTMLPDF

Abstract

Based on the novel prescription for the power function, (x+iy)m, the new expression for Ψ(x,y|m), the eigenfunction of the operator of the third component of the angular momentum, M^z, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x+iy)m=ρmeimϕ, are invariant under the rotations at 2π for any, not necessarily integer, m—the eigenvalue of M^z. For any real m the functions Ψ(x,y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M^z. The eigenfunctions and eigenvalues of the angular momentum operator squared, M^2, derived for the two different prescriptions for the square root, (m2)1/2, (m2)1/2=|m| and (m2)1/2=±m, are reported. The normalizable eigenfunctions of M^2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M^2.

DOI
10.3390/physics4020043

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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.

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Eigenfunctions and eigenvalues of the operator of the square of the angular momentum are studied. It is shown that neither from the requirement for the eigenfunctions be normalizable nor from the commutation relations it is possible to prove that the eigenvalues spectrum is a set of only integer numbers (in units $\hbar=1$). We present regular, normalizable eigenfunctions with the non-integer eigenvalues thus demonstrating that a non-integer angular momentum is admissible from the theoretical viewpoint.

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Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as i iI ΨΨ=∑ =1 ˆ (33.4) Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives fˆ ,iΨ ∗Ψi ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV ΨΨ=ΨΨΨΨ= n ˆˆ Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5) which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of fˆ .iΨ matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain ∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6) which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have fˆ ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7) that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as iΨ ,fˆ (33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4) k kki f λ ˆ This can be rewritten as = n 1 λ which reads (33.9) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ a a a a a a fff fff fff …… … ……… … … λ Equation (33.6) is called the matrix eigenvalue equation and consists of n linear homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ……… ( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes ( ) ( ) ( ) ( ) 01ˆˆdet = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − =− λ λ λ λ nnnn fff fff fff f … ……… … … (33.11) If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.

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•Sep 7, 2023•1 citation
Based on the novel prescription for the power function, (x+iy)m, the new expression for Ψ(x,y|m), the eigenfunction of the operator of the third component of the angular momentum, M^z, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x+iy)m=ρmeimϕ, are invariant under the rotations at 2π for any, not necessarily integer, m—the eigenvalue of M^z. For any real m the functions Ψ(x,y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M^z. The eigenfunctions and eigenvalues of the angular momentum operator squared, M^2, derived for the two different prescriptions for the square root, (m2)1/2, (m2)1/2=|m| and (m2)1/2=±m, are reported. The normalizable eigenfunctions of M^2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M^2.
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.
Eigenfunctions and eigenvalues of the operator of the square of the angular momentum are studied. It is shown that neither from the requirement for the eigenfunctions be normalizable nor from the commutation relations it is possible to prove that the eigenvalues spectrum is a set of only integer numbers (in units ℏ=1\hbar=1ℏ=1). We present regular, normalizable eigenfunctions with the non-integer eigenvalues thus demonstrating that a non-integer angular momentum is admissible from the theoretical viewpoint.
Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as i iI ΨΨ=∑ =1 ˆ (33.4) Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives fˆ ,iΨ ∗Ψi ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV ΨΨ=ΨΨΨΨ= n ˆˆ Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5) which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of fˆ .iΨ matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain ∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6) which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have fˆ ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7) that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as iΨ ,fˆ (33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4) k kki f λ ˆ This can be rewritten as = n 1 λ which reads (33.9) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ a a a a a a fff fff fff …… … ……… … … λ Equation (33.6) is called the matrix eigenvalue equation and consists of n linear homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ……… ( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes ( ) ( ) ( ) ( ) 01ˆˆdet = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − =− λ λ λ λ nnnn fff fff fff f … ……… … … (33.11) If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials