Standing Waves1

Standing waves1 Part I O(j) is not fully equivariant if j is odd Representations of SO3 Eigenfunctions of the Laplacian Attempts to explain the periodic table Attempts to explain atomic spectra Part II Three corrections to the fine structure A discussion of perturbation theory Half-integer values of l { first explanation Generalities about the Hamiltonian formulation Schroedinger's function W Half-integer values of l { other explanations The atom Schroedinger's choice of solutions Residues The canonical divisor and j quantum number Intuitive description of polarization Relation between smooth and holomorphic Laplacian operators A coincidence about the periodic table Linear systems of wave forms Hodge theory The Levi-Civita action is not Lagrangian for its Riemann metric Foundations Unification of the various coupling schemes Discussion of the problem of non-convergent integrals The ethics of science First concepts of the Lamb shift A question about Pauli exclusion The notion of probability Concluding remark False fine structure Electron spin and Pauli exclusion It was not the right conclusion John Atwell Moody Coventry, 2014 1The javascript mentioned in this article is at http://spectrograph.uk 1 Part I This note is a discussion of the first part of Schroedinger's 1926 review paper (in English) [2] which arose in begin- ning to explain spectral lines of hydrogen as frequency differences, and to approach some relativistic ideas of Bruce Westbury. Schroedinger posited for a an electron of mass µ and charge e at the point (x; y; z) with the nucleus at the origin, a potential function V = −e2=r with r the distance from the origin. He wishes to have a function W (x; y; z; t) with gradient (px; py; pz; −E); and chose W (x; y; z; t) = −Et + S(x; y; z); imagining a standing electromagnetic wave with ampli- tude A(x; y; z) given 2π Ψ = A(x; y; z)sin( W (x; y; z; t)) h with A; S are real functions, h Planck's constant. He stated that the `standard wave equation' it should satisfy has 1 @ ∆Ψ = ( )2Ψ u2 @t with u the speed of motion of the level sets given by Et = S(x; y; z): Here u is the ratio between the time partial derivative and the magnitude of the spacial gradient E=jgrad(W )j which calculates to E=p2µ(E − V ) if µ is the electron mass. 2 If we define constants −8π2µe2 β = h2 −8π2µE α = h2 the condition they must satisfy, therefore, is that α+β=r is an eigenfunction of the Laplacian. The connected rotation group of three space operates naturally on the Riemann sphere, which can be viewed as the cosets of a Borel subgroup, and all the representations we will look at extend to the full connected automorphism group, the group holomorphic automorphisms of the Riemann sphere, after complexification if necessary. The vector bundles O(j) for j an even number are com- pletely natural, they are equivariant for the full automorphism group of the Riemann sphere, because they are tensor powers of the canonical line bundle. More- over, again for even values j = 2l; the global sections are the complexification of the 2l + 1-dimensional space of real harmonic polynomials which are homogeneous of degree l: 3 Note that this implies that the complex polynomials of degree 2l in two variables are the complexification of a space of real polynomials of degree l in three variables. To see that the dimensions are right, you can use the fact that any homogeneous polynomial of degree l in three variables is uniquely a a sum of powers of r2 times homogeneous harmonic polynomials of lower degree [1]. Thus the sequence of dimensions of the homogeneous harmonic polynomials of degree l is 3 4 5 3 6 4 1; ; −1; − ; −( −1)−1; ::: 2 2 2 2 2 2 which is indeed the sequence of odd numbers. It follows that all are irreducible. 4 O(j) is not fully equivariant if j is odd. There is no similar interpretation of the global sections of O(j) when j is odd, and indeed the automorphism group of the projective line, nor even the special orthogonal group which preserves the Riemannian metric, do not act in a natural way on the total space of the line bundle with section sheaf O(j) for j odd. The automorphism group of any line bundle maps to the automorphism group of the base manifold with kernel automorphisms of the line bundle which fix the zero section, just nonzero scalars under multiplication in the case of a projective variety, resulting in a central extension group acting on the line bundle, not the original group; the resulting central extension is nontrivial when j is odd. Each automorphism g does lift to a pair of linear automorphisms of determinant one of C2; negatives of each other, and choosing to label one of them s(g) one has s(g)s(h) = z(g; h)s(gh) for the appropriate central cocycle z taking values in f1; −1g: This gives an action on global sections of O(1) which induces an action on the vector bundle itself and therefore all tensor powers. Such a cocycle adjusts what is called the `gauge' in Weyl's book. Or, another way of viewing the global sections of O(j) for odd j as representations having to do with the Riemann sphere is to interpret them as representations of the Lie algebra rather than the Lie group, that is, representations of the global sections of the an- ticanonical line bundle. 5 The complex vector spaces Γ(P1; O(j)) for j odd, how- ever, are even dimensional, and if we interpret them as representations of the three dimensional Lie algebra of vector fields of P1 they restrict to real irreducible representations of twice the dimension 2(j + 1) (a multiple of 4) whose endomorphism algebra is just the field C: That is, the complex number field could arise as the endomorphism ring of the global sections of O(1) as a representation of the vector field Lie algebra of the Rie- mann sphere, and this is so even if initially we do not use complex numbers, and consider vector fields preserving the Riemannian metric. We will see later a more natural way that complex numbers will arise in this context. 6 Representations of so3 Although this is trivial, it makes sense to go through some of the representation spaces just to settle the no- tational conventions. As one lists the orbitals according 1 3 to values of l = 0; 2; 1; 2 ::: the names s; p; d; f; ::: are given to the integral values of l: The wave functions for l = 0; those type s; are the constants. 1 For l = 2 they need a cocycle z(g; h) or can be inter- preted as representations of the Lie algebra; and belong to a four dimensional vector space with a complex structure. For l = 1; those of type p correspond to just real linear forms on three space (of dimension three), and the complexification is the space of holomorphic vector fields on the Riemann sphere (dual to the canonical line bundle). For l = 3=2; they need a cocycle and belong to an eight- dimensional real vector space, with a complex structure. For l = 2 the so-called type d, are the five-dimensional space of real harmonic functions which are among the homogeneous polynomials of degree two in three variables. This is the orthogonal complement of the line spanned by r2 where r is the distance from the origin, under an invariant quadratic form, and its complexification is the global sections of O(3); 7 For l = 5=2 is a representation of dimension 12 with a complex structure. For l = 3 those of type f there is the seven dimensional space of harmonic polynomials which are homogeneous of degree three in three variables, orthogonal to r2 times haromonic homogeneous of degree one. For l = 7=2 is just a sixteen dimensional representation with a complex structure. For l = 4 those of type g, the nine dimensional space of harmonics polynomials which which are homogeneous of degree four and which are orthognal both to r4 and to r2 times the harmonics homogeneous of degree two, and so-on. 8 Eigenfunctions of the Laplacian The action of the Laplacian on products sf(r) with s a homogeneous polynomial of degree l which is a harmonic and f analytic in r is easily calculated. The action of ∆ on the powers of r in k variables is ∆(ri) = i(i − 2 + k)ri−2; and so for s harmonic homogeneous of degree l ∆(ris) = ( i(i − 1) + (k − 1 + 2l) i )ri−2s Then if f is an analytic function of r and s a harmonic analytic function the Laplacian operator acts on the product fs by d k − 1 + 2l(s) d = ( )2 + ( ): dr r dr where the term l(s) can be understood as the Euler derivation acting on the factor s while commuting with functions of r (the actual Casimir operator in that sense is l(l + 1)). Now ∆ has an eigenfunction of the form α + β=r just when r∆ has eigenfunction αr + β: 9 The action of r∆ on functions sf where where s is harmonic homogeneous of degree l is d d r( )2 + (k − 1 + 2l)( ): dr dr Take k = 3; the number of variables. For any number λ, r∆ acts on rmeλrs by λ2r + 2λn + lower order for n = m + l + 1: If we take α = λ2 the difference r∆ − αr − β preserves the filtration by the span of eλrs; reλrs; :::; rmeλrs; acting the associated graded vector space by 2λ(m + l + 1) − β in degree m: For any m we may take β = 2λ(m + l + 1) and the span maps onto the span of just eλrs; :::rm−1eλrs with one dimensional kernel spanned by n−l−1 1 i 1 β r X n + l (nβ r) e 2n s: n − l − 1 − i i! i=0 β2 The relation α = 4n2 above gives energies −2π2µe4 E = h2n2 Note that β is negative.

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