Quantum Coulomb Problem: Fock's Theory & 4D Laplace

• Fock’s fundamental theory of the hydrogen atom considers it in momentum space.
• The theory allows a 3D sphere's rotation group to be realized in 4D space.
• Fock's theory gets modified, abandoning momentum space descriptions
• Invariant tensor methods of electrostatics in 3D and 4D are used to transform and simplify the theory.
• A coordinate 4D space is found where the Schrödinger equation becomes the 4D Laplace equation.
• The move from harmonic 4D polynomials to original 3D physical space is algebraic, involving derivatives with respect to a coordinate interpreted as time.
• A differential equation for eigenfunctions in momentum space is obtained with its solutions.
• Vector ladder operators are underconsideration
• A concise calculation of the quadratic Stark effect is provided.
• The Schwinger resolvent is derived by the method of harmonic polynomials.

Quantum Coulomb Basics

• The quantum Coulomb problem remains fundamental in quantum theory by allowing calculation of the spectrum of a system of two opposite charges
• Founders of 20th-century physics like Bohr, Sommerfeld, Pauly, Schrödinger, and Fock are associated with it.
• It has been thoroughly studied via special functions, beginning with its introduction toatomic spectra theory.
• Its simplicity, along with its underlying symmetry- the group SO(4) of rotations in four dimensional space makes it useful tool for theoretical physics concepts
• SO(4) symmetry simplicity allows constructing various concepts.

Unclear Complexities

• The quantum Coulomb problem has some questions that have not been fully clarified.
• Calculating the quadratic Stark effect involves unexplained complexities.
• Unmatched simplicity makes perturbation unexpected.
• Fock's result raises questions.
• SO(4) symmetry realized in momentum space is wrapped into a 3D sphere including how excursion relates to 4D space.

Historical Background and Fock's Approach

• Two classical vector integrals: angular momentum and Laplace-Runge-Lenz vector in quantum mechanics correspond to vector operators
• These vector operators commute with the energy operator/Hamiltonian.
• An analysis of their commutators generates a Lie algebra, coinciding with a Lie algebra of operators of infinitesimal rotations in 4D space.
• The correspondence implies a transformation maps the original quantum Coulomb problem into free motion of a particle over a 3D sphere embedded in 4D space.
• The energy operator remains invariant under rotations of the 3D sphere
• Fock's theory struck contemporaries
• Schrödinger's equation (SE) in momentum space serves as the theory's starting point.
• The space can be considered a 3D plane in 4D space.
• Fock uses stereographic projection so 3D plane transforms into a 3D sphere

Stereographic Projection and Transformations

• Stereographic projection has been known since antiquity
• Stereographic projection conveniently transforms a globe into a flat map.
• There's a surmise on the factor for psi functions to turn an original integral equation into one for spherical functions on a 3D sphere
• This equation is rarely used in physics but is well-known in special functions theory, remaining invariant under rotations in 4D space.
• Fock does not explain the transformation’s physical meaning.
• SO(4) symmetry relates to wrapped momentum space instead of coordinate space.
• How the electron relates with stereographic projection remains unanswered.

Modified Approach

• An inverse 4D Fourier transform modifies Fock's approach applicable to eigenfunctions extended harmonically to 4D momentum space.
• This constitutes transition to new coordinate space as Fock transforms SE into 4D momentum space.
• In 4D coordinate space, modified SE becomes 4D Laplace equation.
• Simultaneously, eigenfunctions manifest as harmonic 4D tensors/homogeneous polynomials.
• Projections (contractions with numerical tensors) yield solid spherical 4D functions.
• Transition to solutions of original SE in Fock's theory is simple algebra with help of differentiation.
• SO(4) symmetry is realized in coordinate space regarded closer than space derived by Fock.
• Structure of functions preserves the remarkably simple inverse transformation, of importance for theoretical concepts.

Function Replacement and Tensor Usage

• Polynomials replace spherical functions, historically linked with electrostatics since Maxwell's time and associated with multipole moments.
• Polynomials are favorable for calculations with differential operators and introduced from the beginning.
• Spherical coordinates  , , and  ,  , are not involved.
• Dipole and quadrupole moments appear because fundamental physics concepts.
• Invariant polynomial tensors in Cartesian coordinates simplifies calculations.
• Rules for using harmonic symmetric tensors are demonstrated in Section 3.
• Trace over any pair of indices vanishes.

Tensor Significance and Variations

• Select tensor properties make analytic calculations more compact, reduce factorials, and formulate fundamental theory questions.
• Using such tensors allow advantage in perturbation theory.
• The quadratic Stark effect can also be computed.
• Derivation of the Schwinger resolvent can be done.
• The derivation of the integral equation obtained by Fock is presented with reference to special functions theory, using electrostatics methods in 3D/4D spaces.
• Derivation of vector ladder operators in the Coulomb problem with their relation to inversion with respect to a sphere is presented.
• Using harmonic tensors in moving to 4D coordinate space simplifies Fock theory modifications with complex transformations.
• SO(4) symmetry is inherent in invariant 4D tensors.
• Singling out one coordinate (complex time) is the final algebraic transition to the physical space.
• Multiplication by imaginary unit i yields real time with equation to vector 'absorbed' by original SE solution, concealing SO(4) symmetry.
• Extension from Fock sphere into 4D space is where harmonic tensors are found
• Spatial 4D inversions instead of stereographic projection of the sphere are used
• Discarding delta functions in final transformation
• These simplify original transformation
• SO(4) symmetry preserves to algebraically relate SE eigenfunctions to harmonic 4D tensors
• Physical symmetry of the problem generates state correspondence which is not in the Laguerre and Gegenbauer polynomials theory.

Harmonic Polynomials

• Angular factors are solid spherical functions, the harmonic degree-l.
• Homogeneous coordinate polynomial Pl(x) demonstrate properties such as
  • ΔPl(x) = 0
  • Pl(cx) = clPl(x)
  • (x∇)Pl(x) = lPl(x)

Euler's Theorem and Operator Interpretation

• Euler's theorem applies to all homogeneous functions, not just polynomials.
• Homogenous equations may also apply to equations of factor 1/r and its powers.
• The operator is interpreted as the angular momentum modulus operator: l= (x∇)

Equation Solutions

• (2) and similar equations have solutions represented as Pl(x)F(r).
• Substituting harmonic degree-l polynomial into (2) invokes a Laplace operator.
• Properties like (3) give rise to radial function equation as factor Pl(x) is canceled:
  • Δ[Pl(x)F(r)] = [ΔPl(x)]F(r) + 2[∇Pl(x)][∇F(r)] + Pl(x)ΔF(r) = (2l/r)[∂F(r)/∂r] + Pl(x){1/r[∂/∂r(rF(r))]}

Confluent Hypergeometric Function

• Transformation leads to the Gauss function:
  • F(α,β,r) = 1 + αr/β 1! + α(α+1)r2/β(β+1) 2! + α(α+1)(α+2)r3/β(β+1)(β+2) 3!
• A discrete spectrum has the α parameter as negative integer, polynomial as its series convert.
• In linear scaling of Coulomb, we take argument x into account.
• Eigenfunctions are considered in the form Pl(x)F(-k,2l+2,2r)exp(-r), n=l+k+1.
• Number k matches degree of 2nd factor in Eq. (7) and hence degree of polynomial is n-1.
• Normalization is unimportant here; must specially stipulate otherwise.

Simultaneous Eigenfunctions

• If there is no preferred direction, both direction and spectrum are independent of m, entire eigenfunction set be considered simultaneously.
• Rank-l tensors use rotation group SO after using transformations developed in electrostatics.
• Dipole invariant harmonic / Coulomb
• Quadropole state is similar

Fock's Theory - Schrödinger Equation

• Schrödinger equation is considered when moving to the momentum representation. - In which, Ψnl(x)= (1/(2π)3)Sanl(p)ei(px)d3p
• Contains a convolution with respect to momenta.
• SE in the mometum space is nonlocal and expresses as: - This is nonlocal - This relates to potential 1/r in the pass and 4π/2/p
  • ( p2+1)anl(p)− (2n/2π2)S(anl(p)/ |p-p'|2)d3p'
• The 1ˢᵗ step in the theory is to multiply the function anl(p) (done the explanation) by (1+p2)2 .

Wrapping the Plane

• The 2nd step involves wrapping the 3D plane into 3D sphere employing (ξ,ξ0) with 4 coordinates.
• Stereographic projection has a projecting the 3D p onto a unit radius sphere.
• Straight line with tangent of slope of projecting relates to tangent of tangent.

The Math

• It flows that:
  • ξ=tan2φ=(2|p|)/(1+p^2)
  • similarly ξ0=cos2φ=(p^2-1)/p^2+=-1).
  • The above are in relation
  • . ξ2+ ξ02=1.

Stereographic Projection

• Stereographic projection doubles the tilt angle φ, which produces essential effect, and the flat drawing correctly reflects 4D transformation.
• In the new variables, with Fock's factor accounting eigenfunction.
• projection must be conformal, where angles intersecting must remain.
• The metric in p-space coordinates on sphere is expressed - 4(dp)2/(p^2+1)^2
• Hence, the p-space contraction coefficient is (1+p^2)/2.
• Integral kernel is transformed with some additional steps

Harmonic, Multipole Tensors

• Multipole potentials arise when point charge potential expands in power coordinates r
• Using xoi, the vector to the radius vector is the Maxwell Pole with a 1/|r-r0|

Multipole Expansion

• Multipole moment is expressed as = Σl (-1)l (r0∇)l 1/r=∑(r0l*x0.. /𝑙!𝑟^2𝑙+1 = ∑l(r0l,Mi l (r))/l!r2 l+1
• Following notations applies for lth tensor for radius: - xoi…ok=rol
• Symmetric tensor of rank 1 is represented - Mi…k (r )

Tensor and Properties

• Equation 19 tensor is homogeneous harmonic polynomial
• Contracting over any indices and scalar produces Laplace operator that equates to zero.
• Multipole harmonic arises if divided Equation 20 tensor by r2l +1 - Mi… (r ) =r2i+1