Fock's Theory: Quantum Coulomb Problem

• Fock's theory regarding the hydrogen atom within momentum space is analyzed.
• Fock’s theory can be utilized to realize the rotation group of a 3D sphere within a 4D space.
• Fock’s theory is modified, moving away from momentum space descriptions using invariant tensor methods in 3D and 4D electrostatics.
• There is a coordinate 4D space where the Schrödinger equation transforms into the 4D Laplace equation.
• The shift from harmonic 4D polynomials to 3D physical space is algebraic, utilizing derivatives with respect to a time-interpreted coordinate.
• A differential equation is derived for eigenfunctions in momentum space and then solved.
• The quadratic Stark effect can be calculated concisely, and the Schwinger resolvent is found using harmonic polynomials.
• Vector ladder operators are also taken into consideration.
• Keywords from the document: Fock’s theory, quantum Coulomb problem, harmonic polynomials, Fourier transform, and ladder operators.

Essence of the Quantum Coulomb Problem

• The quantum Coulomb problem is fundamental for computing the spectrum of a two-charge system.
• The quantum Coulomb problem has been examined using special functions, and is foundational to atomic spectra theory.
• Important figures associated with the quantum Coulomb problem include N. Bohr, A. Sommerfeld, V. Pauly, E. Schrödinger, and V. Fock.
• The SO(4) group, representing rotations in 4D space, simplifies the quantum Coulomb problem.
• Computing the quadratic Stark effect is complex, despite perturbation simplicity.
• SO(4) symmetry in Fock's result is realized in momentum space within a 3D sphere, using a 4D space.
• Classical vector integrals of angular momentum and the Laplace-Runge-Lenz vector relate to vector operators that commute with the Hamiltonian.
• Commutators generate a Lie algebra, coinciding with operators of rotations in 4D space.
• Transformation of variables and operators maps the quantum Coulomb problem to a particle's free motion on a 3D sphere in 4D space.
• In Fock's theory, the starting point is the integral Schrödinger equation (SE) in momentum space, considered a 3D plane within a 4D space.
• Fock uses stereographic projection to wrap a 3D plane into a 3D sphere.

Fock’s Approach Modification

• Fock's approach is modified via an inverse 4D Fourier transform, applicable to eigenfunctions extended harmonically to 4D momentum space.
• The modified Schrödinger equation transforms into the 4D Laplace equation within the new 4D coordinate space.
• In this space, eigenfunctions become harmonic 4D tensors, or homogeneous polynomials, and their projections become solid spherical 4D functions.
• Transitioning to original Schrödinger equation solutions involves differentiation-based algebra.
• SO(4) symmetry exists in a coordinate space closer than Fock's, the inverse transformation is simple.
• Spherical functions are replaced with electrostatics-related polynomials, specifically multipole moments.
• Tensor polynomials xi and (3xi xk - r²δik) generate dipole and quadrupole moments respectively, used for l=1, 2.
• Invariant polynomial tensors in Cartesian coordinates simplify calculations.
• Harmonic symmetric tensor rules are from their properties, reflected in special functions, and trace over any index pair vanishes.

Tensor Properties

• Tensor properties make calculations compact, reduce factorials, and allow questions to be correctly formulated.
• Using tensors in perturbation theory benefits quadratic Stark effect calculation and Schwinger resolvent derivation.
• The integral equation derived by Fock is presented with reference to special functions and electrostatics in 3D and 4D spaces.
• Vector ladder operators in the Coulomb problem relate to sphere inversion.
• Harmonic tensors' form is invariant in 4D coordinate space under complex transformations.
• The algebraic transition singles out a coordinate, which is complex time, multiplied by i, and equated to the radius vector's length.
• Variations includes extending Fock sphere into 4D space, using spatial 4D inversion, and discarding delta functions.
• The transformation algebraically relates eigenfunctions to harmonic 4D tensors
• The correspondence is generated by the problem's physical symmetry, not known in the theory of the Laguerre and Gegenbauer polynomials.

Harmonic Polynomials

• Schrödinger equation for eigenfunctions with atomic units exhibits a form dependent on orbits.
• Eigenfunctions in momentum representation have scaled argument p' = np.
• The angular factor is degree-l polynomial Pl(x) with properties ∆Pl(x) = 0, Pl(cx) = clPl(x), and (x∇)Pl(x) = lPl(x).
• lˆ = (x∇) is the angular momentum modulus operator.
• Solutions are of the form Pl(x)F(r), properties yield radial function equation: ∆[Pl(x)F(r)] = (2l/r)Pl(x) ∂F(r)/∂r + Pl(x)[∂²(rF(r))/∂r²].
• Gauss function provides a transformation after substitution, where discrete spectrum parameter a converts series into polynomial.
• Eigenfunctions in the Coulomb problem have the form Pl(x)F(-k, 2l+2, 2r)exp(-r), with n = l+k+1.
• Degree of second polynomial factor is k, degree of entire polynomial is n-1.
• Using rank-l tensors invariant under SO(3) is suitable for the transformations as the spectrum is independent of quantum number m and is non directional.
• Quadrupole state solution form is (3xi xk - r²δik)F(-k,6,2r) exp(-r), n=3+k.
• Octupole state solution form is 3(5xi xk xl - xiδkl - xlδik - xkδli)F(-k,8,2r) exp(-r), n=4+k.
• Dipole state solution is xiF(−k, 4,2r) exp(-r), general form of invariant harmonic tensor are used in the Coulomb problem.

Fock's Theory

• Transitioning to momentum representation, Schrödinger equation has a convolution with respect to momenta
• Schrödinger equation in momentum space is nonlocal: ( p² + 1)anl(p) - (2n/2π²) ∫ anl(p')d³p' / |p - p'|² = 0.
• The function anl(p) is multiplied by (1 + p²)².
• The 3D plane is wrapped into a 3D sphere.
• The tangent of the slope of the projecting straight line is 1 / |p|.
• Stereographic projection doubles the tilt angle φ and gives a effect that gives validity to the 4D transformation.
• Eigenfunction in the new variables, accounting for Fock's factor: bnl(ξ, ξ0) = (p² + 1)² anl(p).
• Preservation of angles is required, which given by a conformal transformation.
• Metric on the sphere in the momentum-space coordinates as 4(dp)² / (p² + 1)².
• The contraction coefficient for elements of the p-space is (1 + p²)/2.
• Volume element in formula (10) is expressed in terms of the 3D surface element: d³p = (1/8)(1 + p²)³dS.
• Kernel of integral can be transformed to 1/|p - p'|² = ½ * (p² + 1) * [(ξ - ξ')² + (ξ0 - ξ'0)²] * (p'² + 1).
• Spherical function and a sum of functions with index (n-1) corresponds to n in original SE, and is substituted into equation.
• A rotation with factor Ylm(θ,φ) goes into eigenfunction with the same factor if conformal.

Harmonic and Multipole Tensors

• Multipole potentials appear when the potential of point charge is expanded in powers of coordinates x0i of radius vector r0.
• The lth tensor power of the radius-vector written as x0i...x0k = r0l.
• For a harmonic tensor of rank l: M(l)i...k(r) = M(il).
• Tensor is a homogeneous harmonic polynomial with properties as a contraction over indices is zero.The tensor is divided by r^{2l+1}.
• Multipole harmonic tensor M(l)i...k(r)/r^(2l+1), which is a homogeneous harmonic function with degree -(l+1).

Harmonic tensor

• The multipole function is found to involve dipole, quadrupole and octupole tensors, and higher-rank ones are not used due complexities.
• Contraction provides a method for calculating amplitudes of potential without spherical functions of tensors, which will reduce amount of work required.
• Power-law theorem can determine the electrostatic moments of a distribution of charges.
• The power tensor, is a collection of the vector charges.

Calculating the Tensor

• Substitution of new parameters, relates the total derivative to the second electrical sum.
• Tensors are more easily demonstrated using ladder operators in the Laplace domain.
• The result is a general class of results following application of electrostatics.

4D Harmonic Tensors

• The potential of a point charge in 4D is r^2.
• It has been shown through this paper, that it remains possible to create multipole for 4 dimensional potentials: M(i,k)/r^(2n+2)
• It has also been shown that these tensors form a similar structure to that shown earlier for the 3 dimensions.
• Certain tensors of rank =1, must be set to zero to be able to evaluate them. D harmonic tensors are generally easier than 3D tensors. They do howver require that the eigen tensors are multiplied. If this is the case 4D tensors will be used instead of 3D.

Decomposition

• The process allows you to correctly expand the tensors in harmonic coordinates.
• The key with this method is to correctly calculate the axis of the trace by two indices.
• The more complex expression will often require that more terms are considered to estimate the overall state.
• The tensor is more easily estimated.

Stark's quadratic effect

• Formula exists for determining the strak effect.
• These are determined by passing parameters to parabolic coordinates.
• From this perspective the dipole moment is propotional to the field.
• With the electric polarizability in the ground state.

Harmonic Tensors

• A consideration is mode for the Stark effect and using symmetry.
• An original formulation for the first state, will use the form: (∇+Eo+1/r)Φ=EzΦ.
• The second energy is dependent on the linear parameter as follows.
• The second term to account for linear correction is found using Φ2=(φ,Ez,φ00).

Derivation and Solution

• Method of harmonic tensors is applied to the Schodinger equation in momentum Space.
• In this approach the equation follows the traditional form of the first equation.
• The resulting eigenfunctions account for aspects of the gegenbauer polynomial.
• The differential expression leads to simple solutions and derivation.
• Following this result an equation containing the (∆-1)^2 parameter from the function is produced.

Group Meaning

• Expression solves the 4d equation with a 4d SO(4) -invariant laplace equation, for the case when the momentum fock is spherical.
• These properties include the fact that their radius square is == to 1.
• Eigenvalues can be generated through the fock expression - that in terms of fock.
• These transformations allow us to compare these expressions and solve the schrodinger equation.

Modifying Fock’s Theory

• This extends the momentum into the 4d space.
• This is achieved by having the harmonic tensors in spherical form.
• No repeats of the SO(4) symmetry are made. And the main property is that between 4d tensors is that transition is preserved.
• This was shown when coincident point between the +ve axis and points are == set.

Proof

• The solid harmonics follow expression in its transformation.
• Factor is added, which does not require a re-scaling.
• The overall result is a very standard expression.

Transition

• Transformation occurs and the fock transformation takes place and can be applied in generalized sense for four coordinates.
• The numerator of the equations contains the polynomial.
• The transformation is achieved by approximating contour in the lower half, or using a set of generalized coordinates.

Electrostatic Mehtods

• The potential of charges on a charge distribution, can be approximated on the surface of a unit sphee, in the case when the density is spherical.
• In simple electrostatic the tensor are used, and take on a simple form - at which point the electric density is considered inside and outside the sphere.
• the potential remains non-continous at a boundary, and is equal to : 4 *PI * sigma.
• By subbing and integrating by parts we obtain a new integral of 3d schrodinger functions, this is the same as the analogous forms in 4d spherical harmonics.

4D Space

• On the surface of charge distribution, the charge can be written, for every situation.
• This is the basic tensor relation - related to its harominc polynomial state (N-1). Where that is simply the surface on which the sphere exists.
• Therefore The result has a discontinuity of 4 PI. The above is then also used to demonstrate the 3d -> 4d case - by deriving the fock expressiouns - with a limit on the surface.
• Therefor these derivations are more easily achieved in 3d space rather than momentum space.

Operator Raising

• Operators that can operate up the spectrum ladder.
• With a set of equations for coherent set states, in respect to the "creation" vs "annihilation" paradox.
• This creates a square angular momentum expression - leading to (lˆ+1).
• • This forms ladder like structure in the harmonic space, by taking on the form of harmonic tense EigenState.
• This result can be used with different forms written throughout each harmonic tensor structure.
• Each operator follows the same rules: