Fock's sphere in theory of hydrogen atom

The quantum Coulomb problem , which allows calculating the spectrum of a system of two opposite charges, is still fundamental in quantum theory. . The names of the founders of twentieth physics are associated with it: N. Bohr, A. Sommerfeld, V. Pauly, E. Schrödinger and V. Fock. The introduction in the theory of atomic spectra begins with it, and it has been thoroughly studied using methods of the theory of special functions. Due to its simplicity and underlying symmetry- the group SO(4) of rotations in four dimensional space – it is an extremely useful and fine tool of theoretical physics for constructing various concepts .

Despite the apparent exhaustive treatment of the quantum Coulomb problem, there are still some questions that have not been fully studied. Fock’s result is surprising: why is the SO(4) symmetry realized in the momentum space wrapped into three –dimensional (3D) sphere, with an extension to the four-dimensional space ?

Essence of the problem

Let us recall the background proceeding Fock’s accomplishment. Two classical vector integrals, the angular momentum and the Laplace-Runge-Lenz vector, in quantum mechanics correspond to vector operators that commute with the energy operator, i.e.,with the Hamiltonian. An analysis of their commutators carried out in shows that they generate a Lie algebra (a linear space with a commutation operation) coinciding with a Lie algebra of operators of small (infinitesimal) rotations in 4D space .

For physics, the correspondence means that some transformation of variables and operators maps the original quantum Coulomb problem into the problem of free motion of a particle over a 3D sphere embedded in a 4D space. The energy operator is then invariant under rotations of the 3-D sphere . This is reminiscent of the remarkable effect of Lewis Carroll’s soaring grin of the Cheshire cat.

Fock's approach struck contemporaries.The starting point in his theory is integral Schrödinger’s equation (SE) in momentum space. The space can be considered as 3D plane in a 4-D space. Fock then wraps 3D plane into a 3D sphere using stereographic projection, known since antiquity as convenient transformation of a globe into a flat map. (Fock’s globe is three-dimensional as is the map.) At the same time, additionally, Fock surmises the factor for wave functions such that original integral equation turns into an equation for spherical functions on 3D sphere (to be distinguished from functions on two-dimensional sphere.)

This equation, rarely used in physics but well-known in the theory of special functions, , is invariant under rotations in 4D space. Fock does not explain the physical meaning of transformation he found . As a result, the fundamental questions remain: why is the SO(4) symmetry realized in the wrapped momentum space rather than in the position space, and how has the electron 'learned about the stereographic projection?'

Recently, Fock's theory is further developed with the help of transformation of eigen-functions from the momentum space into 4-D position one. It was found that final transition of 4D spherical harmonics into physical space is algebraic and does not need an integration at all

Fock's theory

The Schrödinger equation for eigenfunctions (SE), using atomic units (the unit of energy being ${\frac {Z^{2}me^{4}}{\hbar ^{2}}}$ and the unit of length being Bohr's radius ${\frac {\hbar ^{2}}{Zme^{2}}}$ ) , has the form

$\left(-{\frac {1}{2}}\Delta -{\frac {1}{r}}\right)\Psi _{nlm}=-{\frac {1}{2n^{2}}}\Psi _{nlm}$.

In what follows, it is convenient to reduce the orbits of all radii $na_{B}$ to a single radius , i. e., change the radius vector for each eigen function as $\mathbf {r'} ={\frac {\mathbf {r} }{n}}$. The equation then takes a deceptively simple form.

$(-\Delta +1)\Psi _{nlm}(\mathbf {r} )={\frac {2n}{r}}\Psi _{nlm}(\mathbf {r} ),$

where $r$ is the modulus of the vector $\mathbf {r}$. The eigenfunctions in the momentum representation then have scaled argument $\mathbf {p'} =n\mathbf {p}$ .

The Schrödinger equation (with $(\hbar =1)$)when moving to the momentum representation:

$\Psi _{nlm}(\mathbf {r} )={\frac {1}{(2\pi )^{3}}}\int \ a_{nlm}(\mathbf {p} )e^{(i\mathbf {pr} )}d^{3}\mathbf {p}$,

contains a convolution with respect to momenta. Because the potential ${\frac {1}{r}}$ goes to ${\frac {4\pi }{\left|\mathbf {p^{2}} \right|}}$, the SE in the momentum space is nonlocal:

$(\left|\mathbf {p^{2}} \right\|+1)a_{nlm}(\mathbf {p} )-{\frac {2n}{2\pi ^{2}}}\int {\frac {a_{nlm}(\mathbf {p'} )}{\left|\mathbf {p-p'} \right|^{2}}}d^{3}\mathbf {p'} .$

Fock's sphere

The first step of the theory is to multiply the function $a_{nlm}(\mathbf {p} )$ (done without an explanation) by factor $(1+\mathbf {p^{2}} )^{2}$ . The second step is to wrap the 3D plane into 3-D sphere (with four coordinates $({\boldsymbol {\xi }},{\mathit {\xi _{0}}})$; see Fig.1). Figure 1 shows that the tangent of the slope of the projecting (red) straight line is $\tan \varphi ={\frac {1}{\left|\mathbf {p} \right|}}.$

File:Scheme of projection.png

Fig.1 Stereographic projection of $3D$ $\mathbf {p}$-plane on $3D$ sphere

Hence follow the formulas

$\left|{\boldsymbol {\xi }}\right|=\sin 2\varphi ={\frac {2\left|\mathbf {p} \right|}{(1+\mathbf {p^{2}} )}},$

${\boldsymbol {\xi }}={\frac {2\mathbf {p} }{(1+\mathbf {p^{2}} )}},$

${\mathit {\xi _{0}}}=\cos 2\varphi ={\frac {(\mathbf {p^{2}} -1)}{(\mathbf {p^{2}} +1)}},$

${\boldsymbol {\xi }}^{2}+\xi _{0}^{2}=1.$

The stereographic projection doubles the tilt angle $\varphi$ and this is the effect it produces. The flat drawing correctly reflects the 4-D transformation.

In the new variables, with Fock’s factor taken into account, the eigen-function becomes

$b_{nlm}({\boldsymbol {\xi }},\xi _{0})=(\mathbf {p^{2}} +1)^{2}a_{nlm}(\mathbf {p} ).$

It is essential that the projection be given by a conformal transformation. In this case, the angles between intersecting curves are preserved. The metric on the sphere in the momentum space coordinates ( of the p-plane) is expressed as

${\frac {4}{\mathbf {(p^{2}+1)^{2}} }}(d\mathbf {p} )^{2}.$

Hence, the contraction coefficient for elements of the p-space is ${\frac {(\mathbf {p^{2}} +1)}{2}}$ . The volume element in Fock's equation can be expressed in terms of the 3D surface element of sphere:

$d^{3}\mathbf {p} ={\frac {1}{8}}(\mathbf {p^{2}} +1)^{3}dS_{\xi }$

Additionally, the kernel of the integral can be (very fortunately but not obviously) transformed as

${\frac {1}{\mathbf {(p-p')^{2}} }}={\frac {2}{\mathbf {(p^{2}+1)} }}{\frac {1}{[{\boldsymbol {(\xi -\xi ')^{2}}}+(\xi _{0}-\xi '_{0})^{2}]}}{\frac {2}{\mathbf {(p'^{2}+1)} }},$

which doesn't follow from the conformal property. These relations allow Fock to obtain the integral equation

$b_{nlm}({\boldsymbol {\xi }},\xi _{0})-{\frac {n}{2\pi ^{2}\ }}\int {\frac {b_{nlm}({\boldsymbol {\xi '}},\xi '_{0})}{[{\boldsymbol {(\xi -\xi ')^{2}}}+(\xi _{0}-\xi '_{0})^{2}]}}dS_{\xi '},$

where as can be seen from the figure, the surface element of the unit 4D sphere with the volume $2\pi ^{2}$ is

$dS_{\xi }={\frac {d{\xi _{1}}d\xi _{2}d{\xi _{3}}}{\xi _{0}}}={\frac {dV_{\xi }}{\xi _{0}}}$.

Next, V. Fock refers to the theory of spherical functions in 4-D space . These functions contain 3D spherical functions times the Gegenbauer polynomials of the argument $\xi _{0}$.

Found equation is rather complicated while Gegenbauer polynomials are very simple and useful for physicists.

See also

• Second quantization
• Fock state
• Hartree-Fock method
• Fock-Lorentz symmetry
• Fock-Schwinger gauge

Further reading

Fock, V.A. (2004) V.A. Fock-Selected Works:Quantum Mechanics and Quantum Field Theory. CRC Press.https://doi.org/10.1201/9780203643204

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