Chern Surfaces and the Wave Model Of The Atom

                                                        

                                                                 Fig. 1: Assorted Polyhedra  

                         Fig. 2. Chern surfaces and class number

Chern surfaces, developed for use in global differential geometry, don't exactly roll off the tongue or conjure up cocktail table discussions, but they have immense importance in math as well as in physics. Developed by Shiing-Shen Chern, the Chern class number to be associated with a given surface was originally conceived as the number of times a closed surface could be wrapped around another closed surface. Before getting to a physics application, let's consider the math origins.

Basically these can be traced to the polyhedral theorem of Leonard Euler. The Table shown with representative polyhedra (Fig. 1) can serve as a basis for a more concrete appreciation. Starting from the left a distinct polyhedron is identified (e.g. tetrahedron) then simply illustrated in a no frills way, then its vertices (V), edges (E) and faces (F) identified before giving the "Euler characteristic" E in the final column on the right. This is given by the simple form: V - E + F.

Thus, the tetrahedron with V = 4, E = 6 and F = 4 will have characteristic:

V - E + F = (4) - (6) + (4) = (-2) + (4) = 2

As one can see from the table, Euler's theorem basically asserts the same number, 2, results for all cases in which his formula is applied - whether for regular or irregular convex polyhedra. Obviously, the more faces, vertices, edges added the more the particular surface is smoothed out to approach a sphere (e.g. with infinite edges, faces) then one has the Euler theorem transferring to the Gauss- Bonnet theorem, formulated and proven in the 19th century.

Now, a simple physics application of the Chern surface and class number is to Bohr "quantization", such as depicted in the accompanying graphic (Fig. 2) for two cases: left - where the quantization is not satisfied (i.e. the outer wave surface doesn't complete itself an integral number of times around a given Bohr orbit, and right - the case where it does, leading to the quantum number (n) being identified with the integer number of waves completed, in this case 4.

The beauty is that the radius is scaled into n (standing) waves of de Broglie wavelength l_D . A visual reference for this wave-orbiting electron atom can be represented as shown in the same right side of the graphic, with the de Broglie wavelength spanning the distance between successive "humps", and emphasizing that an integral number of such wavelengths form the circumference of the atomic orbit, as required by 2π r = n l_D . 

By extension of the concept of these standing waves, but for different n and r, one can arrive at the probabilistic wave model of the atom.   The wave model from Bohr’s uses the following quantized relationship:

m vr  = nh/ 2p  = L

where L is the angular momentum.  Re-arranging:

h/ mv =  2p r/ n = l_D    or  2p r =  n l_D

Showing the circumference is scaled into n (standing) waves of wavelength l_D   as shown below (Fig. 3):

Fig. 3 : Standing waves for a hydrogen atom

  This wave-orbiting electron atom still has a radius r,  but with waves each separated by one de Broglie wavelength , l_D.

     Thereby  an integral number of such wavelengths form the circumference of the atomic orbit, as required by  the condition:  2p r =  n l_D.

  For example, in the case of hydrogen the first three of these cloud-wave regions are shown in Fig. 4.

Fig. 4: Hydrogen orbitals

Let us ‘zoom in’ on the more spherical n= 1 configuration, and the probability for the electron in this space as depicted in Fig. 4 below:

Fig. 5: The n = 1 electron orbital for hydrogen

    This diagram more than any other dispenses with the notion that hydrogen electron occupies a definite position. Instead, it’s confined someplace within a “cloud” or probability space (b) but that probability can be computed as a function of the Bohr radius (a_o = 0.0529 nm).  The probability P_1s for the 1s orbital is itself a result of squaring the “wave function” for the orbital.  If the wave function is defined y (1s) = 1/Öp (Z/ a_o) exp (-Zr/ a_o), and the probability function is expressed:

P = ½y (1s) y (1s) *½

     Where y (1s) * is the complex conjugate, then the graph shown in Fig. 4 is obtained. Inspection shows the probability of finding the electron at the Bohr radius is the greatest, but it can also be found at distances less than or greater than  0.0529 nm.

Chern surfaces, as we see in this post, have a wide applicability including to quantum mechanics.

See Also:

Brane Space: An Introduction to Quantum Mechanics (3)