Fine-structure constant (spiral)

H atom energy levels emerge from hyperbolic spiral

Electron orbit in the H atom can also be mapped as a gravitational orbit. Although this approach maps only the electron-as-mass and so is limited to a simple radial component of the orbital on a 2-D plane (when compared to the more comprehensive Schrödinger equation), it does exhibit interesting properties; quantisation (the integer values) of the (principal quantum number) n-shells naturally emerge as a direct function of pi, as well as the transition frequencies between the n-shells . This suggest that quantisation could have geometrical origins.

Electron transition path radius (n=1 to n=9)

$\varphi =(2)\pi ,\;r=4a_{0}$ (360°)

$\varphi =(8/3)\pi ,\;r=9a_{0}$ (360+120°)

$\varphi =(3)\pi ,\;r=16a_{0}$ (360+180°)

$\varphi =(16/5)\pi ,\;r=25a_{0}$ (360+216°)

$\varphi =(10/3)\pi ,\;r=36a_{0}$ (360+240°)

$\varphi =(7/4)\pi ,\;r=49a_{0}$

$\varphi =(7/2)\pi ,\;r=64a_{0}$ (360+270°)

The electron is treated, not as a distinct entity but instead as an (electric) wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates). This orbit can then be simulated as a gravitational orbit by plotting these (mass) points.

By applying the same to (incoming) photons, transition between (principal quantum number) n-shells can also be mapped as a (semi-classical) gravitational orbit transition. Semi-classical because transitions occur in steps, the steps as the duration of 1 electron-proton (reduced mass) wavelength. As the electron continues orbiting the nucleus (proton) during transition, a specific hyperbolic spiral path emerges.

Hyperbolic spiral

Hyperbolic spiral

A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center. As this curve widens (radius r increases), it approaches an asymptotic line (the y-axis) with the limit set by a scaling factor a (as r approaches infinity, the y axis approaches a).

The spiral shape that the electron maps can be represented in Cartesian coordinates. Periodically the angles converge to give integer radius, the general form (beginning at the outer limit ranging inwards) gives;

$x=a^{2}{\frac {cos(\varphi )}{\varphi ^{2}}},\;y=a^{2}{\frac {sin(\varphi )}{\varphi ^{2}}},\;0<\varphi <4\pi$

radius = ${\sqrt {(}}x^{2}+y^{2})r$

$\varphi =(2)\pi ,\;4r$ (360°)

$\varphi =(4/3)\pi ,\;9r$ (240°)

$\varphi =(1)\pi ,\;16r$ (180°)

$\varphi =(4/5)\pi ,\;25r$ (144°)

$\varphi =(2/3)\pi ,\;36r$ (120°)

Theory

Principal quantum number n

Electron at different n level orbitals

The H atom has 1 proton and 1 electron orbiting the proton, in the Bohr model (which approximates a gravitational orbit), the electron can be found at select radius (the Bohr radius) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level.

The principal quantum number n denotes the energy level for each orbital. As n increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as n increases, the electron orbit is further from the nucleus). Each shell can accommodate up to n² (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2n² electrons. As these energy levels are fixed according to this integer n, the orbitals may be said to be quantized.

(Bohr) orbital

The basic orbital radius has 2 components, dimensionless (the fine structure constant alpha) and dimensioned (electron + proton wavelength);

wavelength = $\lambda _{H}=\lambda _{p}+\lambda _{e}$

radius = $r_{orbital}=2\alpha n^{2}(\lambda _{H})$

As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius

electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference

length = $l_{orbital}={\frac {1}{r_{orbital}}}$

Duration = 1 step per wavelength and so velocity

velocity = $v_{orbital}={\frac {1}{2\alpha n}}$

Giving period of orbit

period = ${\frac {2\pi r_{orbital}}{v_{orbital}}}=2\pi 2\alpha 2\alpha n^{3}\lambda _{H}$

As we are not mapping the wavelength component, a base (reference) orbital (n=1)

$t_{ref}=2\pi 4\alpha ^{2}$ = 471964.356...

The angle of rotation depends on the orbital radius

$\beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha }}}}$

Photon orbital model

The electron can jump between n energy levels via the absorption or emission of a photon. In the Photon-orbital model, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that $orbital\;radius+photon=zero$ (cancel).

The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path.

The photon is actually 2 photons as per the Rydberg formula (denoted initial and final).

$\lambda _{photon}=R.({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}})={\frac {R}{n_{i}^{2}}}-{\frac {R}{n_{f}^{2}}}$

$\lambda _{photon}=(+\lambda _{i})-(+\lambda _{f})$

The wavelength of the ($\lambda _{i}$) photon corresponds to the wavelength of the orbital radius. The (+$\lambda _{i}$) will then delete the orbital radius as described above (orbital + photon = zero), however the (-$\lambda _{f}$), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+$\lambda _{i}$) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

$r_{orbital}=r_{orbital}+(\lambda _{i}-\lambda _{f})$

However, the ($\lambda _{f}$) has the longer wavelength, and so after the ($\lambda _{i}$) photon has been absorbed, and for the remaining duration of this ($\lambda _{f}$) photon wavelength, the orbital radius will be extended until the ($\lambda _{f}$) is also absorbed. For example, the electron is at the n = 1 orbital. To jump from an initial $n_{i}=1$ orbital to a final $n_{f}=2$ orbital, first the ($\lambda _{i}$) photon is absorbed ($\lambda _{i}+\lambda _{orbital}=zero$ which corresponds to 1 complete n = 1 orbit by the electron, the orbital phase), then the remaining ($\lambda _{f}$) photon continues until it too is absorbed (the transition phase).

$t_{ref}\sim 2\pi 4\alpha ^{2}$

$\lambda _{i}=1t_{ref}$

$\lambda _{f}=4t_{ref}$ (n = 2)

After the ($\lambda _{i}$) photon is absorbed, the ($\lambda _{f}$) photon still has $\lambda _{f}=(n_{f}^{2}-n_{i}^{2})t_{ref}=3t_{ref}$ steps remaining until it too is absorbed.

orbital transition during orbital rotation

This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of $r_{incr}$ (both photon and orbital radius may be considered as constructs from multiple units of this geometry);

$r_{incr}=-{\frac {1}{2\pi 2\alpha r_{wavelength}}}$

In summary; the ($\lambda _{i}$) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in step

$r=r_{orbital}$

WHILE ($\lambda _{i}>0$)

$r=r+r_{incr}$

//$\lambda _{i}$ photon

Conversely, because of its minus term, the ($\lambda _{i}$) photon will simultaneously extend the orbital radius accordingly;

WHILE ($r<4r_{orbital}$)

$r=r-r_{incr}$

//$\lambda _{f}$ photon

The model assumes orbits also follow along a timeline z-axis

$t_{orbital}=t_{ref}{\sqrt {1-{\frac {1}{(v_{orbital})^{2}}}}}$

The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius;

$t_{transition}=t_{ref}{\sqrt {1-{\frac {1}{(v_{transition})^{2}}}}}$

Transition

Spiral angle

For an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength). In this example the electron transition starts at the initial (n_i = 1) orbital

$\varphi =0,\;r_{orbital}=2\alpha$

Bohr radius during ionization, as the H atom electron reaches each n level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)

For each step during transition;

$\beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha }}}}$

$\varphi =\varphi +\beta$

Setting t = step number (FOR t = 1 TO ...), we can calculate the radius r and $n_{f}^{2}$ at each step.

$r=r_{orbital}+{\frac {t}{2\pi 2\alpha }}$ (number of increments t of $r_{incr}$)

$n_{f}^{2}=1+{\frac {t}{2\pi 4\alpha ^{2}}}$ ($n_{f}^{2}$ as a function of t)

$\varphi =4\pi {\frac {(n_{f}^{2}-n_{f})}{n_{f}^{2}}}$ ($\varphi$ at any $n_{f}^{2}$)

We can then re-write ($n_{f}$ is only an integer at prescribed spiral angles);

$\beta ={\frac {1}{{r_{orbital}}^{2}n_{f}^{3}}}$

Giving integer values at these spiral angles

$\varphi =(2)\pi ,\;r=4r_{orbital}$ (360°)

$\varphi =(8/3)\pi ,\;r=9r_{orbital}$ (360+120°)

$\varphi =(3)\pi ,\;r=16r_{orbital}$ (360+180°)

$\varphi =(16/5)\pi ,\;r=25r_{orbital}$ (360+216°)

$\varphi =(10/3)\pi ,\;r=36r_{orbital}$ (360+240°)

$\varphi =(7/4)\pi ,\;r=49r_{orbital}$

$\varphi =(7/2)\pi ,\;r=64r_{orbital}$ (360+270°)

Transition frequency

Experimental

Experimental values for H(1s-ns) transitions (n the principal quantum number).

H(1s-2s) = 2466 061 413 187.035 kHz

H(1s-3s) = 2922 743 278 665.79 kHz

H(1s-4s) = 3082 581 563 822.63 kHz

H(1s-∞s) = 3288 086 857 127.60 kHz (n = ∞)

R = 10973731.568157 (Rydberg constant)

α =137.035999177 (inverse fine structure constant

The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton.

$r_{wavelength}=\lambda _{H}={\frac {2c}{\lambda _{e}+\lambda _{p}}}$

Dividing (dimensioned) wavelength ($r_{wavelength}$) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). The $(n^{2}-1)$ term gives the number of orbital wavelengths in the transition phase;

$h_{(1s-ns)}=(n^{2}-1){\frac {\lambda _{H}}{H(1s-ns)}}$

$h_{(1s-2s)}$ = 1887839.82626...

$h_{(1s-3s)}$ = 4247634.04874...

$h_{(1s-4s)}$ = 7551347.55306...

Simulation

The following example simulates an electron transition, the electron begins at radius $r=r_{orbital}$ and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process.

The period $t_{sim}$ and length $l_{sim}$ are measured at integer $n^{2}r$ (n = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the angles; (360°(1r), 360°(4r), 360+120°(9r); 360+180°(16r), 360+216°(25r), 360+240°(36r) ...), and so we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period $t_{sim}$ can then be used to calculate the transition frequencies.

In this example, the nucleus = 249 mass points (start x, y co-ordinates close to 0, 0) and the electron = 1 mass point (at radius x = r, y = 0), t_sim = period and l_sim = distance travelled by the electron ($l_{orbital}=l_{sim}$ at n=1), the radius coefficient r_n = radius divided by $r_{orbital}$. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the x, y co-ordinates for the electron, nucleus center and barycenter.

H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8

$j_{atom}=250$ (atomic mass)

$i_{nucleus}=j_{atom}-1=249$ (relative nucleus mass)

$r_{wavelength}=2({\frac {j_{atom}}{i_{nucleus}}})^{2}$ = 2.0160965

$r_{orbital}=2\alpha \;*\;r_{wavelength}$ (radius) = 552.5556

$t_{n}={\frac {t_{sim}}{r_{wavelength}}}$

$l_{n}={\frac {l_{sim}}{l_{orbital}}}-l_{orbital}$

$r_{b}=r_{sim}-{\frac {r_{sim}}{j_{atom}}}$

$r_{n}={\frac {r_{b}}{r_{orbital}}}$

table 1. Electron transition (mass = 250; r_n= 1 to 5)

rn	tsim	ln	angle	x, y (electron)	x, y (nucleus)	x, y (barycenter)
1	471957.072	0.9999897	360°	550.334, 0.0036	-2.2102, -0.00002	-0.00004, -0.00001
4	1887867.293	2.000012	359.952489°	2202.8558, 0.0001	-7.9565, -1.9475	0.8868, -1.9397
9	4247689.502	4.000014	119.92712°	-2473.180, 4296.283	13.558, -10.325	3.611, 6.901
16	7551439.538	6.000014	179.91669°	-8815.254, 12.818	25.636, 13.303	-9.728, 13.301
25	11799118.905	8.000014	215.9122°	-11158.64, -8081.13	16.580, 39.083	-28.118, 6.602

Comparison of the spiral angle at r_n = 4, 9, 16 (360, 360+120, 360+180) with different mass (m = 64, 128, 250, 500, Rydberg). For the proton:electron mass ratio; m = 1836.15267...

table 2. Spiral angle at $r_{n}$ = 4, 9, 16

mass	rn = 4	rn = 9	rn = 16
m = 64	359.80318°	119.70323°	179.66239°
m = 128	359.90394°	119.85415°	179.83377°
m = 250	359.95249°	119.92711°	179.91669°
m = 500	359.97706°	119.96501°
Rydberg	360°	360+120°	360+180°

Further information: Quantum_gravity_(Planck)

External links

• Gravity at the Planck scale
• The Programmer God
• Simulation hypothesis modelling at the Planck scale using geometrical objects