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The Fine Structure Constant and Bohr Atom Orbitals

Practically every physical constant consists of a number multiplied by one or more physical units. e.g.

c = 3e8 m/sec. (1)

The number part changes with a different choice of units. e.g.

c = 3e10 cm/sec.      (2)

Of all the physical constants, one stands out in that it is dimensionless. That constant is called The Fine Structure Constant. It is symbolized by “a” and equals approximately 1/137. More precisely

a = e2/2e0hc.    (3)

Since it is dimensionless, a is the same in every system of units.

Many physical quantities can be expressed as a times the physical units of choice. For example the speed of a Bohr atom ground state electron can most simply be expressed as

v = ac.      (4)

Or more generally

vN = ac/N.                   (5)

A second useful constant is

a’ = h/(2pmca),                 (6)

where m is the electron’s mass. a’ has the dimension “meter”, and the radius of the Nth electron orbital is

rN = N2a’.           (7)

Appendix A contains a Python program that displays the values of several physical constants in m-kg-sec units.

Appendix A

import math

#Physical Constants

c=2.99792458e8  #Speed of light

print('c=',c,' m/sec')

h=6.626068e-34    #Planck constant (m*m*kg/sec

print('h=',h,' m*m*kg/sec')

eps0=8.85418782e-12    #Permittivity constant (nt*m*m/coul*coul)

print('eps0=',eps0,' nt*m*m/coul*coul')

q=1.60217662e-19    #Elementary charge

print('q=',q,' coul')

me=9.10938356e-31   #Electron rest mass

print('me=',me,' kg')

alpha=q**2/(2*eps0*h*c) #Fine Structure Constant

print('alpha=',alpha)

alpha1=h/(2*math.pi*me*c*alpha)

print('alpha1=',alpha1,'m')

ve1=alpha*c

re1=h/(2*math.pi*me*c*alpha)

N=1

while (N>0):

    N=input('N?')

    N=int(N)

    vN=c*alpha/N

    rN=N**2*alpha1

    print('vN=',vN,', rN=',rN)