__Bohr's
H Atom Revisited (Part 1)__

In 1913 Niels Bohr
introduced his model of the Hydrogen atom. No doubt influenced by Rutherford’s
work, he suggested that the atom consists of a central, ** resting** proton (nucleus)
with an electron orbiting in a circle. According to the latest information on
the Internet, in the atom’s lowest energy state, re1, the radius of the
electron’s orbit is re1= 5.291772 × 10

Now according to Coulomb F,
the force exerted on the electron by the proton, is generally F=q^{2}/4pe_{o}(re^{2}). And according to Newton 2 F=(me)(ve^{2})/(re).
This yields ve=q^{2}/(4pe_{o}(re)(me))^{1/2}=2187691
m/sec.

Let us now consider
Bohr’s assumption that the proton is at rest at the origin of an inertial
rectangular coordinate system. Looking down from the positive z-axis, we imagine
that the electron is at (-re,0,0) and is orbiting CCW around the origin. This
would mean that at any given instant the entire atom’s momentum would simply
equal the electron’s momentum. But the electron’s momentum varies in
direction from moment to moment. Yet in Bohr’s model (and presumably by
observation) the atom’s total momentum is **zero***
*at
any given moment.
The only way this condition can be satisfied is if

But our critique of
Bohr’s model is not yet done for the following reason: the electron should
radiate energy, since any charged particle going around a circular path
radiates. Indeed Lorentz had found that a charge, going in a circle of radius r
and at speed v, should experience a “radiation reaction” force opposite at
any moment to the direction of ** v**.
The formula for this force is F

What might be the
explanation for the needed counteraction to F_{RadReact} acting on the
orbiting electron? It cannot be a magnetic force, since such a force would act
perpendicular to ** v**.

Now unknown to Bohr, later
theorists worked out the formula for the electric field of a charge whose
kinetic history is known. This formula entails terms such as the source
charge’s *retarded* velocity and
acceleration, which pertain to a time *prior*
to their electric force on some distant charge. The key to this phenomenon is
that electromagnetic fields are not instantly felt at distant points. Their
influence travels at the speed of light from source charge to subject charge.
The field felt by a subject charge at time t is a function of kinematical
variables at the source charge at some earlier time t_{retarded} = t_{present}–L/c,
where L is the distance between the object charge and the source charge’s
position at the time t_{retarded}. (Note in this case that L is a
displacement, and is not angular momentum.) As mentioned earlier, the formula
for the radiation reaction force, acting on the Bohr model electron at time t=0,
is F_{RadReact}=q^{2}w^{3}(re)/6pe_{o}c^{3}. And the
formula for the proton-engendered electric force acting on the electron is the
negative of F_{RadReact}. This force provides the needed counterforce to
F_{RadReact}. If the sum of these two forces is zero, then the net
radiation emitted by the electron is zero.

We may determine where the
proton was by “inching up” to
values for L and q using a computer (See Appendix A for a schematic, and Appendix B for a
typical “inching up” program written in Python). Note that the radius of the
proton’s orbit has been greatly exaggerated in the schematic for clarity.

It is worth mentioning that
Bohr’s postulate, L=Nh/2p, does not apply to H atom protons. The formula that works in the proton
case is Lp=N(hp)/2p, where the constant hp is
a companion to h. hp has the value hp = 2p(Lp)/N = 3.608669e-37 = h(me/mp).

__Appendix
A__

The
Bohr Atom at a Typical Delayed Time

__Appendix
B __

import math

c=2.99792458e8 #Speed
of light

h=6.626068e-34
#Planck constant

eps0=8.85418782e-12
#Permittivity constant

q=1.60217662e-19
#Elementary charge

ElectronMass=9.10938356e-31
#Electron rest mass

me=ElectronMass

ProtonMass=1.6726219e-27
#Proton rest mass

mp=ProtonMass

ElectronOrbitRadius1=.529177211067e-10

re1=ElectronOrbitRadius1

ve1=q/math.sqrt(4*math.pi*eps0*me*re1)

L1=me*ve1*re1

print('L1=',L1)

L1=h/(2*math.pi)

print('L1=',L1)

input("OK?")

#Any of the lowest 10 energies can be computed.

N=0

while N<=10:

N=int(input('level=?(>10
to exit) '))

if(N<11):

print('Computing level ',N)

#ElectronOrbitRadius=N**2*ElectronOrbitRadius1
#N'th level electron orbit radius

L=N*L1

ve=q**2/(4*math.pi*eps0*N*L1)

print('ve1=',ve1,', ve=',ve)

input('OK?')

#ElectronSpeed1=h/(2*math.pi*ElectronMass*ElectronOrbitRadius1)
#Bohr rule

ve=h/(2*math.pi*me*re1*N)

print('ve=',ve)

input('OK?')

#ElectronKineticEnergy1=ElectronMass*ElectronSpeed1**2/2
#Electron kinetic energy

#ElectronKineticEnergy=ElectronMass*ElectronSpeed**2/2

re=N*L1/(me*ve)

taue=2*math.pi*re/ve

nue=1/taue #Orbital
fequency

omega=2*math.pi*nue #Angular
frequency

vp1=me*ve1/mp #total momentum=0

vp=me*ve/mp

rp1=vp1/omega

rp=vp/omega

#ProtonKineticEnergy=ProtonMass*ProtonOrbitalSpeed**2/2

#PE=-q**2/(4*math.pi*eps0*(ElectronOrbitRadius+ProtonOrbitalRadius))
#Potential energy

#TotalEnergy=ElectronKineticEnergy+ProtonKineticEnergy+PE
#Total energy of atom

maxN=1e8 #Maximun
number of iterations in orbital computation

#DelayedTime=(ElectronOrbitRadius1+ProtonOrbitalRadius1)/c

DelayedTime=(re+rp)/c

dt=-DelayedTime/maxN #Time
increment

loopN=-1

theta=0. #See
diagram

RadReactF=q**2*omega**3*re/(6*math.pi*eps0*c**3)

RetardedProtonSpeed=vp
#Proton travels at constant speed

ar=RetardedProtonSpeed**2/rp

Fy=0. #Initial
trial force of proton on electron

#Find parameters when forces on electron sum to zero

while RadReactF+Fy>=0:

loopN=loopN+1

#If infinite loop, Print params and exit by clicking X in display

if loopN>maxN:

print('Possible infiite loop.')

print('loopN, RadReactF, Fy',loopN,'; ',RadReacF,', ',Fy)

print('Abort by clicking
X in your display.')

input('OK?')

#If necessary, keep 'inching up’ to loop termination condition.

RetardedTime=-loopN*dt

theta=omega*RetardedTime

RetardedVelocityX=vp*math.sin(theta)
#See diagram

RetardedVelocityY=vp*math.cos(theta)

RetardedAccelerationX=-(vp**2)/rp*math.cos(theta)

RetardedAccelerationY=vp**2/rp*math.sin(theta)

L=math.sqrt((re+rp)**2-(rp*math.sin(theta))**2)
#See diagram

alpha=math.atan(rp*math.sin(theta)/L)

Lx=-L*math.cos(alpha)

Ly=L*math.sin(alpha)

#Compute electric field at electron (see Griffith's text)

ux=c*Lx/L-RetardedVelocityX

uy=(c*Ly/L-RetardedVelocityY)

u=math.sqrt(ux**2+uy**2)

ux=-u*math.cos(alpha)

uy=u*math.sin(alpha)

Ey=q/(4*math.pi*eps0)*L/(Lx*ux+Ly*uy)**3

Ey=Ey*(uy*(c**2-RetardedProtonSpeed**2)-(Lx*(ux*RetardedAccelerationY-uy*RetardedAccelerationX)))

Fy=-q*Ey #Interactive
force on electron

#When Fy+RadiationReactionForce=0, display results.

L=math.sqrt((re+rp)**2-(rp*math.sin(theta))**2)
#See diagram

alpha=math.atan(rp*math.sin(theta)/L)

Lx=-L*math.cos(alpha)

Ly=L*math.sin(alpha)

print("L=",L)

print("alpha=",alpha)

print("Lx=",Lx)

print("ve=",ve," vp=",vp," rp=",rp)

print(N*h/(2*math.pi*me*re*ve)) #should be 1.

print('RadReactF,Fy= ',RadReactF,', ',Fy)

#Do another level.

#Reference: Griffiths, ** Introduction to Electrodynamics**.