H Atom Revisited (Part 1)
In 1913 Niels Bohr
introduced his model of the Hydrogen atom. No doubt influenced by Rutherfordís
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 atomís lowest energy state, re1, the radius of the
electronís orbit is re1= 5.291772 ◊ 10−11 meters. And the value for the electronís mass is me=
9.10938356 ◊ 10-31 kg. Bohr postulated that the
electron's angular momentum, L=(me)(ve)(re), equals the constant Nh/2p, where N=1, 2, 3... (N=1 is the lowest
"ground" state of the atom.) Thus according to Bohrís postulate the most probable
value for the ground state ve is 2187691 m/sec.
Now according to Coulomb F,
the force exerted on the electron by the proton, is generally F=q2/4peo(re2). And according to Newton 2 F=(me)(ve2)/(re).
This yields ve=q2/(4peo(re)(me))1/2=2187691
Let us now consider
Bohrís 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 atomís momentum would simply
equal the electronís momentum. But the electronís momentum varies in
direction from moment to moment. Yet in Bohrís model (and presumably by
observation) the atomís total momentum is zero
any given moment.
The only way this condition can be satisfied is if the proton is not actually at rest. Assuming the atomís center
is at rest at the origin, the proton must rotate CCW around the atomís center
quite like the electron does. And
the electron and proton must be diametrically opposed at every instant. (Of
course the proton orbital radius would be less than that of the electron, owing
to the protonís mass being greater than the electronís.) Together the two
should orbit around the origin at a common angular frequency, w.
But our critique of
Bohrís 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 ďradiation reactionĒ force opposite at
any moment to the direction of v.
The formula for this force is FRadReact = q2w3(re)/(6peoc3). If this
force is not counteracted then the charge should radiate and spiral into an
increasingly smaller-radius quasi-circle. However, if it is counteracted by a
component of the electric force from another charge, then the charge should not
radiate. It should continue to travel at a constant speed in a constant-radius
What might be the
explanation for the needed counteraction to FRadReact acting on the
orbiting electron? It cannot be a magnetic force, since such a force would act
perpendicular to v. It
must be a component of an acting electric force engendered by the rotating
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
chargeís 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 tretarded = tpresentĖL/c,
where L is the distance between the object charge and the source chargeís
position at the time tretarded. (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 FRadReact=q2w3(re)/6peoc3. And the
formula for the proton-engendered electric force acting on the electron is the
negative of FRadReact. This force provides the needed counterforce to
FRadReact. 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 ďinching upĒ to
values for L and q using a computer (See Appendix A for a schematic, and Appendix B for a
typical ďinching upĒ program written in Python). Note that the radius of the
protonís orbit has been greatly exaggerated in the schematic for clarity.
It is worth mentioning that
Bohrís 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).
Bohr Atom at a Typical Delayed Time
#Electron rest mass
#Proton rest mass
#Any of the lowest 10 energies can be computed.
to exit) '))
print('Computing level ',N)
#N'th level electron orbit radius
#Electron kinetic energy
vp1=me*ve1/mp #total momentum=0
#Total energy of atom
number of iterations in orbital computation
#Proton travels at constant speed
trial force of proton on electron
#Find parameters when forces on electron sum to zero
#If infinite loop, Print params and exit by clicking X in display
print('Possible infiite loop.')
print('loopN, RadReactF, Fy',loopN,'; ',RadReacF,', ',Fy)
print('Abort by clicking
X in your display.')
#If necessary, keep 'inching upí to loop termination condition.
#Compute electric field at electron (see Griffith's text)
force on electron
#When Fy+RadiationReactionForce=0, display results.
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.