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Bohr's
First Big Mistake
by Miles
Mathis
First
published October 21, 2005
In this paper
I will reveal the first major mathematical error of Quantum
Mechanics. As promised in other papers, I have gone back to the
very beginning, to rerun all the equations. Before the Heisenberg
Uncertainty Principle, before the problem of superposition,
before the problem of the twoslit experiment and all the other
theoretical problems, there existed the foundational math of
Quantum Mechanics. This was the basis and expression of the
theory, and still is. The math I will analyze here is still
taught to this day as the bedrock of QED. All physics students
learn this math when they are first taught the Bohr Model.
Of course the Bohr Model has long been superceded, but the math
below has never been corrected. It still infects QED at the
foundational level, since this very math is used to obtain all
the existing maths of Quantum Theory. This math underlies and
infects Schrodinger's equations and all subsequent maths and
theories.
Bohr's Math
In
determining the allowed orbits, Bohr first equated the
centripetal electrostatic force on the electron to the
centripetal acceleration, by this equation: F = ma =
mv^{2}/r mv^{2}/r = ke^{2}/r^{2} mv^{2}
= ke^{2}/r
Bohr next uses the following equations
to express the angular momentum of the electron: L = rmv L
= nh/2π Where h is Planck's
constant rmv = nh/2π ke^{2}/r = m[nh/2πmr]^{2} r
= h^{2}n^{2}/4π^{2}mke^{2} r_{1}
= h^{2}/4π^{2}mke^{2}
This is
the Bohr radius, which we are told is in agreement with the
observed size of hydrogen atoms. And we can go back and find the
equation for v: v = [h/2πm][4π^{2}mke^{2}
/h^{2}] = 2πke^{2}/h = 2.18 x 10^{6}
m/s
Next, we find the energy of the orbit: E = K + U =
mv^{2}/2 + U mv^{2} = kZe^{2}/r
(where Z is the atomic number) E = kZe^{2}/2r –
kZe^{2}/r E = kZe^{2} /2r E_{1}
= 13.6 eV
The Problem
The problem
is that for some reason Bohr has used the rotational momentum but
not the rotational kinetic energy. He assumes that the momentum
of the electron will be expressed by an equation that includes
the radius (L = mrv), so that we have an angular momentum. But
when he finds the kinetic energy of the electron, he uses a
straight translational equation—one that does not include the
radius. You will say that it includes the radius after we make
the substitution, but that does not count. The first equation is
a translational equation that contains the variable v, not the
variable ω.
The substitution is made on the variable v, which is not thought
to be an angular velocity in either equation. In the equation mv^{2}
= kZe^{2}/r,
v is supposed to be the instantaneous tangential velocity. In the
equation E = mv^{2}/2
+ U, v is a straightline velocity. We can make the substitution
precisely because they are both assumed to be translational
variables. Current theory
always assures us that rotational motion must be expressed by
rotational variables. In the chapters in physics books on
rotational kinetic energy, we are given this equation for
rotational kinetic energy: K = mv^{2}/2 v
= rω
K = m(rω)^{2}/2
= Iω^{2}/2
Why didn't Bohr use this equation? Because he couldn’t figure
out how to make it yield the right experimental numbers.
You will say, "You can use either equation, since, as you
just showed, they are equivalent: m(rω)^{2}/2
= mv^{2}/2."
I answer, "Are they? Are you telling me that in the equation
K = mv^{2}/2,
v = 2πr/t?" You will no doubt throw up your hands in
frustration, and say, "Yes, just do the substitution, you
fool!" But you wilfully miss my point. I know they are
transferableI just showed thatbut they are not the same. You
can't measure
a straightline velocity with pi and a radius, since you aren't
given pi and a radius. And you can't measure a rotational
velocity with a distance, since you require pi and a radius.
I have said
before that mainstream physics has lost sight of Newton's
original notation somewhere along the way, and we see it again
here. For instance, the Lagrangian can be written
F
= – GMm/r^{2 }+ mv^{2}/2r
Newtons
equation from Proposition LXVI in the Principia is
F(r)
=
(GM_{0}m_{0}/r^{2})
+ m_{0}ω^{2}r
Newton
explicitly wrote the velocity as an angular or orbital velocity,
but somewhere along the way that has been replaced by a
tangential or linear velocity, compromising the math.
A
compensation of errors has saved Bohr in all the equations above.
He has achieved the right experimental numbers only by finessing
the math to fix the conceptual mistakes. The fundamental
conceptual mistake is in assuming that the v variable in v =
2πr/t is an instantaneous tangential velocity. It is not. It is
an orbital velocity. This velocity describes an arc of the
circle; it curves; it cannot be a tangential velocity. The
tangential velocity vector is a straightline vector with its
tail at the tangent. It does not follow the curve of the arc over
any interval, even an infinitesimal interval. Newton never
claimed that the v variable in his equation was the tangential
velocity. In my paper on the equation a =
v^{2}/r,
I show exhaustively that this is so, by quoting Newton directly
from The
Principia,
and by rerunning his versine derivation. Even at the ultimate
interval, or at the limit, the tangential velocity and the
orbital velocity are not equal.
The orbital velocity and the angular velocity are actually the
same thing. The orbital velocity is just the angular velocity at
a given radius. They both
curve. The only difference is that the angular velocity is
measured in radians and the orbital velocity is measured in
meters. That is why the difference between them is just r. The
variable v, as used in most places above, is no more a tangential
velocity than the variable ω
is. I suggest a new letter to denote orbital velocity, and I
suppose for this paper w is good enough. v = x/t ω
= 2π/t w = 2πr/t a = w^{2}/r
L = rmw
The last two equations show how Bohr was saved
from his first mistake. The variable is an orbital velocity in
both places, which saves his substitution. Neither velocity
variable is a translational velocity nor a tangential velocity,
although current textbooks still tell us they both are.
His
energy equations are saved in a different way. You can see that
he is trying to do a straight substitution of w for v, which
cannot work. But he should
be seeking a rotational kinetic energy for his electron, not a
translational kinetic energy. That is to say, he should be
writing his energy equations in terms of w, not v. So the
equations should run like this: mw^{2}
= kZe^{2}/r
E = mw^{2}/2
+ U In which case the
equations work as before.
I know that most physicists
will see my corrections as caviling. They will not care about
these subtleties, since the equations yield the right numbers,
whether they are in the old form or my new form. They will
dismiss my points as semantics or metaphysics, or some other easy
epithet of contemporary science.
But my correction here is strictly mathematical. Part of applied
mathematics is variable assignment. Sloppy variable assignment
must be fudged over later with sloppy equation assignment, so
that you end up with proofs like those above. The misassigned
variables were misassigned in the same way in Bohr’s derivation
of the radius, so that he didn’t have to do anything else to
make the equations work. But with the kinetic energy, he had to
make a further misassignment to make them work out. He had to
misassign an entire equation, giving his electron translational
kinetic energy instead of rotational kinetic energy.
No one seems to have been embarrassed by this since then, but it
is more than a cosmetic failure. As I will show in subsequent
papers, it is just this sort of error that has brought us to the
theoretical impasses we now face. Bohr’s compensations were
minor. In the equations above he needed only very small
mathematical finesses. But very soon these finesses snowballed on
QED. Before long the math was requiring huge finesses like
renormalization. Most contemporary physicists seem not to be
embarrassed by renormalization. But they should be. I will show
that if you are rigorous in your math at all points, you don’t
need any renormalizing. The trick is to keep the equations
"normal" from the start.
In my paper on Newton linked above, I predicted a "kinetic
energy meltdown" due to the form of F = ma = mv^{2}/r.
I said that an improper substitution was being begged by not
properly differentiating between the orbital velocity and the
tangential velocity. The term mv^{2}/r
is a strictly rotational term, but there is nothing but subtle
conceptual theory to keep someone from applying it to a
translational kinetic energy situation. Bohr has only just
avoided this catastrophe, since his electron really is in
circular motion. But what if an electron in orbit ejected a
photon, and that photon was assumed to be ejected from the
tangent? The photon would now be in linear motion, not circular
motion. What is its kinetic energy? You can see the
problem. Furthermore, there is
actually not even subtle conceptual theory to keep anyone from
making this improper substitution, since current theory does not
theoretically disallow it. Only my
theory disallows it. The subtle conceptual theory is so far mine
alone. Current theory believes
that the velocity variable in a = v^{2}/r
is an instantaneous tangential velocity. That would make it the
same as a translational velocity. That is why current theory uses
the same variables for both. And that is why the proofs of Bohr
followed the form they did and why they have never been
corrected. That is why all modern physics textbooks conflate
tangential velocity and orbital velocity.
And that is why I repeat my prediction: this improper
substitution will be found to be at the heart of some
mathematical impasse in contemporary physics. If I do not root it
out of QED very soon, I will be quite surprised.
Update:
I have now reworked Bohr's equation completely, finding a new
value for the Bohr radius. Coulomb's constant is shown to be an
expression of the Bohr radius. See my paper The
Bohr Magneton.
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