De Broglie's Mistake
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 16 Aug 2005 08:09:31 -0700
>>From Osher Doctorow mdoctorow@xxxxxxxxxxx
COPYRIGHT NOTICE
De Broglie's Mistake
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
Prince Louis de Broglie is considered to have given the major impetus
to quantum theory beyond Planck and Einstein with his equation:
1) lambda = h/p
where lambda is the wavelength and p the momentum of a particle.
Wavelength is also known as period (T), or time for one complete cycle
or vibration of wave motion. However, from the circular representation
of harmonic motion, the period T is an angle theta made by a moving
point moving with constant angular velocity w along a circle in the x-y
plane, the angle being that between the radius vector and the positive
unit x-axis vector, and the frequency f or number of complete
vibrations per unit time is:
2) f = 1/T
or in the quantum theory (e.g., for light), writing f = nu (which I'll
abbreviate to v):
3) v = c/T
The circular representation shows that the angle T or theta is related
to a displacement and to a position x on the x axis with the circle
centered at the origin, namely position x is the projection of the
point on the circle onto the x axis immediately below it, and x moves
to left or right as the point on the circle moves counterclockwise or
clockwise respectively for example along the circle. Calling the
moving point on the cicle Q and the projection at x P, P makes one
complete "vibration" (back and forth motion) for each revolution of Q,
and so v or f is also the number of such vibrations as horizontal (x)
displacements represented by position x, as well as the number of
complete revolutions T per unit time.
The above construction is well known, but it is usually not realized
that x as position is as accurate a description as x as displacement
with the circle centered at the origin. If w is the angular velocity,
we have:
4) T = theta = wt
5) x = R cos(wt), R radius of circle
If we cumulate or add /x/ as the total distance travelled by P, this
total distance increases linearly proportional to time t, and hence it
is "arbitrary" whether we consider that time or distance is involved,
and if the distances are cumulatively strung along the x axis
successively, it is also arbitrary whether time or position is involved
in regard to the latter scenario. This is despite the fact that:
6) w = v = 2pi f
7) x = R cos(2pi f)
since there are 2pi radians in a complete revolution.
Let's write x for position rather than using some symbol for cumulative
/x/ as above, in which case De Broglie's equation (1) becomes:
8) x = k/p
or:
9) xp = k
where k is a positive constant which is very small. This is the Weak
Heisenberg Uncertainty Principle that I defined in my last few threads.
Why didn't anybody notice this? The objects or "wave-particles"
involved are not visual for us as macroscopic observers! Microscopic
or quantum physics is one step removed from direct observation!
Moreover, the "convention" that there is no real trajectory even though
harmonic and trigonometric machinery is used almost impels one to
ignore displacement and distance in the above scenario! Finally, it
could be argued that there is something less "real" about x or /x/
cumulated than about t, but this is silly since time is even less
directly "observed" than space at the quantum level if not equally!
Isn't projection one step removed from circular motion, and isn't
circular motion one step removed from simple harmonic or general wave
motion? This is irrelevant since the question is whether spatial (x
axis) or time scales should be used. The point is that a spatial scale
for x is as "valid" as using time t in de Broglie's equation.
Osher Doctorow
.
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