One Dimension Gravity Well‹Another Flawed Interpretation
according to Nesvizhevsky et al., a step in the transmission
of falling neutrons through a variable-height channel comprising a
mirror on the bottom and an absorber at the top occurs at a height of
13 um because neutrons fall in quantized jumps
In reality,
for widths greater than 13 um which is equal to 1/2(h/m sub
n)^(-2/3)(g)^(-1/3), the height is greater that the de Broglie
wavelength corresponding to the scattering of the falling neutron
from the mirror to the absorber; thus, a step in the transmission of
failing neutrons occurs at 13 um (See below.)
PZ is wrong again. But, this time it is particularly
embarrassing given all the "huffing and puffing" that this experiment
proves that the one dimensional well of QM is correct over CQM
wherein such abstractions are merely that‹nonexistent mathematical
abstractions. How could anyone believe that something falls in jumps
anyway? Physical principles are to be followed, not pure mathematics.
Given that a medical doctor could immediately see the obvious
error in the Nesvizhevsky experiment, perhaps PZ should ask one of
his students to look over his comments before he posts.
Regarding the H-(1/2) Hyperfine Lines paper and related
papers, it is also advised that when commenting on this experimental
data, PZ should actually read the papers and research the field
before he denigrates scientists, referees, and journals when they are
in fact correct and he is in the wrong. It may be understandable
since he has been "out to pasture" for a while and not actively
involved in research. However, his misses are time consuming, and it
is advised that he seek outside advise.
He also should not misrepresent the abilities of QM. For
example, the "stunning success" of QM theory regarding high energy
scattering is merely an empirical curve fit of the data to fudge
factor corrections due the influences of hypothetical (never
observed) virtual particles.
Unprofessional behavior without considering or regard to
evidence and facts and misrepresentations of evidence and facts to
advance vested interests is also potentially very destructive. It is
particularly true when a negative campaign comes from someone that
the public has vested with trust and authority and recognizes as an
expert. PZ made his intentions clear in his farewell diatribe last
August 14th. As part of a regurgitation of the by then stale,
previously-dealt-with issues, he stated he would drive a stake in the
heart of CQM. PZ should be reminded that the public actually expects
publicly paid scientists and officials to engage in professional
scientific discourse in the interest of advancing science and
technology, not kill off a competing theory or technology for their
own benefit.
We still do not know the repercussions of the overt actions
of PZ. Dr. Zimmerman represented in an abstract on the APS (American
Physical Society) website that he was speaking on behalf of the U.S.
State Department, and that the U.S. State Department and the U.S.
Patent Office had fought back with success against BlackLight Power.
As is public record, the same USPTO withdrew BlackLight's
Chemical Patents without regard to the evidence or facts of the case
since it was missing from the Patent Office at that time.
PZ's abstract is of public record in documents that
BlackLight filed in its law suit against the Patent Office to right
the Patent Office's withdrawing BlackLight's chemical patents from
issuance.
Nesvizhevsky et al. [1] claim that they created a potential
well for falling neutrons formed by the Earth's gravitational field
and a horizontal mirror. According to Nesvizhevsky et al., "we now
consider how to demonstrate that bound states exist for neutrons
trapped in the Earth's gravitational field. The gravitational field
alone does not create a potential well, it can only confine particles
by forcing them to fall along field lines. We need a second 'wall'
to create the well." Supposedly, a neutron falling in the Earth's
gravitational field hits the bottom mirror, is reflected, and the
neutron wavefunction interferes with itself. The self-interference
creates a standing wave in the neutron density: the probability of
finding a neutron at a given height exhibits maxima and minima along
the vertical direction which is a function of the quantum number of
the bound states. The quantum mechanical probability wave problem is
solved as a particle on a box or one-dimensional well problem [2].
Nesvizhevsky et al. [1] give the standing waves as asymmetric
sinusoidal waves‹the claimed distortion due to the argument that "the
gravitational field is much softer than a infinite sharp wall; as a
result, the gravitational well extends in the opposite direction to
the gravity with increasing quantum number."[Footnote 1.]
Consequently, the neutron wavefunctions are deformed upwards, and the
energy differences between states become very slightly smaller as the
quantum numbers increase. For example, the energy of the n=1 state
is 1.4 peV, and that of the n=4 state is 4.1 peV, rather than 5.6 peV
for a linear relationship. For comparison, the classical potential
energy V of a neutron lifted a height of z=15 um against the Earth's
gravitational field is given by
V=mgz=(1.67X10^-27 kg)(9.8 m/s^2)(15X10^-6 m)=1.5X10^-12
eV=1.5 peV (1)
where m is the mass of the neutron and g is the acceleration due to gravity.
Nesvizhevsky et al. [1] directed ultracold neutrons with a
horizontal velocity of about 10 m/s through a parallel plate channel
wherein the top plate was a neutron absorber and the bottom plate was
a neutron mirror. The neutrons were selected by a collimator that
projected the neutrons at a slightly upward angle such that they
followed a parabolic trajectory in the Earth's gravitational field.
The neutron vertical velocity at the peak height of the parabola
corresponded to classical result of zero, and increased as the
neutron fell to the bottom mirror. The vertical velocity component
was limited by the variable height of the vertical neutron absorber.
For example, a vertical velocity of 1.7X10-^2 m/s corresponded to a
parabolic height of z=15 um wherein the kinetic energy K given by
K=1/2mv^2=(1.67X10^-27 kg)(1.7X10^-2 m/s)^2=1.5 peV
(2)
was converted to gravitational potential energy given by Eq. (1).
The neutron as well as the proton and electron are
fundamental particles with a de Broglie wavelength. They demonstrate
interference patterns during diffraction as given in the Electron
Scattering by Helium section. The observed far-field position
distribution is a picture of the particle's transverse momentum
distribution after the interaction. The momentum transfer is given
by (hbar)(k) where k is the wavenumber (2Pi/lambda). The relevant
wavelength lambda is the de Broglie wavelength associated with the
momenta of the particles which is transferred through interactions.
An example is the interference pattern for rubidium atoms given in
the Wave-Particle Duality is Not Due to the Uncertainty Principle
section. Also see the Electron in Free Space section.
The de Broglie wavelength lambda is given by
lambda=h/p=h/(mv) (3)
where h is Planck's constant, m is the mass of the neutron, and v is
the neutron velocity in the direction of the wavelength. In the
Nesvizhevsky experiment, a neutron with an initial vertical velocity
of 1.7X10-^2 m/s has zero velocity at the top of the parabolic
trajectory. The corresponding velocity of the falling neutron at the
mirror before reflection is negative 1.7X10-^2 m/s, and after
reflection, it is positive 1.7X10-^2 m/s. The de Broglie wavelength
of the neutron in the vertical direction corresponding to the
momentum acquired by falling from the top of the trajectory and
undergoing momentum reversal at the mirror is given by
lambda=h/deltap=h/(2mv)=6.63X10^-34 Js/(1.67X10^-27
kg)(2)(1.7X10^-2 m/s)=11.7 X10^-5 m=12 um (4)
which is less than z=15 um corresponding to the initial vertical
velocity of about 1.7X10-^2 m/s.
The time scale for the collision of a neutron with the bottom
mirror was much less than the transit time t(t) of the neutron
through the slits which is given by the ratio of the channel length
(0.1 m) and the horizontal speed (10 m/s).
t(t)=0.1 m/10 m/s=0.01 s (5)
The time scale t(d) for the fall of a neutron with a parabolic height
of z=15 um was also much less than the transit time of a neutron
through the slits.
t(d)=SQRT(2z/g)=SQRT((2)(15X10^-6)/9.8 m/s^2)=1.7X10^-3 s (6)
The interaction scale in the vertical direction is the de Broglie
wavelength for the neutron-mirror collision; thus, neutron
transmission through the slits is limited by the height of the
absorber relative to the de Broglie wavelength. The de Broglie
wavelength is inversely proportional to the initial velocity (Eq.
(4)). And, from Eqs. (1) and (2), the parabolic height increases as
v^2. Then, the slit-width for transmission threshold z1 is the de
Broglie wavelength that equals the parabolic height corresponding to
the initial kinetic energy. The de Broglie wavelength is larger than
the slit width for widths less than z1, and the opposite relationship
occurs for slits wider than z1. The velocity given by equating the
initial kinetic energy (Eq. (2)) and the corresponding gravitational
potential energy (Eq. (1)) is
v=SQRT(2gz1) (7)
The corresponding de Broglie wavelength given by Eqs. (4) and (7) is
lambda=z1=1/2(h/m)^2/3(g)^-1/3=12.6 um (8)
Nesvizhevsky et al. [1] flowed neutrons between the mirror
below and the absorber above and recorded the transmission N
(counts/s) as a function of the width delta z if the slit formed by
the mirror and the absorber. Thus, the width delta z acted as a
vertical velocity selector. The expected classical prediction is
that there is some transmission at a slit width greater that of the
neutron cross section for neutrons propagating with no vertical
velocity component. This was in fact observed. For neutrons with a
vertical velocity component, no transmission of neutrons is expected
until the slit width is greater than the vertical de Broglie
wavelength corresponding to momentum reversal at the mirror. This is
due to the interaction of the reflected neutrons with the absorber
with a separation less than this length. From Eq. (8), the slit
height at which neutrons are predicted to be transmitted is about 13
um. This was exactly what was observed. At this point, the
detection rate N should increase as a linear function of the slit
width corrected for any changes in the vertical component of the
neutron velocity due to changes in the acceptance angle for neutrons.
Nesvizhevsky et al. [1] give a correction factor of z^.5 to N due to
the increase in the accepted spread of velocities. Thus, the
classically predicted transmission as a function of slit width delta
z is
N=c(z-z1)^1.5 (9)
where c is a constant dependent on the neutron flux and z1 is the
vertical de Broglie wavelength given by Eq. (8). There was
remarkable agreement between the experimental data of Nesvizhevsky et
al. and the classical quantum mechanical prediction given by Eq. (9).
In contrast, the experimental data did not match critical
predictions of quantum mechanics. According to Nesvizhevsky et al.
[1], "we expect a stepwise dependence of N as a function of delta z.
If delta z is smaller than the spatial width of the lowest quantum
state, then N should be zero. When delta z is equal to the spatial;
width of the lowest quantum state, then N should increase sharply.
Further increase in delta z should not increase N as long as delta z
is smaller than the spatial width of the second quantum state. Then
N should again increase stepwise." In contrast to these predictions,
some transmission was observed at a slit width of an order of
magnitude less than that of the predicted transmission threshold.
Also, no stepwise transmission between quantum states was observed.
Nesvizhevsky et al. [1] erred by not considering the vertical de
Broglie wavelength in the cutoff for transmission.
Moreover, at sufficiently large slit width delta z,
Nesvizhevsky et al. [1] predict that the classical dependence N
proportional to delta z should be approached. Their data shows that
their erred classical prediction actually coincides with the data at
the n=3 state‹a far cry from the point at which the quantum and
classical results are expected to coincide based on the
one-dimensional-well problem of quantum mechanics. (The two are not
to converge until the quantum number n becomes very large and
approaches infinity [4].) Their results further point to the
tendency to misinterpret data in order to support quantum theory when
in fact the data disproves it.
Footnote 1. How, the particle "knows" that the field extends beyond
the reflecting barrier" is not addressed. Nor is the internal
inconsistency that the Standard Model attributes the force of gravity
to exchange of gravitons and not to a classical field. Ironically,
even though gravity is a ubiquitous force, gravitons have never been
observed after 70 years of searching. In addition, quantum
electrodynamics requires that the vacuum is filled with an infinite
number of virtual particles which occupy quantum states. The
consequences such as the prediction of an infinite cosmological
constant and the failure of quantum mechanics to provide a successful
quantum gravitational theory are also not addressed. See Mills
article [3].
1. V. V. Nesvizhevsky, H. G. Borner, A. K. Petukhov, H. Abele, S.
Baebler, F. J. Rueb, T. Stoferele, A. Westphal, A. M. Gagarski, G. A.
Petrov, A. V. Strelkov, "Quantum states of neutron's in the Earth's
gravitational field", Nature, Vol. 415, (2002), pp. 297-299.
2. McQuarrie, D. A., Quantum Chemistry, University Science Books,
Mill Valley, CA, (1983), pp 77-101.
3. R. Mills, "The Nature of Free Electrons in Superfluid Helium--a
Test of Quantum Mechanics and a Basis to Review its Foundations and
Make a Comparison to Classical Theory", Int. J. Hydrogen Energy, Vol.
26, No. 10, (2001), pp. 1059-1096.
4. Beiser, A., Concepts of Modern Physics, Fourth Edition,
McGraw-Hill, New York, (1987),. pp. 147-149.
Randy Mills
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