One Dimension Gravity WellÂ‹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 thatÂ‹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 wavesÂ‹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 stateÂ‹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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