*Continued from Part One:*

*3. Bohm’s Version of Quantum Mechanics*
Regarding
the various violations of the Bell Inequality, David Bohm with colleague Brian Hiley developed an
alternative form of quantum mechanics to integrate it within existing
observations. This began by treating the de Broglie wave as a physically real
entity not merely a statistical one.
Also, Bohm and Hiley refined the concept of the pilot wave, originally
proposed by Louis de Broglie

If matter waves, or de Broglie waves, can
be “piloted” then clearly they will have far more theoretical impact than if
mere products of random encounters or observations. Hence, Bohm and Hiley ‘s
development of a “pilot wave” theory to accompany the acceptance of B-waves
physical reality.

To
understand the “clocking” guidance system for these waves, we begin with the basic
energy definition for the quantum given some rest frequency, f

_{o. }
Then the energy quantum associated with this frequency is, by Planck’s
equation:

E

_{o}= h f_{o}
Where h is the Planck
constant. Then we can also write:

f

_{o}= E_{o}/ h
In the relativistic limit, for photons: f

Now, change to angular frequency w

_{o }= m_{o}c^{2 }/ hNow, change to angular frequency w

_{o}to make the synchronous mechanism consistent with that proposed by de Broglie and Bohm, Hiley[1]. Then:
2p f

_{o }= m_{o}c^{2 }/ h
Replacing the Planck constant by

**ħ**= h/ 2p , the Planck constant of action:
2p f

_{o }= m_{o}c^{2 }/**ħ**
Then:

w

_{o}= m_{o}c^{2 }/ ħ
Which is the “

**clock frequency**” in the photon rest frame.
There is also an additional condition, known as the

**, for the clock to remain in phase with the pilot wave:***Bohr-Sommerfield condition*
∮ p dx = n ħ

Now, the momentum p = m

_{o}c , so that the integral becomes:
2p x (m

_{o}c) = n ħ
And
the de Broglie wavelength emergence (l

_{D}= h/ p) is evident in the equation. In this sense, we have:
2p x = n
(h/ m

_{o}c) = n l_{D}
Or
the same expression (2p
r = n l

_{D}) for the standing waves in an atom. In Bohm’s own development, the procession of B-waves is actually enfolded within a “packet” of P-waves[2]. A basic diagram of the arrangement is shown below.The axis labeled E is actually the real part of the electric field component, E

_{z}. The width of the p-wave packet is denoted by the spread:

Dk = p / (x – x

_{o})
Where
x

_{o}denotes the center point of the wave packet. In other words, if the center point is at x_{o}= 0, the packet width is just Dk = p / x.
The
wavelength, l = 2p / Dk then is

**much less**than the width of the packet. E.g. if Dk = p / x then:
l = 2p / (p / x) = 2x. so if x = 1
nm, then l = 2 nm and:

Dk = p / x = p / (1 nm) = p nm, but p (nm) > 2 nm.

The maximum of the wave packet is
approximated closely by the square of the amplitude:

[
E

_{z}]^{2}= 4 sin^{2}Dk (x – x_{o}) / (x – x_{o})^{2}
We can check the limits of the preceding.
Let x

_{o}= 0 then:
[
E

_{z}]^{2}= 4 sin^{2}Dk x / x^{2}
And:

[
E

_{z}]^{2}= 4 sin^{2}Dk / x
Conversely,
let x

_{o}= x, then:
[ E

_{z}]^{2}= 0 = 4 sin^{2}Dk (x – x) / (x – x)^{2}
Thus,
the p-wave packet ceases to exist as a discrete or localized entity and thereby

**loses its particle properties**. But what about the photon mass?
Mass
can be derived as a basis for the wave packet spread Dx = (x – x

_{o}). Thus, a “particle” is represented as a finite wave packet with wavelength-based spread Dl, such that, according to the Heisenberg Uncertainty Principle:
Dx = D k =
D (1/~~l~~) ~ 1

Now: ~~l~~ = h/ m vg

Where:
g = (1
– v

^{2}/c^{2})^{-½}
Recall
we saw the basic clock frequency (in the zero reference frame):

w

_{o}= m_{o}c^{2 }/ ħ
Which
may now be generalized for relativistic speeds:

w = m g c

^{2 }/ ħ
In
terms of the Compton
wavelength:

=
ħ/ mc

w = g (c/ x )

Given
a spread in the velocity D
v = D k =
D (vg) then the Heisenberg
Uncertainty Principle states:

D x D p » ħ
where D
p =
D (m vg)

Therefore:

*D**x**D**(m v**g**)**»**ħ*
Or,
since the “ultimate” lower limit on D x is of the order of the Compton wavelength,
i.e. D x
» x or ħ/
m c, we have:

m
» ħ/

__D____x__c
Where

__D____x__denotes a lower limit to D x. And it can be shown that the quantity 1/ x possesses the additive and inertial properties of mass.**4**.

**Bohm’s Version of the Uncertainty Principle:**

In
Bohm’s case, the Heisenberg relations are embodied in his theory as a limiting
case

*over a certain level of intervals of space and time*. However, the potential exists for the fields to be averaged over smaller intervals and hence,*subject to a greater degree of self-determination than is consistent with the Heisenberg principle*. As Bohm concludes[3]:*From this, it follows that our new theory is able to reproduce, in essence at least, one of the essential features of the quantum theory, i.e. Heisenberg’s principle, and yet have a different content in new levels*

Bohm
is primarily concerned with the canonically conjugate field momentum, for which
the associated coordinates, i.e. Dt,
Df

_{k}fluctuate at random. Thus, we have, according to Bohm:__p__

_{k}= a (Df

_{k}/Dt)

Where a is a constant of proportionality, and Df

_{k}is the fluctuation of the field coordinate. If then the field fluctuates in a random way the region over which it fluctuates is;__(__

__d__

__D__

__f__

_{k}__)__

^{2}= b (Dt)

Taking
the square root of both sides yields:

__(__

__d__

__D__

__f__

_{k}__)__= b

^{1/2}(Dt)

^{1/2}

Bohm
notes that

__p___{k}also fluctuates at random over the given range so:
d p

_{k}= a b^{1/2}/ (Dt)^{1/2}
Combining
all the preceding results one finally gets a relation reflective of the
Heisenberg principle, but time independent:

d p

_{k}(d Df_{k }) = ab
This
is analogous to Heisenberg’s principle, cf.

dp dq

__<__ħ
Where
the product

*ab*plays the same role as ħ**Problems:**

1) Consider the details of the Einstein ‘box’
thought experiment in which he attempted to out wit Bohr. Based on the
hypothetic measured quantities given, how much uncertainty would be expected in
the mass? In the weight? (Find the mass uncertainty in kg, and the weight in
N.)

2)An
Aspect-style experiment is carried out and th e
result is found to be:

S
= (A1,A2)I + (A1,A2)II + (A1,A2,)III + (A1,A2)IV =

2.65

__+__0.10
Explain
whether Bell ’s
inequality was confirmed or not. If not, explain why not.

3)
Consider a theoretical wavelength, l = 2p / Dk, where Dk is the expected width
of the wave packet. If Dk = 0.5 nm, and x = 1 nm
with x

_{o}= 0.5 nm, then compute the E-field amplitude: E_{z }
4) In a particular experiment to test Bohmian
quantum mechanics on a computer, the uncertainty in one input turns out to be: Dt = 10

^{-39}s and in the other, Df_{k}= 10^{-51}m. From this data, find the quantity b. Then compose a form of the Uncertainty principle and obtain the product ab. (Where ab plays the same role as ħ in the conventional form of the Uncertainty principle.)
Comment on how your product ab compares to ħ.

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