The EPR Paradox and Quantum Nonlocality (Part 2)

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_o  =   m_o  c² / h

Now, 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  c² / h

Replacing the Planck constant by ħ = h/ 2p , the Planck constant of action:

2p  f_o  =   m_o  c² /  ħ  

Then:

w_o =  m_o  c² /  ħ 

Which is the “clock frequency”  in the photon rest frame.

     There is also an additional condition, known as the Bohr-Sommerfield condition, for the clock to remain in phase with the pilot wave:

∮ 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 sin² 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 sin² Dk x / x ²

And: 

[ E_z ] ² =      4 sin² Dk /  x

Conversely, let x_o = x, then:

 [ E_z ] ² = 0 =  4 sin² 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/c²)  ^-½

Recall we saw the basic clock frequency (in the zero reference frame):

w_o =  m_o  c² /  ħ 

Which may now be generalized for relativistic speeds:

w =  m g  c² /  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 vg)    » 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  Df_k) ²  = b (Dt)

Taking the square root of both sides yields:

(d  Df_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.

---

.[1] Bohm and Hiley:  Foundations of Physics, (12), No. 10, p. 1001.

[2] See, e.g. Bohm, D: Quantum Theory, Dover, p. 169, 1951.

[3] Bohm, op. cit.,. 91.