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Abstract

The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls for a re-examination of the meaning of “quantum trajectories”. We will review the arguments that have been assumed to have established that a trajectory has no meaning in the context of quantum mechanics. We show that the conclusion that the Bohm trajectories should be called “surreal” because they are at “variance with the actual observed track” of a particle is wrong as it is based on a false argument. We also present the results of a numerical investigation of a double Stern-Gerlach experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations where the appearance of the quantum potential is open to direct experimental exploration.
entropy
Article
Quantum Trajectories: Real or Surreal?
Basil J. Hiley * and Peter Van Reeth *
Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, UK
*Correspondence: ubap727@mail.bbk.ac.uk (B.J.H.); p.reeth@ucl.ac.uk (P.V.R.)
Received: 8 April 2018; Accepted: 2 May 2018; Published: 8 May 2018


Abstract:
The claim of Kocsis et al. to have experimentally determined “photon trajectories” calls
for a re-examination of the meaning of “quantum trajectories”. We will review the arguments that
have been assumed to have established that a trajectory has no meaning in the context of quantum
mechanics. We show that the conclusion that the Bohm trajectories should be called “surreal” because
they are at “variance with the actual observed track” of a particle is wrong as it is based on a
false argument. We also present the results of a numerical investigation of a double Stern-Gerlach
experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations
where the appearance of the quantum potential is open to direct experimental exploration.
Keywords: Stern-Gerlach; trajectories; spin
1. Introduction
The recent claims to have observed “photon trajectories” [
1
3
] calls for a re-examination of what
we precisely mean by a “particle trajectory” in the quantum domain. Mahler et al. [
2
] applied the Bohm
approach [
4
] based on the non-relativistic Schrödinger equation to interpret their results, claiming
their empirical evidence supported this approach producing “trajectories” remarkably similar to those
presented in Philippidis, Dewdney and Hiley [
5
]. However, the Schrödinger equation cannot be
applied to photons because photons have zero rest mass and are relativistic “particles” which must be
treated differently. In fact details of how to treat photons and the electromagnetic field in the same spirit
as the non-relativistic theory have already been given in Bohm [
6
], Bohm, Hiley and Kaloyerou [
7
],
Holland [
8
] and Kaloyrou [
9
], but this work seems to have been ignored.
Flack and Hiley [10]
have
re-examined the results of the experiment of Kocsis et al. [
1
] in the light of this electromagnetic
field approach and have reached the conclusion that these experimentally constructed flow lines
can be explained in terms of the momentum components of the energy-momentum tensor of the
electromagnetic field. What is being measured is the weak value of the Poynting vector and not the
classical Poynting vector suggested in Bliokh et al. [11].
This leaves open the question of the status of the Bohm trajectories calculated from the
non-relativistic Schrödinger equation [
4
,
5
] for particles with finite rest mass. The validity of the
notion of a quantum particle trajectory is certainly controversial. The established view has been
unambiguously defined by Landau and Lifshitz [
12
]:—“In quantum mechanics there is no such
concept as the path of a particle”. This position was not arrived at without an extensive discussion
going back to the early debates of Bohr and Einstein [
13
], the pioneering work of Heisenberg [
14
] and
many others [15]. We will not repeat these arguments here.
In contrast to the accepted position, Bohm showed how it was possible to define mathematically the
notion of a local momentum,
p(r
,
t) = S(r
,
t)
, where
S(r
,
t)
is the phase of the wavefunction.
From this definition it is possible to calculate flow-lines which have been interpreted as ‘particle
trajectories’ [
5
]. To support this theory, Bohm [
4
] showed that under polar decomposition of the wave
Entropy 2018,20, 353; doi:10.3390/e20050353 www.mdpi.com/journal/entropy
Entropy 2018,20, 353 2 of 18
function, the real part of the Schrödinger equation appears as a deformed Hamilton-Jacobi equation,
an equation that had originally been exploited by Madelung [16] and by de Broglie [17].
Initially this simplistic approach was strongly rejected as it seemed in direct contradiction to the
arguments that had established the standard interpretation, even though the approach was based
on the Schrödinger equation itself with no added new mathematical structures. However, recently
this approach has received considerable mathematical support from the extensive work that has been
ongoing in the literature exploring the deep relation between classical mechanics and the quantum
formalism which has evolved from a field called “pseudo-differential calculus”. Specific relevance of
this work to physics can be found in de Gosson [18] and the references found therein.
In this paper we want to examine one specific criticism that has been made against the notion
of a “quantum trajectory”, namely the one emanating from the work of Englert et al. [
19
] (ESSW).
They conclude, “the Bohm trajectory is here macroscopically at variance with the actual, that is:
observed track. Tersely: Bohm trajectories are not realistic, they are surreal”. A similar strong criticism
was voiced in Scully [
20
] who added that these trajectories were “at variance with common sense”.
However the claim of an “observed track" in the above quotation should arouse suspicion coming
from authors who claim to defend the standard interpretation as outlined in Landau and Lifshitz [
12
] .
The first part of the ESSW argument involved what they called the ‘standard analysis’ of a
gedanken experiment consisting of several Stern-Gerlach magnets, an experiment that is discussed in
Feynman [
21
]. It is this part of the argument that we examine in this paper. We show that they arrive at
the wrong conclusion because they have not carried through the analysis correctly. Although Hiley [
22
]
and Hiley and Callaghan [
23
] have presented a detailed criticism of this topic before in a different
context, the point that we make in this paper is new. The standard use of quantum mechanics itself
shows that what ESSW call the “macroscopically observed track” is identical to what has been called
the “Bohm trajectory”. We support our arguments with detailed simulations of potential experiments
that are being planned at present with our group at UCL.
2. Re-Examination of the Analysis of ESSW
2.1. General Results Using Wave Packets
The ESSW paper [
19
] contains an error in their analysis of the Stern-Gerlach experiment as shown
in Figure 1which is similar to the set-up shown in Figure 4 appearing in ESSW [
19
]. It depicts the
tracks of spin one-half particles entering two Stern-Gerlach (SG) magnets. The particles enter along
the
y
-axis with their spins initially pointing along this axis. The orientation of the magnetic field in
each SG magnet is as shown in the figure, the second SG magnet being twice the length of the first.
Figure 1. Sketch of Particle Tracks Presented in ESSW.
Entropy 2018,20, 353 3 of 18
On entering the first magnet, the wave packet begins to split into two wave packets which move
apart in the magnetic field. The packet,
ψ+
, moves in the
+z
direction while the other,
ψ
, moves in
the
z
direction. Thus the
ψ+
packet follows the upper track, while the
ψ
packet follows the lower
track. Note here it is the wave packet we are discussing, not the particle.
To account for the
z
-motion of the packets, we use standard quantum mechanics as in ESSW [
19
],
where the spin-dependent Hamiltonian is
H=1
2mP2+E(t)σzF(t)zσz,
where
E(t)σz
is the magnetic energy at
z=
0 and
F(t)zσz
is the energy due to the inhomogeneous field.
The two components of the wave function are initially chosen to be
ψ+(z, 0) = ψ(z, 0) = (2π)1/4 (2δz0)1/2 exp "z
2δz02#,
where
δz0
is the initial spread in
z
which is assumed small compared with the eventual maximum
separation of the two beams.
At a later time, the equations of motion of the two wave packets are
ψ±(z,t) = A(t)exp B(t)[zz]2±i
¯h[zp+¯h
2Φ(t)],
where
A(t)=(
2
π)1/4 h2δz0+i¯ht
2mδz0i1/2
and
B(t) = 1
4δz0(δz0+i¯ht
2mδz0)
. In arriving at this expression
we have used the impulse approximation as presented in Bohm [
24
]. Here
p(t) = R0tdt0F(t0)
is
the momentum transferred to the “up” wave packet. The actual magnitude is not relevant to our
discussion; the interested reader is referred to the original ESSW paper for these details. The magnitude
of Φ(t) = 2/¯hR0tdt0E(t0)is again not relevant to our argument.
Since no measurement has been made and the two beams are still coherent, the wave function
after it has traversed the magnet is written in the form
|Ψi=|ψ+i| +zi+|ψi| zi. (1)
This gives the final probability density as
ρ(z,t) = |ψ+(z,t)|2+|ψ(z,t)|2,
showing that there is no interference as the wave packets no longer overlap.
The z-component of the current is given by
j(z,t) = ¯h
2im Ψ
zΨΨ
zΨ(2)
=¯h
m(ψ
+ψ++ψ
ψ)C(t)z+(ψ
+ψ+ψ
ψ)C(t)z+p
¯h
where
C(t) = ¯ht/[
2
m((δz0)4+ (¯ht/
2
m)2)]
. Note that the probability density is symmetric about the
z=
0 plane, while
(ψ
+ψ+ψ
ψ)
is anti-symmetric, showing that the probability current is therefore
antisymmetric, therefore,
ρ(z,t) = ρ(z,t)with j(z,t) = j(z,t). (3)
Also, as
(ψ
+ψ+ψ
ψ)=
0 on the
z=
0 plane,
j(z
,
t) =
0 at
z=
0. Until this stage we agree
totally with the calculations of ESSW using standard quantum mechanics based on conventional wave
Entropy 2018,20, 353 4 of 18
packet calculations, but it should be noted that this argument only holds when the incident spin is in
the
y
-direction as in the ESSW thought experiment. Particle trajectories have not been discussed so far.
2.2. What Can Be Said about the Behaviour of Individual Particles?
Now we turn to consider what can be inferred about the behaviour of the individual particles,
if anything. To answer this question let us return to Landau and Lifshitz [
25
] who argue that although
we cannot talk about a precise particle trajectory, we can talk about the probability of finding a particle
in a volume
V
, provided the volume is large enough so that we avoid any problems associated with
the uncertainty principle. Particles will flow into and out of the volume by crossing the boundary of
the small volume. In this process we must ensure that probability is conserved.
To see how this works in detail, let us write the well-known conservation of probability equation
in integral form. Thus
d
dt Z|Ψ|2dV =Z.jdV =IjdΣ(4)
where at the last stage we have used Stokes’ theorem. Here
j
is the probability current density used
to ensure probability conservation. The integral of this current over the surface
Σ
is the probability
that a particle will cross the surface in unit time. By considering a series of connected volumes we
can construct what can be regarded as a “macroscopic particle track”. Mott [
26
] has given a deeper
analysis of this process.
Let us now apply this analysis to the situation shown in Figure 1. Construct a surface
Σ
comprising
the
z=
0 plane and a surface enclosing the upper half of the figure so as to include the upper parts of
the magnet. Since the current density is zero everywhere on the
z=
0 plane, no particles can cross
this plane. Thus the particles that arrive in the upper-half of the experimental setup must remain
in the upper-half and can never cross the
z=
0 plane as long as the wave packets remain coherent.
This clearly shows that the continuation of the trajectories sketched in Figure 4 of the ESSW paper
(as in Figure 1here) is not correct.
In Figure 5 of their paper, ESSW show more explicitly the spin directions together with a sketch of
two Bohm trajectories. This shows that their spin wave packets cross the
z=
0 axis whereas the Bohm
trajectories do not. ESSW take this to mean that at first, part of the Bohm trajectories follow one of the
wave packets and then, after their spin wave packets cross this axis, the trajectories follow the other
wave packet. We will show in Section 4.3 the behaviour of their wave packets is not correct because
they have not included spin correctly into the Bohm model.
3. The Bohm Approach When Spin Is Included
To give an account of the behaviour of a particle with spin in the non-relativistic limit, we must
widen the scope of the Bohm approach. An extended model for a spin-half particle based on the
Pauli equation has already been presented in Bohm, Schiller and Tiomno (BST) [
27
]. Full details
of this model have also been discussed in a series of papers by Dewdney et al. [
28
31
] and by
Holland [
32
]. This simple model has been applied to neutron diffraction and a single Stern-Gerlach
magnet, the results being reported in [
29
,
30
]. It should be noted that none of this work is referred
to in the ESSW paper and yet this is clearly significant as the Stern-Gerlach magnets operate on the
magnetic moments of the particles.
If they had been aware of this work they would not have made the statement that in the Bohm
theory a particle has a position and nothing else. In the BST extension, not only do we have position,
but also the orientation of the spin vector. Here the Euler angles
(θ
,
φ
,
ψ)
are used to specify the spin
direction. This is essentially the precursor of the flag picture of the spinor presented in Penrose and
Rindler [
33
]. Bell [
34
] has a simpler model which was also based on the three components of the
spin vector. A more general approach using Clifford algebras in which the Pauli spin matrices play a
Entropy 2018,20, 353 5 of 18
fundamental role has been presented in Hiley and Callaghan [
35
]. This approach shows how the BST
model emerges as a particular representation using Euler angles.
3.1. Spin and the Use of the Pauli Equation
We start with the Pauli equation
i¯hξ
t=Hξ, (5)
where ξis the two-component spinor which we write in the form
ξ=Rei(ψ/2) cos(θ/2)ei(φ/2)
isin(θ/2)ei(φ/2)!. (6)
Here (θ,φ,ψ)are the three Euler angles.
The Hamiltonian His then written in the form
H=¯h2
2mie
2mA2
+µσ.B+V, (7)
where µis the magnetic moment of the particle.
The original physical idea here was to assume the particle is a spinning object whose orientation
is specified by the three Euler angles
(θ
,
φ
,
ψ)
. The probability of the particle being at a given position,
(r
,
t)
, is
ρ(r
,
t) = R2(r
,
t) = |ξ(r
,
t)|2
. This means the properties of the Pauli particle are specified
by four real numbers
(ρ
,
θ
,
φ
,
ψ)
given at the point
(r
,
t)
. The time evolution of these parameters is
determined by the Pauli Equation (5) as we will now show.
It is more convenient to rewrite the wave function in the form
ξ(r,t) =
R+eiS+
¯h
ReiS
¯h
,
where
θ=2 tan1R
R+;ψ=S++S
¯hπ/2; φ=S+S
¯h+π/2. (8)
To find the velocity of the particle, let us first write the quantity
ξξ
in terms of the Euler angles,
ξξ=RR+i
2R2ψ+i
2cos θR2φ.
Then following Hiley [36] we can define a complex local velocity
v=vRe +ivIm =i¯h
m
ξξ
ξξ
where the probability density is given by R2=ξξ.
The real part of the local velocity is
vRe =¯h
2mˆ
z(ψ+cos θφ)(9)
Entropy 2018,20, 353 6 of 18
which replaces
v(r
,
t) = S(r
,
t)/m
defined for the spin-less particle. The imaginary part, which was
not discussed by Bohm in his original paper (but see Bohm and Hiley [
37
]) is called the “osmotic
velocity” and has the form
vIm =i¯h
mˆ
zR
R. (10)
We will now use Equations (9) and (10) to simulate the detailed behaviour of the particles and
their spin orientations as they traverse the set-up illustrated in Figure 1.
4. Detailed Calculation of the Trajectories
4.1. One Stern-Gerlach Magnet
We begin by simulating the behaviour of the particles having passed through a single
Stern-Gerlach magnet. For simplicity we use the impulse approximation given in Bohm [
24
] to
analyse the evolution of a wave packet as it leaves the magnet (A full treatment using Feynman
propagators is being prepared by Hiley and Callaghan. This allows us to calculate trajectories inside
the SG. Preliminary results confirm the results presented here.).
In the Hamiltonian given in Equation (7), we replace
B
by the field in the SG magnet, which we
write as Bµ(B0+zB0
0), where B0
0is the field gradient inside the magnet and set Aand Vto zero.
Following Dewdney et al. [30] and Holland [32], we choose the initial wave function to be
ξ0=ξ++ξ=f(z)(c+u++cu) = (2π)1/2 Zg(k)(c+u++cu)eikz dk,
where
g(k) = (
2
σ2/π)1/4ek2σ2
is a normalised Gaussian packet centred at
k=
0 in momentum space.
Here
u+
and
u
are the eigenstates of the spin operator
σz
. The solution of the Pauli equation at time
t
after the particle has left the SG magnetic field is
ξ= (2π)1/2 Zdkg(k)c+u+exp i+ (k0)z¯ht
2m(k0)2
+cuexp i+ (k+0)z¯ht
2m(k+0)2
where
=µB0th
,
0=µ0B0
0th
and
t
is the time spent in the field. Carrying out the integral
we find
ξ(z,t) = (2πs2
t)1/4c+u+exp[(z+ut)2/4σst]exp hi(+ (z+1
2ut)0)i
+cuexp[(zut)2/4σst]exp hi(+ (z1
2ut)0)i. (11)
Here st=σ(1+i¯ht/2mσ2), and u=¯h0/m. We now write ξ(t)in the form
ξ(z,t) = c+R+eiS+/¯hu++cReiShu(12)
where
R±=2πσ21/4 (1+¯h2t2/4m2σ4)1/4 exp (z±ut)2
4σ2(1+¯h2t2/4m2σ4)!(13)
Entropy 2018,20, 353 7 of 18
and
S±h=(z±1
2ut)01
2tan1(¯ht/2mσ2) + ¯ht(z±ut)2
8mσ4(1+¯h2t2/4m2σ4). (14)
We are now in a position to calculate the local velocities from the specific solution given
by Equation (12). Since the real part of the local velocity is given by Equation (9), namely,
¯h(ψ+cos θφ)/2m
, we need only evaluate
∂ψ/z
and
∂φ/z
since we are only considering the
motion along the
z
-direction. In order to find these derivatives, and those required for the osmotic
velocity and the quantum potential, we express the parameters
(ρ
,
θ
,
φ
,
ψ)
in terms of
(R+
,
R
,
S+
,
S)
using Equations (8), (13) and (14), and obtain,
∂ψ
z=htz
8mσ4(¯h2t2/4m2σ4+1),
∂φ
z=20+hut2
8mσ4(¯h2t2/4m2σ4+1),
∂θ
z=sin θut
σ2(¯h2t2/4m2σ4+1),
and
1
R
R
z=z+ut cos θ
2σ2(¯h2t2/4m2σ4+1).
The Bohm velocity given by Equation (9) then becomes
vRe =¯hˆ
z
2m 20cos θ+¯ht[z+ut cos θ]
2mσ4(¯h2t2/4m2σ4+1)!. (15)
Note here that the second term in the above expression corresponds to the spreading of the wave
packet and contributes little to the overall behaviour. The main effect of the field comes from the
first term
0cos θ
, which reveals clearly how the velocities and therefore the trajectories are strongly
affected by the behaviour of the spin vector. This term depends implicitly on
(z
,
t
,
u)
and is responsible
for the splitting of the beam.
The imaginary part or osmotic local velocity given in Equation (10), namely,
vIm =i¯h[R/R]/m
,
now becomes
vIm =2¯h
m
ˆ
z[z+ut cos θ]
σ2(¯h2t2/4m2σ4+1). (16)
Note there is no explicit dependence on the magnetic field gradient but there is an implicit
dependence through uand cos θ.
These results enable us to calculate specific trajectories and spin vectors for various particle initial
positions and for various values of
(c+
,
c)
should that become necessary. The choice of the latter
determine the initial value of the spin vector direction
θ
which, in our case was chosen to be along the
y
-direction, hence
(c+
,
c) =
1
/2
. The results shown in figures below are calculated for parameters
listed in Table 1.
Entropy 2018,20, 353 8 of 18
Table 1. Parameters used in the numerical investigation.
Atom Ag
Mass 1.8 ×1025 Kg
Width of magnets 4 and 8 ×104m
Length of magnets 1 and 2 ×102m
Velocity of atoms vy=y/t=500 m/s
Time within magnets t=2 and 4 ×105s
Magnetic field strength at centre B0=5 Tesla
Magnetic field gradient B0
0=1000 Tesla/m
Wave packet width σ=1×104m
Wave packet speed u=µBB0
0t/m=1 m/s
0=µBB0
0t¯h=muh0=1.714 ×109m1
4.2. Numerical Values for Single Stern-Gerlach Magnet
Integrating Equation (9) will give us the Bohm trajectories. In Figure 2we show the ensemble of
Bohm trajectories and the spin orientations as they leave the Stern-Gerlach magnet, shown in brown at
the LHS of the figure. The background colours show the probability density, black being the greatest,
while blue is zero.
Figure 2. Trajectories with spin vectors immediately on exiting the Stern-Gerlach (SG) magnet.
The dark background shows how the wave packets diverge along straight lines, as do the
trajectories. Superimposed on the trajectories are the spin orientations.
Notice that, contrary to the conventional view, the atoms do not immediately “jump” into one or
other
z
-spin eigenstates, rather the spin vectors undergo continuous evolution until they reach their
final
z
-spin eigenstate. This occurs once the two wave packets
ψ+(z
,
t)
and
ψ(z
,
t)
have separated and
have no significant overlap. The upper beam will contain only atoms with spin “up" in the
z
-direction
while those in the lower beam will all be “down” in the
z
-direction. Notice also that the rotational
changes occur in a magnetic field-free region . We can also see that the alignment of the spin vector at
Entropy 2018,20, 353 9 of 18
a given
y
value close to the magnet depends on
z
, with the spin associated to trajectories closer to
the
z=
0 axis rotated least. In Section 4.7 we will see that the cause of these behaviours is a torque
produced by the quantum potential. These results for a single magnet confirm what was already found
in Dewdney et al. [2931].
Figure 3shows the effect of the osmotic velocity, which we have represented by arrows. They are
responsible for maintaining the wave packet profile and will be discussed further in Section 4.6.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
y(m)
z(mm)
Figure 3. The osmotic flow vectors immediately on exiting the SG magnet.
4.3. Two Stern-Gerlach Magnets
Having seen how the atoms behave in a single SG magnet, let us now move on to consider two
SG magnets with opposite field directions as shown in Figure 1. Note here the second SG magnet is
double the length of the first.
The method is similar to the case of the single magnet, except now we use, as initial wave packet,
the inverse Fourier transform of the wavefunction at the second magnet at time
t=t1
. We obtain the
real part of the local velocity as
v2Re =¯hˆz
2m
20
220
1
¯h2(t1+t)2
4m2σ4+1
cos θ(17)
+¯h(t1+t)
2mσ4¯h2(t1+t)2
4m2σ4+1[z+u2tcos θ]
(18)
and the osmotic velocity as
v2Im =¯hˆz
m
1
2σ2¯h2(t1+t)2
4m2σ4+1[z+(u1(t1+t) + u2t)cos θ](19)
Entropy 2018,20, 353 10 of 18
where
t=
0 at the exit of the second magnet. In Figure 4we have plotted the trajectories together with
the spin orientations as the atoms pass through two SG magnets. The details of the parameters used in
the calculations are again as listed in Table 1. The position of the second magnet is as indicated by the
brown bar between y=0.1 m and y=0.12 m.
Figure 4. Spins emerging from two Stern-Gerlach magnets.
There are several features of the ensemble of trajectories that are noteworthy. Firstly, at the exit
of the second magnet, the wave-packets are refocused toward the
y
-axis until the inner edge of the
packets reaches the axis at y0.22 m at which point they diverge again.
Secondly, no trajectories are found to cross the
z
= 0 plane. This should, in fact, not be surprising
since
vRe
can also be obtained from
j(z
,
t)/ρ(z
,
t)
. This means that the “Bohm trajectories” are identical
to the probability flow lines and, as we have seen, the probability flow lines do not cross the
z=
0
plane. Thus there is no experimental difference between the Bohm approach and standard quantum
mechanics at this stage. It could be argued that it is quantum mechanics that is “at variance with
common sense”!
Thirdly, notice once again that the spins do not immediately “jump” into the eigenstates as
assumed by the standard theory. Rather they take a small but finite time to reach the final eigenstate as
discussed above in Section 3.1. Furthermore note that when the beams are refocused close to the
z
= 0
plane, at about
y=
0.22, the spin vectors are rotated so that they all become aligned with the
y
axis
before being rotated again until they end up anti-parallel to the direction with which they entered the
second magnet. This rotation is very surprising but is generated by the quantum torque that arises
from the quantum potential as we show in the next section in Equation (22).
Furthermore this is in contradiction with Figure 5 of ESSW where they argue that the Bohm
trajectories are not realistic because in order to get the observed final spin state, their particles must
cross the
z=
0 axis. Therefore the present work shows clearly the importance of coupling the spin and
the centre of mass motion in order to obtain a correct and consistent analysis of the problem.
Figure 5shows the direction of the osmotic velocity in the two SG magnets case. Its behaviour is
again exactly the same as in the one SG magnet case.
Entropy 2018,20, 353 11 of 18
0.1 0.2 0.3 0.4 0.5
y(m)
-0.6
-0.4
-0.2
0.2
0.4
0.6
z(mm)
Figure 5. The osmotic velocity superimposed on the trajectories for two Stern-Gerlach magnets.
To return the packet to its original state with all the spins pointing in the
y
-direction, we have
to add a third magnet as indicated in the original diagram in Feynman et al. [
38
]. Thus the Bohm
approach gives a complete account of the average behaviour of the individual quantum processes.
4.4. The Appearance of the Quantum Torque
Now let us show the source of the quantum torque. We start by examining the real part of the
Pauli Equation (5) under polar decomposition of the wave function, which can be written in the form
1
2¯h∂ψ
t+cos θ∂φ
t+1
2mv2+QP+2µ
¯hσ.s+V=0. (20)
Here once again we see, as in the case of the Schrödinger equation, an extra energy term,
QP
,
the quantum potential energy, appears. In the present case QPtakes the form
QP=(¯h22R)/2mR ¯h2
8m[(θ)2+sin2θ(φ)2]. (21)
The first term will be recognised as the quantum potential found in the Schrödinger equation.
The second term determines the evolution of the spin vector which is given by
s=1
2¯hξσξ=1
2(sin θsin φ, sin θcos φ, cos θ).
The equation of motion for the spin vector s, is then found to be
ds
dt =T2µ
¯h(s×B). (22)
Here Bis an external magnetic field and
T= (mρ)1s×
i
xiρs
xi. (23)
It is the quantum torque,
T
, that acts on the individual atoms, rotating their spin vectors and the
flag plane.
Entropy 2018,20, 353 12 of 18
4.5. Detailed Calculation of the Quantum Potential
To understand better the role played by the quantum potential, let us examine in more detail its
mathematical structure as shown in Equation (21). We restrict our analysis to the case of a single magnet.
As the quantum Hamilton-Jacobi Equation (20) is an equation that conserves energy, the appearance of
Q
implies that some of the kinetic energy of the particle is transferred to the quantum potential energy
Q. As we see from Equation (21), the quantum potential energy has two components
Qtrans =¯h22R
2mR and Qs pin =¯h2
8m[(θ)2+sin2θ(φ)2].
We will examine the two terms independently. First consider
Qtrans
. Since the particle is moving
in one-dimension
2R2R
z2=2
bd 2R(z+ut cos θ)2
bd +R14u2t2
bd sin2θ,
where we have written
b= ¯h2t2
4m2σ4+1!and d=4σ2.
Then
Qtrans =¯h22R
2mR =¯h2
bdm 2
bd [(z+ut cos θ)2+2u2t2sin2θ] + 1.
Now we turn to evaluate the spin part of the quantum potential,
Qspi n
, where we need to evaluate
φ=20+hut2
8mbσ4and θ=sin θut
bσ2.
This gives
Qspi n =¯h2sin2θ
8bm u2t2
σ420¯hut2
2mσ+402 ¯h2t2
4m2σ4+1!!.
The expression for the total quantum potential,
Q=Qtrans +Qspin
is rather complex so it will
be helpful if we can make an approximation without significantly altering the final result. This can
be done by noticing the magnitude of
b=¯h2t2
4m2σ4+1
1. This means that we are assuming the
wave packet does not spread significantly during the flight times considered. We arrive at the final
expression for the total quantum potential:
Q¯h2
mσ2
mσ4[(z+ut cos θ)2+2u2t2sin2θ] + 1
+¯h2sin2θ
8mu2t2
σ420¯hut2
mσ4+402.
4.6. Numerical Details: Quantum Potential Single Stern-Gerlach Magnet
In Figure 6below we plot the transverse quantum potential
Qtrans
and the spin quantum potential,
Qspi n
for the single SG magnet. The end of the SG magnet is again along the
z
-axis at
y
= 0, with the
atoms flowing along the y-axis out of the page.
Entropy 2018,20, 353 13 of 18
Figure 6. Transverse (left) and spin (right) quantum potential at exit of a single SG magnet.
The atoms initially experience the first part of the quantum potential where the beam begins to
split into two as shown in Figure 2. Both quantum potentials split symmetrically into two parts about
the
y
-axis. The two “domes” of
Qtrans
, shown in the left hand of the figure, cover each beam as they
separate. The width of each dome characterises the spreading wave packet as it evolves in time. Also,
when compared to the osmotic velocities shown in Figure 3, we can see how these velocities are related
to the gradient of
Qtrans
. The trajectories are seen to follow paths of constant gradient and the osmotic
velocities are constant along the trajectories in Figure 3. Furthermore, those trajectories in the wings
of the wave packets experience a more steep gradient and the osmotic velocities are indeed found to
be larger there. At the maximum of the packet, the osmotic velocity is zero. An interpretation of the
Qtrans
would therefore be that it gives rise to a force, which is anti-parallel to the osmotic velocity and
restricts the spreading of the wave packet.
The spin part of the quantum potential
Qspi n
is shown in the right hand of Figure 6. The upward
slope produces the quantum torque that rotates the spin vectors of the atoms as the two beams separate.
This rotation continues until the two packets are completely separate. When this happens all the
spins point “up” in the upper beam, while they all point “down” in the lower beam. At this stage the
Qspi n
0 ensuring the atoms remain in their final spin eigenstates. Figure 7shows the projection of
the
Qspi n
of Figure 6on the trajectories and spin orientation. Note also that the trajectories close to
the
y
-axis do not experience the same steepness of
Qspi n
as do those which are off-axis. This explains
why, as remarked earlier, the spin vectors closer to the
y
-axis take longer to align themselves either up
or down.
0.00 0.05 0.10 0.15 0.20
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Y(m)
Z(mm)
Figure 7.
Trajectories with spin vectors overlaid on the spin quantum potential immediately on exiting
a single SG magnet.
Entropy 2018,20, 353 14 of 18
4.7. Numerical Details: Quantum Potential in Two SG Magnet Case
Now let us consider the case when the two Stern-Gerlach magnets are in place. The positions of
the magnets are shown in brown. Recall here that the inhomogeneities in the magnetic fields oppose
each other.
In Figure 8we show both
Qtrans
and
Qspi n
for the case of two magnets. The gap in each figure
corresponds to the position of the second magnet. The quantum potential after the second magnet is
similar to that of the single SG magnet as shown in Figure 6. These results give a detailed picture of the
expected evolution of a non-relativistic atom with spin one-half as it goes through both SG magnets.
Figure 8. Qtrans (left) and Qs pin (right) quantum potential for a two SG magnets system.
Figure 9shows the projection of the spin quantum potential superimposed on the trajectories.
Notice that the quantum torque is strongest well outside the second SG magnet in the magnetic
field-free region, producing a 180 degree rotation of the spin vector. It is at this point that the wave
packets begin to interfere strongly. In fact the quantum torque continues to act outside the magnet
until the two wave packets
ψ+(z
,
t)
and
ψ(z
,
t)
cease to overlap. Notice once again how the spin
does not immediately ‘jump’ into one of the two spin
z
-eigenstates, but undergoes a well-defined time
evolution. Such a behaviour would have, perhaps, been welcomed by Schrödinger himself [39].
0.0 0.1 0.2 0.3 0.4 0.5
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Y(m)
Z(mm)
Figure 9.
Trajectories with spin vectors overlaid on the spin quantum potential for a two SG
magnet system.
Entropy 2018,20, 353 15 of 18
Once they no longer overlap, each atom remains in one or the other spin eigenstates. Again,
as was the case with the single SG magnet, the spin vectors along the trajectories close to the
y
-axis,
especially at the point where the two beams are refocused, experience less of the gradient of
Qspi n
.
Thus it is clear that the quantum torque arises from the interference region, implying it is an internal
feature of the overall behaviour, suggesting a kind of dramatic re-structuring of the underlying process.
Bohm was intuitively well aware of this possibility and it was one of the reasons why he
abandoned the view that the atom only had a local, “rock-like” property. He preferred to regard
the atom as a quasi-local region of energy undergoing a new type of process that he described in
more general terms as an “unfolding-enfolding” process, comparing it to a gas near its critical point,
the particle itself constantly forming and dissolving, as in critical opalescence [
40
,
41
]. In other words,
the quantum evolution involves an entirely new re-ordering process which should not be regarded as
a particle following a well defined trajectory.
This view of the evolving quantum process becomes even more compelling since Hiley and
Callaghan [
42
] and Takabayasi [
43
] have shown that the local momentum and energy are actually
related to the energy-momentum tensor, Tµν , through the relations
ρpj(r,t) = T0j(r,t)and ρE(r,t) = T00(r,t),
a feature of which Schwinger [
44
] was well aware. The question of which particular trajectory a
specific atom actually takes cannot be answered because the experimenter has no way of choosing or
controlling the initial position of the particle. The final result is also totally independent of the observer.
A detailed discussion of the role of the experimenter in the Bohm approach can be found in Bohm and
Hiley [
45
,
46
]. A more recent paper by Flack and Hiley [
47
] shows how the Bohm trajectories emerge
from an averaging over this deeper process.
We can see from Figure 9, the above simulations predict some interesting structure in near field
behaviour of the atoms after they leave the second SG magnet. This could be experimentally explored
through weak measurements as suggested in [
48
]. At present, our group [
49
] is attempting to measure
the weak values of momentum and spin which, if successful, would ultimately enable us to not only
construct these flow lines, but also to measure the time evolution of the angle
θ(y
,
t)
of the spin vector.
We are also exploring the possibility of using the techniques we are developing to check the
results shown in Figure 2. At present we are on the edge of what is technically possible and
if we are successful, the experiments will show that the quantum potential energy appearing in
Equation (20) has an observable experimental consequence and therefore cannot be ignored in
analysing quantum phenomena.
5. Conclusions
In this paper we have shown that the differences that are claimed to exist between the standard
approach to quantum mechanics and the Bohm approach do not exist when both are applied correctly.
Indeed it is hard to imagine how there could be any differences in the predicted experimental results
since both approaches use exactly the same mathematical structure. For the type of experiments
considered by ESSW [
19
], the probability current plays a key role. In both approaches the probability
current is considered as a particle flow, the conventional approach regarding it as a measure of particles
flowing out of a small region,
V
, of space, whereas the Bohm approach assumes the probability current
arises from the velocities of individual particles through the relation
j(r)/ρ(r) = S(r)/m=p(r)/m
.
In the Bohm model this is taken as the definition of the local momentum,
p(r)
. Clearly the behaviour
of the probability currents is identical to the local momentum. This is what ESSW failed to recognise.
Notice that this disagreement arises before the addition of any device to measure which path the
particle actually took.
The inclusion of a which-way detector into the discussion merely confuses the issue. Traditionally
it is assumed that any measurement to determine which path a particle actually takes brings about
Entropy 2018,20, 353 16 of 18
the “collapse” of the wave function. Suppose a position measurement is made after the atom has left
the second SG magnet as shown in Figure 1. The wave function (1) will not then be the pure state
but instead will be a mixture which must be described by a density matrix
ρ
with
ρ26=ρ
. This means
there is no interference between the two wave packets
ψ+
and
ψ
in which case the particles actually
cross the
z=
0 plane as shown in Figure 1. Exactly the same thing happens in the Bohm model as was
discussed in detail in Hiley [
22
] and Hiley and Callaghan [
23
]. We will not repeat the argument again
in this paper but refer the interested reader to the original papers. Our conclusion is that the standard
quantum mechanics produces exactly the same behaviour as the Bohmian approach so it cannot be
used to conclude the Bohm trajectories are “surreal”.
Since these earlier objections were raised, an entirely new way of experimentally constructing the
“Bohm particle trajectories” has been developed by Kocsis et al. [
1
] as discussed in the introduction.
Furthermore in the case of atoms the claim that these are “particle trajectories” has been re-examined
recently by Flack and Hiley [
47
] who have concluded that the flow lines, as we shall now call them,
are not the trajectories of single atoms but an average momentum flow, the measurements being taken
over many individual particle events. In fact they have shown that they represent an average of the
ensemble of actual individual stochastic Feynman paths.
The calculations we have presented in this paper provide a detailed background to the experiments
of Monachello et al. [
49
] and Morley et al. [
50
]. This means that we will not have to rely on theoretical
arguments alone to reach an understanding of the behaviour reported in this paper but we hope to be
able to provide experimental evidence to further clarify the situation.
Author Contributions:
Both authors contributed in the same manner to the research and wrote the paper together.
Acknowledgments:
Special thanks to Bob Callaghan, Robert Flack and Vincenzo Monachello for their helpful
discussions. Thanks also to the Franklin Fetzer Foundation for their financial support.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Kocsis, S.; Braverman, B.; Ravets, S.; Stevens, M.J.; Mirin, R.P.; Shalm, L.K.; Steinberg, A.M. Observing the
Average Trajectories of Single Photons in a Two-Slit Interferometer. Science 2011,332, 1170–1173.
2.
Mahler, D.; Rozema, L.; Fisher, K.; Vermeyden, L.; Resch, K.; Braverman, B.; Wiseman, H.; Steinberg, A.M.
Measuring Bohm trajectories of entangled photons. In Proceedings of the CLEO: QELS-Fundamental Science,
Optical Society of America, San Jose, CA, USA, 8–13 June 2014; p. FW1A-1.
3.
Coffey, T.M.; Wyatt, R.E. Comment on “Observing the Average Trajectories of Single Photons in a Two-Slit
Interferometer”. arXiv 2011, arXiv:1109.4436.
4.
Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables I. Phys. Rev.
1952,85, 166–179.
5.
Philippidis, C.; Dewdney, C.; Hiley, B.J. Quantum Interference and the Quantum Potential. Nuovo Cimento
1979,52, 15–28.
6.
Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables II. Phys. Rev.
1952,82, 180–193.
7.
Bohm, D.; Hiley, B.J.; Kaloyerou, P.N. An Ontological Basis for the Quantum Theory: II—A Causal
Interpretation of Quantum Fields. Phys. Rep. 1987,144, 349–375.
8.
Holland, P.R. The de Broglie-Bohm Theory of motion and Quantum Field Theory. Phys. Rep.
1993
,224,
95–150.
9. Kaloyerou, P.N. The Causal Interpretation of the Electromagnetic Field. Phys. Rep. 1994,244, 287–385.
10.
Flack, R.; Hiley, B.J. Weak Values of Momentum of the Electromagnetic Field: Average Momentum Flow
Lines, Not Photon Trajectories. arXiv 2016, arXiv:1611.06510.
11.
Bliokh, K.Y.; Bekshaev, A.Y.; Kofman, A.G.; Nori, F. Photon trajectories, anomalous velocities and weak
measurements: A classical interpretation. New J. Phys. 2013,15, 073022.
12.
Landau, L.D.; Lifshitz, E.M. Quantum Mechanics: Non-Relativistic Theory; Pergamon Press: Oxford, UK, 1977; p. 2.
Entropy 2018,20, 353 17 of 18
13.
Einstein, A. Albert Einstein: Philosopher-Scientist; Schilpp, P.A., Ed.; Library of the Living Philosophers:
Evanston, IL, USA, 1949; pp. 665–676.
14.
Heisenberg, W. Physics and Philosophy: The Revolution in Modern Science; George Allen and Unwin: London,
UK, 1958.
15. Jammer, M. The Philosophy of Quantum Mechanics; Wiley: New York, NY, USA, 1974.
16. Madelung, E. Quantentheorie in hydrodynamischer Form. Z. Phys. 1926,40, 322–326.
17.
de Broglie, L. La mécanique ondulatoire et la structure atomique de la matière et du rayonnement.
J. Phys. Radium
1927,8, 225–241.
18.
de Gosson, M. The Principles of Newtonian and Quantum Mechanics: The Need for Planck’s Constant; Imperial
College Press: London, UK, 2001.
19.
Englert, J.; Scully, M.O.; Süssman, G.; Walther, H. Surrealistic Bohm Trajectories. Z. Naturforsch.
1992
,47,
1175–1186.
20.
Scully, M. Do Bohm trajectories always provide a trustworthy physical picture of particle motion? Phys. Scr.
1998,76, 41–46.
21.
Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures on Physics III; Addison-Wesley: Reading, MA,
USA, 1965; Chapter 5.
22.
Hiley, B.J. Welcher Weg Experiments from the Bohm Perspective, Quantum Theory: Reconsiderations of Foundations-3,
Växjö, Sweden 2005; Adenier, G., Krennikov, A.Y., Nieuwenhuizen, T.M., Eds.; AIP: College Park, MD, USA,
2006; pp. 154–160.
23.
Hiley, B.J.; Callaghan, R.E. Delayed Choice Experiments and the Bohm Approach. Phys. Scr.
2006
,74,
336–348.
24. Bohm, D. Quantum Theory; Prentice-Hall: Englewood Cliffs, NJ, USA, 1951.
25.
Landau, L.D.; Lifshitz, E.M. Quantum Mechanics: Non-Relativistic Theory; Pergamon Press: Oxford, UK, 1977;
pp. 56–57.
26. Mott, N.F. The Wave Mechanics of α-Ray Tracks. Proc. R. Soc. 1929,126, 79–84.
27.
Bohm, D.; Schiller, R.; Tiomno, J. A causal interpretation of the Pauli equation (A). Nuovo Cimento
1955
,1,
48–66.
28.
Dewdney, C. Particle Trajectories and Interference in a Time-dependent Model of Neutron Single Crystal
Interferometry. Phys. Lett. 1985,109, 377–384.
29.
Dewdney, C.; Holland, P.R.; Kyprianidis, A.; Vigier, J.-P. Spin and non-locality in quantum mechanics. Nature
1988,336, 536–544.
30.
Dewdney, C.; Holland, P.R.; Kyprianidis, A. What happens in a spin measurement? Phys. Lett. A
1986
,119,
259–267.
31.
Dewdney, C.; Holland, P.R.; Kyprianidis, A. A Causal Account of Non-local Einstein-Podolsky-Rosen Spin
Correlations. J. Phys. A Math. Gen. 1987,20, 4717–4732.
32.
Holland, P.R. The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum
Mechanics; Cambridge University Press: Cambridge, UK, 1995.
33.
Penrose, R.; Rindler, W. Spinors and Space-Time; Cambridge University Press: Cambridge, UK, 1984; Volume 1.
34.
Bell, J.S. Speakable and Unspeakable in Quantum Mechanics; Cambridge University Press: Cambridge, UK, 1987.
35.
Hiley, B.J.; Callaghan, R.E. The Clifford Algebra approach to Quantum Mechanics A: The Schrödinger and
Pauli Particles. arXiv 2010, arXiv:1011.4031.
36.
Hiley, B.J. Weak Values: Approach through the Clifford and Moyal Algebras. J. Phys. Conf. Ser.
2012
,
361, 012014.
37.
Bohm, D.; Hiley, B.J. Non-locality and Locality in the Stochastic Interpretation of Quantum Mechanics.
Phys. Rep. 1989,172, 93–122.
38.
Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures on Physics III; Addison-Wesley: Reading, MA,
USA, 1965; Chapter 5.2.
39. Schrödinger, E. Are There Quantum Jumps? Part I. Br. J. Philos. Sci. 1952,3, 109–123.
40.
Bohm, D. The Implicate Order: A New Approach to the Nature of Reality; A Talk Given at Syracuse University;
Syracuse University: Syracuse, NY, USA, 1982.
Entropy 2018,20, 353 18 of 18
41.
Bohm, D. A proposed Explanation of Quantum Theory in Terms of Hidden Variables at a Sub-Quantum
Mechanical Level. In Observation and Interpretation, Proceedings of the Ninth Symposium of the Colston Research
Society, Bristol, UK, 1–4 April 1957; Korner, S., Ed.; Butterworth Scientific Publications: London, UK, 1957;
pp. 33–40.
42.
Hiley, B.J.; Callaghan, R.E. The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and
its relation to the Bohm Approach. arXiv 2010, arXiv:1011.4033.
43.
Takabayasi, T. Remarks on the Formulation of Quantum Mechanics with Classical Pictures and on Relations
between Linear Scalar Fields and Hydrodynamical Fields. Prog. Theor. Phys. 1953,9, 187–222.
44. Schwinger, J. The Theory of Quantised Fields I. Phys. Rev. 1951,82, 914–927.
45.
Bohm, D.J.; Hiley, B.J. Measurement Understood Through the Quantum Potential Approach. Found. Phys.
1984,14, 255–264.
46.
Bohm, D.; Hiley, B.J. The Undivided Universe: An Ontological Interpretation of Quantum Theory; Routledge:
London, UK, 1993.
47.
Flack, R.; Hiley, B.J. Feynman Paths and Weak Values. Preprints
2018
, 2018040241,
doi:10.20944/preprints201804.0241.v1.
48.
Flack, R.; Hiley, B.J. Weak Measurement and its Experimental Realisation. J. Phys. Conf. Ser.
2014
,504, 012016,
doi:10.1088/1742-6596/504/1/012016.
49.
Monachello, V.; Flack, R.; Hiley, B.J.; Callaghan, R.E. A method for measuring the real part of the weak value
of spin using non-zero mass particles. arXiv 2017, arXiv:1701.04808.
50.
Morley, J.; Edmunds, P.D.; Barker, P.F. Measuring the weak value of the momentum in a double slit
interferometer. J. Phys. Conf. Ser. 2016,701, 012030.
c
2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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