Chattaraj P.K. (Ed.) Quantum Trajectories
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34 Quantum Trajectories
Note that Equations 2.56 and 2.57 are derived by solving the differential equations
corresponding to the WKB approximation for q q
−
and q q
+
, so that the
coefficients of κ
−1/2
exp ±
q
q
−
dxκ, e.g., A
−
and B
−
in Equation 2.56, cannot be
simultaneously vanishing. In particular, the fact that ψ
D
∝ ψ yields
A
−
B
D
−
− A
D
−
B
−
= 0, A
+
B
D
+
− A
D
+
B
+
= 0. (2.58)
Let us now consider the case in which, for a given E satisfying Equation 2.23, any
solution of the corresponding SE diverges at least at one of the two spatial infinities,
that is
lim
q→+∞
(|ψ(−q)|+|ψ(q)|) =+∞. (2.59)
This implies that there is a solution diverging both at q =−∞and q =+∞.
In fact, if two solutions ψ
1
and ψ
2
satisfy ψ
1
(−∞) =±∞, ψ
1
(+∞) =±∞
and ψ
2
(−∞) =±∞, ψ
2
(+∞) =±∞, then ψ
1
+ ψ
2
diverges at ±∞. On the
other hand, Equation 2.58 rules out the case in which all the solutions in their WKB
approximation are divergent only at one of the two spatial infinities, say −∞. Since,
in the case Equation 2.23, a solution which diverges in the WKB approximation is
itself divergent (and vice versa), we have that in the case Equation 2.23, the fact that
all the solutions of the SE diverge only at one of the two spatial infinities cannot occur.
Let us denote by ψ a solution which is divergent both at −∞ and +∞. In the
WKB approximation this means that both A
−
and B
+
are non-vanishing, so that
ψ
∼
q→−∞
A
−
√
κ
exp
−
q
q
−
dxκ
, ψ
∼
q→+∞
B
+
√
κ
exp
q
q
+
dxκ
.
The asymptotic behavior of the ratio ψ
D
/ψ is given by
lim
q→−∞
ψ
D
ψ
=
A
D
−
A
−
, lim
q→+∞
ψ
D
ψ
=
B
D
+
B
+
.
Note that since in the case at hand any divergent solution also diverges in the WKB
approximation, we have that Equation 2.59 rules out the case A
D
−
= B
D
+
= 0. Let us
then suppose that either A
D
−
= 0orB
D
+
= 0. If A
D
−
= 0, then w(−∞) = 0 = w(+∞).
Similarly, if B
D
+
= 0, then w(+∞) = 0 = w(−∞). Hence, in this case w, and
therefore the trivializing map, cannot satisfy Equation 2.53. On the other hand, also in
the case in which both A
D
−
and B
D
+
are non-vanishing, w cannot satisfy Equation 2.53.
For, if A
D
−
/A
−
= B
D
+
/B
+
, then
φ = ψ −
A
−
A
D
−
ψ
D
= ψ −
B
+
B
D
+
ψ
D
,
would be a solution of the SE whose WKB approximation has the form
φ =
B
−
√
κ
exp
q
q
−
dxκ
, q q
−
,
The Equivalence Postulate of Quantum Mechanics 35
and
φ =
A
+
√
κ
exp
−
q
q
+
dxκ
, q q
+
.
Hence, if A
D
−
/A
−
= B
D
+
/B
+
, then there is a solution whose WKB approximation
vanishes both at −∞ and +∞. On the other hand, we are considering the values
of E satisfying Equation 2.23 and for which any solution of the SE has the property
Equation 2.59. This implies that no solutions can vanish both at −∞ and +∞in the
WKB approximation. Hence
A
D
−
A
−
=
B
D
+
B
+
,
so that w(−∞) = w(+∞). We also note that not even the case w(−∞) =±∞=
−w(+∞) can occur, as this would imply that A
−
= B
+
= 0, which in turn would
imply, against the hypothesis, that there are solutions vanishing at q =±∞. Hence,
if for a given E satisfying Equation 2.23, any solution of the corresponding SE
diverges at least at one of the two spatial infinities, we have that the trivializing map
has a discontinuity at q =±∞. As a consequence, the EP cannot be implemented in
this case so that this value E cannot belong to the physical spectrum.
Therefore, the physical values of E satisfying Equation 2.23 are those for which
there are solutions which are divergent neither at −∞nor at +∞. On the other hand,
from the WKB approximation and Equation 2.23, it follows that the non-divergent
solutions must vanish both at −∞ and +∞. It follows that the only energy levels
satisfying the property Equation 2.23, which are compatible with the EP, are those for
which there exists the solution vanishing both at ±∞. On the other hand, solutions
vanishing as κ
−1/2
exp
q
q
−
dxκ at −∞and κ
−1/2
exp −
q
q
+
dxκ at +∞, with P
2
±
> 0,
cannot contribute with an infinite value to
+∞
−∞
dxψ
2
. The reason is that existence
of the QSHJE requires that {e
2i
S
0
, q} be defined and this, in turn, implies that any
solution of the SE must be continuous. On the other hand, since ψ is continuous,
and therefore finite also at finite values of q, we have
q
b
q
a
dxψ
2
< +∞ for all finite
q
a
and q
b
. In other words, the only possibility for a continuous function to have a
divergent value of
+∞
−∞
dxψ
2
comes from its behavior at ±∞. Therefore, since the
implementation of the EP in the case Equation 2.23 requires that the corresponding
E should admit a solution with the behavior
ψ
∼
q→−∞
A
−
√
κ
exp
q
q
−
dxκ
, ψ
∼
q→+∞
B
+
√
κ
exp
−
q
q
+
dxκ
,
we have the following basic fact
The values of E satisfying
V (q) −E ≥
P
2
−
> 0, q<q
−
,
P
2
+
> 0, q>q
+
,
(2.60)
are physically admissible if and only if the corresponding SE has an L
2
(R) solution.
36 Quantum Trajectories
We now give another proof of the fact that if W is of the type Equation 2.60, then
the corresponding SE must have an L
2
(R) solution in order to satisfy Equation 2.53.
In particular, we will show that this is a necessary condition. That this is sufficient
has already been proved above.
Wronskian arguments, which can be found in Messiah’s book [17], imply that if
V (q) −E ≥ P
2
+
> 0, q>q
+
, then as q →+∞, we have (P
+
> 0)
– There is a solution of the SE that vanishes at least as e
−P
+
q
.
– Any other linearly independent solution diverges at least as e
P
+
q
.
Similarly, if V (q) −E ≥ P
2
−
> 0, q<q
−
, then as q →−∞, we have (P
−
> 0)
– There is a solution of the SE that vanishes at least as e
P
−
q
.
– Any other linearly independent solution diverges at least as e
−P
−
q
.
These properties imply that if there is a solution of the SE in L
2
(R), then any solution
is either in L
2
(R) or diverges both at −∞ and +∞. Let us show that the possibility
that a solution vanishes only at one of the two spatial infinities is ruled out. Suppose
that, besides the L
2
(R) solution, which we denote by ψ
1
, there is a solution ψ
2
which
is divergent only at +∞. On the other hand, the above properties show that there exists
also a solution ψ
3
which is divergent at −∞. Since the number of linearly independent
solutions of the SE is two, we have ψ
3
= Aψ
1
+Bψ
2
. However, since ψ
1
vanishes
both at −∞ and +∞, we see that ψ
3
= Aψ
1
+Bψ
2
can be satisfied only if ψ
2
and
ψ
3
are divergent both at −∞ and +∞. This fact and the above properties imply that
If the SE has an L
2
(R) solution, then any solution has two possible asymptotics:
– Vanishes both at −∞ and +∞ at least as e
P
−
q
and e
−P
+
q
respectively.
– Diverges both at −∞ and +∞ at least as e
−P
−
q
and e
P
+
q
respectively.
Similarly, we have
If the SE does not admit an L
2
(R) solution, then any solution has three possible
asymptotics:
– Diverges both at −∞ and +∞ at least as e
−P
−
q
and e
P
+
q
respectively.
– Diverges at −∞ at least as e
−P
−
q
and vanishes at +∞ at least as e
−P
+
q
.
– Vanishes at −∞ at least as e
P
−
q
and diverges at +∞ at least as e
P
+
q
.
Let us consider the ratio w = ψ
D
/ψ in the latter case. Since any different choice of
linearly independent solutions of the SE corresponds to a Möbius transformation of
w, we can choose
ψ
D
∼
q→−∞
a
−
e
P
−
q
, ψ
D
∼
q→+∞
a
+
e
P
+
q
,
and
ψ
∼
q→−∞
b
−
e
−P
−
q
, ψ
∼
q→+∞
b
+
e
−P
+
q
,
The Equivalence Postulate of Quantum Mechanics 37
were by ∼ we mean that ψ
D
and ψ either diverge or vanish “at least as.” Their ratio
has the asymptotic
ψ
D
ψ
∼
q→−∞
c
−
e
2P
−
q
→ 0,
ψ
D
ψ
∼
q→+∞
c
+
e
2P
+
q
→±∞,
so that w cannot satisfy Equation 2.53. This concludes the alternative proof of the
fact that, in the case Equation 2.60, the existence of the L
2
(R) solution is a necessary
condition in order for Equation 2.53 to be satisfied. The fact that this is a sufficient
condition has been proved previously in deriving Equation 2.55.
The above results imply that the usual quantized spectrum arises as a consequence
of the EP.
Let us note that we are considering real solutions of the SE. Thus, apparently, in
requiring the existence of an L
2
(R) solution, one should specify the existence of a
real L
2
(R) solution. However, if there is an L
2
(R) solution ψ, this is unique up to a
constant, and since also
¯
ψ ∈ L
2
(R) solves the SE, we have that an L
2
(R) solution of
the SE is real up to a phase.
In this chapter we have discussed two main theorems that underlie the formulation
of quantum mechanics from an EP. One should regard this postulate as providing a
novel starting point for formulating quantum mechanics, and ultimately for formu-
lating quantum gravity. It also provides an arena to reexamine many of the tenants of
the conventional approaches, and in this regard we note the related work of several
other authors [18–92], and, in particular, that of Floyd [18–29].
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3
Quantum Trajectories
and Entanglement
Edward R. Floyd
CONTENTS
3.1 Introduction.....................................................................................................41
3.2 Recipe..............................................................................................................42
3.3 Didactic Examples ..........................................................................................46
3.3.1 Wave Packet Spreading ......................................................................46
3.3.2 Simplified Double-Slit Experiment ....................................................47
3.3.3 Quantum Trajectory for EPR-Molecule..............................................49
Bibliography ............................................................................................................50
3.1 INTRODUCTION
Entanglement phenomena offer fine, didactic examples that demonstrate the applica-
tion of quantum trajectories. Quantum entanglement phenomena are tractable to the
quantum trajectory representation. Quantum trajectories render insight into entangle-
ment phenomena of quantum systems. Entanglement is one of the prime features that
distinguishes quantum mechanics from classical mechanics. Entanglement implies
nonlocality as confirmed by the Aspect experiments [1–3]. Entangled quantum sys-
tems have nonclassical correlations that manifest nonlocal behavior for the system.
The quantum trajectory representation can describe how entanglement induces non-
locality [4]. Within the foundations of quantum mechanics, a quantum trajectory rep-
resentation of entangled systems is the key to resolving the nonintuitive behavior of
wave packet spreading [4], the which-way paradox of the double-slit experiment [5],
and the Einstein–Podolsky–Rosen (EPR) paradox [6].
The quantum trajectory representation of quantum mechanics is a nonlocal, phe-
nomenological theory that is deterministic. Herein, “deterministic” means that if
without disturbing a system one can predict with certainty the value of a physical
quantity (the quantum trajectory), then there exists an element of physical reality that
corresponds to such a physical quantity (the quantum trajectory).
The quantum trajectory representation used in this chapter differs from the rep-
resentation of Bohmian mechanics: the two representations have different equations
of motion despite having a common quantum reduced action (Hamilton’s charac-
teristic function) [7–14]. The quantum trajectory representation with its attendant
bipolar Ansatz [9, 12–14], which may be extended to a multipolar Ansatz, is well
41
42 Quantum Trajectories
suited for application to entangled systems. While a bipolar Ansatz is used herein for
convenience, any Ansatz is unnecessary for our quantum trajectory representation.
Faraggi and Matone derived a quantum equivalence principle, where the underlying
quantum Hamilton–Jacobi equation for various systems may be reversibly mapped
into each other by a simple coordinate transform, to render the quantum trajectory
representation without any of the Copenhagen postulates [13].
The goal herein is didactic: to make quantum trajectories more accessible to unini-
tiated workers. Accessibility is provided by incorporating knowledge of the wave
function without sacrificing the profoundness and insight of quantum trajectories.
The techniques developed herein are not limited to entanglement phenomena.
In this chapter, we present a recipe for describing the quantum trajectory of an
entangled system. We consider entangled systems of two components, but the exten-
sion to multicomponent systems is straightforward. We synthesize an “entanglement
molecule” from its entangled components. The entanglement molecule is the entity
that represents the entire entangled system under consideration. This system has entan-
gled components which may be latent quantum anyons or just components (spectral
or otherwise) of an anyon. In the former case, entangled anyons remain latent until
reified by decoherence. In the latter case, the entanglement molecule is an anyon
self-entangled by its own multispectral components. The entanglement molecule rep-
resents the entangled system as a single entity.We deliberately investigate the behavior
of the entanglement molecule rather than the individual behaviors of its two compo-
nents. We develop the entanglement molecule’s wave function in polar form and its
reduced action, both of which manifest entanglement. The reduced action of the entan-
glement molecule is its generator of quantum motion and contains all the information
of the entanglement system. We next apply Jacobi’s theorem to the reduced action
to generate the equation of quantum motion for the entanglement molecule to ren-
der its quantum trajectory. The resultant quantum trajectory manifests entanglement
and has retrograde segments interspersed between segments of forward motion. This
alternating of forward and retrograde segments generates nonlocality and, within the
entanglement molecule, there exists entanglement at a distance. We briefly apply the
quantum trajectory representation to three fundamental, entanglement problems of
quantum mechanics: wave packet spreading, the double-slit experiment, and EPR. In
particular, decomposition of the equation of quantum motion for the EPR entangle-
ment molecule, while rendering the latent classical behavior of the two scattered par-
ticles, also reveals an emergent “entanglon” that maintains the entanglement between
the two scattered latent particles as they become progressively displaced farther from
each other.
3.2 RECIPE
Let us consider two components, P
1
and P
2
, with spatial wave functions , ψ
1
(x) and
ψ
2
(x), that interact for mathematical simplicity through an instantaneous common
repulsive impulse at time t =0. Subsequently, the components are entangled for t>0.
The positions (x
1
, x
2
) of the two components are co-located at the time of impulse
interaction, t = 0atx
1
, x
2
= 0. The masses of P
1
and P
2
are respectively given
Quantum Trajectories and Entanglement 43
by m and α
2
m where α > 0. The system mass is (1 + α
2
)m. The two components,
which may be latent while entangled, subsequently recoil from each other in opposite
directions after impulse at t = 0 with their spatial wave functions given by
ψ
1
(x) = exp(ikx), ψ
2
(x) = α exp(−ikx + iβ); t>0 (3.1)
where k =[2(1 + α
2
)mE]
1/2
/, E is the energy of the entanglement system, and
−π < β ≤ π. The center-of-mass inertial reference frame is chosen so that Equa-
tion 3.1 is true. The term β represents a phase shift between the two components. In
our chosen inertial reference frame, ψ
1
and ψ
2
form a set of independent solutions
of the Schrödinger equation for energy E. For the quantum trajectory represen-
tation, the Wronskian of the set (ψ
1
, ψ
2
) is normalized rather than normalization
in accordance with the Born probability postulate [11]. However, ψ
1
and ψ
2
,as
will be shown later, appear in the generator of motion as a Möbius transformation
(aψ
1
− bψ
2
)/(cψ
1
− dψ
2
), ad − bc = 0 where a, b, c, and d are coefficients [13].
Hence, the relative normalization of ψ
1
and ψ
2
manifested by the factor α suffices.
The choice of inertial reference system, for which ψ
1
and ψ
2
have the wave num-
bers k and −k, induces the relationship x
1
=−x
2
/α
2
for the positions of the two
components for t>0. This is an extension of Fine’s conservation of relative posi-
tion [15]. Conservation of relative position induces loss of parameter independence
and outcome independence [16].
We assume herein that no matter how far apart the two components may become,
they will remain entangled consistent with the Aspect experiments [1–3]. In other
words, the initially entangled system remains “inseparable” even for t 0 and
x
1
− x
2
1.
For entanglement in the wave function representation of quantum mechanics, we
may synthesize an entanglement system as a simple polar wave function, ψ
ent
, from
the pair (bipolar wave function) of entangled components, ψ
1
and ψ
2
, by Bohm’s
method [7]. Bohm’s recipe for expressing complex ψ in polar format is given by
ψ
ent
=[(ψ
ent
)
2
+ (ψ
ent
)
2
]
1/2
exp
i
tan
−1
ψ
ent
ψ
ent
(3.2)
where ψ
ent
=ψ
1
+ψ
2
and ψ
ent
=ψ
1
+ψ
2
. By Bohm, the wave function
for the entangled system, ψ
ent
(x), whose components are specified by Equation 3.1,
is given by Refs [4,8]
ψ
ent
(x) = ψ
1
(x) +ψ
2
(x) =
bipolar wave function
exp(ikx) + α exp(−ikx + iβ)
=[1 +α
2
+ 2α cos(2kx + β)]
1/2
exp
i arctan
sin(kx) −α sin(kx + β)
cos(kx) + α cos(kx + β)
polar wave function is still an eigenfuncion for E =
2
k
2
/[2m(1 + α
2
)]
(3.3)
where we have dropped, for entanglement, the subscript for a component’s
(latent) position x by the extension of conservation of relative position. The above