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Surrealistic Böhm Trajectories
Berthold-Georg Englert1'2, Marian O. Scully3, Georg Süssmann, and Herbert Walther1'2
1 Sektion Physik, Universität München, Am Coulombwall 1, D-8046 Garching, Germany
2 Max-Planck-Institut für Quantenoptik, Ludwig-Prandtl-Straße 10, W-8046 Garching.
3 Department of Physics, Texas A & M University, College Station, TX 77843-4242.
Z. Naturforsch. 47a, 1175-1186 (1992); received September 22, 1992
A study of interferometers with one-bit which-way detectors demonstrates that the trajectories,
which David Böhm invented in his attempt at a realistic interpretation of quantum mechanics, are in
fact surrealistic, because they may be macroscopically at variance with the observed track of the parti-
cle. We consider a two-slit interferometer and an incomplete Stern-Gerlach interferometer, and propose
an experimentum crucis based on the latter.
Introduction
In Bohmian mechanics [1-3], ordinary quantum
theory [4] is supplemented by classical particle trajec-
tories. These trajectories are usually unobserved - in-
deed, they are unobservable in the face of Heisenberg's
uncertainty relation for position and momentum - but
nevertheless they are regarded as realistic, similar to
the tracks seen in a bubble chamber. Yes, such tracks
supposedly exhibit Böhm trajectories within the limi-
tations set by uncertainty relations. More common,
however, is the situation in which a particle is detected
without being observed earlier, and then the particle's
Böhm trajectory can be retrodicted. This retrodiction
assigns reality to the Böhm trajectories in an opera-
tional sense, the more so because the actual trajectory
cannot be predicted owing to the fundamental igno-
rance of the actual initial position. Predictions in
Bohmian mechanics are limited by the probabilistic
knowledge about initial conditions, which knowledge
comes from quantal probability distributions. There-
fore, the predictive power of Bohmian mechanics does
not exceed that of ordinary quantum theory, and so
the alleged superiority of Bohmian mechanics over
ordinary quantum theory is of a purely philosophical
nature.
A particle traverses a bubble chamber and is then
detected. Does the retrodicted Böhm trajectory al-
ways agree with the observed track? Our answer is:
No. For, our considerations show that the Böhm tra-
jectory may be macroscopically at variance with the
Reprint requests to Prof. Dr. G. Süssmann, Sektion Physik,
Universität München, Am Coulombwall 1, W-8046 Gar-
ching bei München.
recorded track. For the sake of simplicity, we do, how-
ever, not study the complicated motion of a charged
particle through a bubble chamber, where many de-
grees of freedom are involved. Instead, we look at
atom interferometers, and a few which-way detectors
replacing the plethora of bubbles. These one-bit
which-way detectors are placed such that they enable
us to distinguish one class of tracks from another,
macroscopically different one. It turns out that the
atom's Böhm trajectory may not belong to the ob-
served class of tracks. We are thus led to the conclu-
sion that the Böhm trajectories, originally introduced
with the aim of arriving at a "realistic interpretation"
of quantum theory, are in fact surrealistic.
The next section contains a concise review of
Bohmian mechanics. Then we turn to gedanken exper-
iments in which atoms pass through double slits, even-
tually supplemented by which-way detectors capable
of recording through which slit the atom went without
thereby disturbing the atom's center-of-mass motion.
We find that there are events when the Böhm trajec-
tory goes through one slit, but the atom through the
other. This is followed by a detailed treatment of mag-
netic atoms traversing an incomplete Stern-Gerlach
interferometer. Here the track recorded by the which-
way detectors is always macroscopically different from
the corresponding Böhm trajectories. We are thus
proposing an experimentum crucis which, according
to our quantum theoretical prediction, will clearly
demonstrate that the reality attributed to Böhm tra-
jectories is rather metaphysical than physical.
In Appendix A we supply a brief description of
quantum-optical which-way detectors, which have
been discussed in detail elsewhere [5].
0932-0784 / 92 / 1200-1175 $ 01.30/0. - Please order a reprint rather than making your own copy.
1176 B.-G. Englert et al. • Surrealistic Böhm Trajectories
The one-bit which-way detectors may evolve dy-
namically in Bohmian mechanics, too. The descrip-
tion is then changed quantitatively but not qualita-
tively. A large fraction of atomic Böhm trajectories
remains macroscopically at variance with the recorded
track. This situation is discussed in Appendix B.
Bohmian Mechanics in a Nutshell
In Bohmian mechanics [1-3] the continuity equation
— Q(t,r) + \-j(t,r) = 0, (1)
obeyed by the probability density g for finding the
particle in the vicinity of the point r and the corre-
sponding probability current density j, is taken as the
starting point for defining, and eventually computing,
trajectories R(t). These Böhm trajectories are inter-
preted as the possible paths along which the particle
could have propagated. The actual trajectory is then
selected by the (initial or rather) final position that is
somehow observed. The trajectories are determined
by Bohm's velocity field v(t, r) that is given by
v(t,r)=j(t,r)/g(t,r). (2)
Fortunately, knowledge of v where g = 0 is never
needed. The differential equation
±R(t) = v(t,R(t)) (3)
in conjunction with a known position at some instant
- usually when the particle is finally detected - then
supplies the actual Böhm trajectory. The probabilistic
predictions of ordinary quantum theory remain un-
altered because an ensemble of trajectories that mim-
icks g at one time will equally well mimick g at any
later time. So, with the understanding that in repeated
experiments initial positions are realized according to
the probabilities implied by g, an ensemble average
over the corresponding trajectories will agree with the
quantum theoretical predictions. In addition, before
the velocity field (2) is available, one must first solve
the Schrödinger equation for the underlying wave
function from which g and j are then calculated. Thus,
the Böhm trajectories cannot provide us with more
information than what is carried by the wave function.
In the sequel we shall confine the discussion to non-
relativistic situations involving one particle of mass m.
The dynamics will by governed by a Hamilton opera-
tor of the form
H = ~p2+V(t,r,A), (4)
2m
where r and p are the position and momentum opera-
tors and A symbolizes additional degrees of freedom
like spin. The wave function may then posses a num-
ber of components ipx{t,r) labelled by a subscript cc.
From these, g and j are constructed in the familiar
way:
Q(t,r) = X\<l>At,r)\2, (5)
a
j(t, r) = — Re £ <A* (t, r) - ') ,
m a I
of which the latter applies only if the p-dependence of
H does not differ from that in (4).
In Bohmian mechanics, a particle has a position
and nothing else, as is particularly emphasized in
Bell's [6] formulation (see also [7]). The properties
associated with A in (4) (or the label a in (5)), such like
spin, are possessed and carried by the wave function
only. Nevertheless, after the trajectory R(t) has been
found, the values of the \j/x(t, R(t)) can be employed to
specify what - in a conscious departure from standard
Bohmian mechanics - could be regarded as the actual
values of the quantities A. For example, if iJ/+ are the
two spin-j components referring to the z direction,
then the unit spin vector at time t is
j I ^ ^ v- \
S(t) = - 2 Im
<K* <A_
(6)
e W+i2-ii/u2/
with g =
| \Js+12
+ | |2 (see (5)), where
ij/±
are evaluated
at r = R(t).
Double-Silt Interferometer
Consider the symmetric double-slit set-up in Fig-
ure 1. The particles coming in from the left all have the
same velocity perpendicular to the slit plate. Thus the
incident wave function is a plane wave with the planes
parallel to the slit plate. In the interference region
between the slit plate and the screen, the wave func-
tion is the coherent superposition of the two contribu-
tions from the slits,
iMt,r) = iMt,r) + iMt,r). (7)
1177 B.-G. Englert et al. • Surrealistic Böhm Trajectories
incident plane wave
z
=
o plane
interference
region screen with
interference pattern
Fig. 1. Double-slit interferometer with indication of phase
fronts for incident plane wave and scattered partial waves ij/>
and ij/< which interfere.
collimated one-bit
incident detectors
plane wave
z=o plane
interference
region SCreen without
interference pattern
Fig. 3. Double-slit interferometer with one-bit detectors
through which the collimated incident plane wave reaches
the slit plate. With which-way information available, the
screen no longer displays the double-slit interference pattern.
slit interference region screen with
plate interference
pattern
Fig. 2. Böhm trajectories in the interference region of a
double-slit interferometer; adapted from [8].
In view of the geometrical symmetry of the arrange-
ment,
iJ/K
is obtained from by reflection at the z = 0
plane,
(t, x, y, z) = i(r, x, y,-z). (8)
Consequently, the probability density g =
\ \\i
|2 as well
as the x- and y-components of the current vector
1 h
j = — Re iJ/*
—
\\jj are even functions of z, in contrast
m i
to the z-component of j, which is odd. Therefore, the
z-component of the velocity field (2) is also an odd
function of z, so that this component vanishes on the
z = 0 plane of symmetry. This has the immediate impli-
cation that the Böhm trajectories do not cross the
z = 0 plane [6]. In other words, when a particle has hit
the upper half of the screen (where z>
0),
then its
retrodicted Böhm trajectory goes through the upper
slit, and those particles arriving on the lower half of
the screen (where z< 0) have, according to Bohmian
mechanics, passed through the lower slit. These sym-
metry considerations are confirmed by the Böhm tra-
jectories computed by Philippidis, Dewdney, and
Hiley [8], see Figure 2.
In ordinary quantum mechanics, the statement that
the particle went through one slit and not the other is,
of course, utterly meaningless, as long as no corre-
sponding observation is performed (as exemplified by
the string of bubbles in a bubble chamber). Quite
consciously, no definite classical history is assigned to
a single event. An adherent of Bohmian mechanics,
however, would proudly announce that he does not
have to give up the concept of a classical particle
trajectory when studying quantal phenomena. And
really, what is the argument about? The particle is not
observed while it traverses the apparatus in Figure 1.
So why can't we all accept the Böhm trajectories as
real, their empirical reality originating in retrodiction
rather than direct observation?
The advanced techniques of modern quantum optics
enable the experimenter to build one-bit detectors
sensitive to the passage of a single atom. For a descrip-
tion consult Appendix A. The details of the detection
mechanism are not relevant for the present discus-
1178 B.-G. Englert et al. • Surrealistic Böhm Trajectories
sions, so we shall symbolize the one-bit recording as a
transition from "no" to "yes". It is important to realize
that this transition happens with virtual certainty and
that the atom's center-of-mass wave function is not
altered noticeably in the process.
The set-up of Fig. 1 is now supplemented by two
such one-bit detectors, one for each slit as in Figure 3.
The detectors will supply us with which-way informa-
tion, not quite in the detailed sense of registering a
whole track (like that seen in a bubble chamber), but
by clearly distinguishing the class of tracks through
one slit from the class through the other. The two-slit
interference pattern on the screen is lost, of course, as
soon as we have this which-way information available,
but that is not the issue here.
With the one-bit which-way detectors in place, the
wave function in the interference region is now
^ = <A>iroS> + <A<iyne°s>, (9)
in which the contribution of the upper slit is correlated
to the which-way information documented in the upper
detector, and likewise for the lower slit and the lower
detector. In contrast to (7), the functions
iJ/>
and
iJ/K
in
(9) are components of the two-component wave func-
tion f. The density g and current j associated with W
are, therefore, different from what is obtained from (8).
In particular, the terms responsible for the two-slit
interference pattern are absent now, as it must be.
However, most important for the matter of interest,
the symmetry properties of
g
and j under reflections at
the z = 0 plane are the same as before: g as well as the
x- and y-component of j and v are unaffected by the
transformation z-y—z, whereas the z-components of
j and v change sign and accordingly vanish on the
symmetry plane z =
0.
Thus it is still true that when the
atom is found on the upper half of the screen its retro-
dicted Böhm trajectory goes through the upper slit,
and the trajectory of an atom found on the lower half
passes through the lower slit.
But through which slit did the atom come? Suppose
that the upper detector says "yes", the lower "no".
Then the probability for finding the atom somewhere
on the screen is proportional to
| \J/>\2,
which does not
vanish on the lower half of the screen. Consequently,
there will be events when the atom goes through the
upper detector and therefore through the upper slit
and then hits the lower half of the screen, so that the
corresponding Böhm trajectory goes through the lower
slit. In other words: the Böhm trajectory is here
macroscopically at variance with the actual, that is:
observed, track. Tersely: Böhm trajectories are not
realistic, they are surrealistic.
As a result of these considerations we disagree with
Bell [6] who says (words adapted to the present discus-
sion): "The naive classical picture [of ordinary quan-
tum mechanics] has the particle, arriving on a given
half of the screen, going through the wrong slit." Per
definition, the "right slit" is, for Bell, the one traversed
by the Böhm trajectory. To state it once more clearly:
for us, the right slit is the one through which the atom
is actually observed going, never mind the naive unob-
servable Böhm trajectory. And if one does not observe
through which slit the atom goes, then the notion of the
right or the wrong slit is meaningless.
Bell [6] has more to say about double-slit inter-
ferometers with which-way detectors. His detectors,
however, are not of the one-bit type, but consist of
very many particles, of which a good fraction gets
macroscopically displaced. Bell considers treating
these particles also according to the rules of Bohmian
mechanics and arrives at the conclusion that, in effect,
either or i//< becomes irrelevant. Thereafter the
symmetry argument no longer applies and the Böhm
trajectory passes always through the same slit as the
atom's track. Bell's reasoning does not apply to the
present scheme, in which no macroscopic displace-
ments occur until the which-way information stored
in the one-bit detectors is finally read off. This reading-
off is done (long) after the atom has hit the screen, so
the relevant Böhm velocity field is, indeed, determined
by the two-component wave function (9). There the
yes/no degree of freedom of the one-bit detectors
appears just as a label, like a in (5). One cannot asso-
ciate continuous position variables with such discrete
degrees of freedom and, therefore, they are not dy-
namical in Bohmian mechanics, in contrast to the
motional degrees of freedom of the many particles that
constitute Bell's macroscopic detector.
One could object that the yes/no label of the one-
bit detector need not really refer to two orthogonal
states of a spin-|-type degree of
freedom.
It could just
as well indicate that some physical system - a har-
monic oscillator, say - is either in its ground state
("no") or in its first excited state ("yes"). Then one does
have a corresponding continuous position variable
with its own Bohmian dynamics. And the Böhm
velocity field for the atom no longer has a definite
symmetry, it changes according to the evolution of the
detector variable. Consequently, now the atom trajec-
tory may cross the z = 0 plane. But, as discussed in
B.-G. Englert et al. • Surrealistic Böhm Trajectories 1179
Appendix B, there remains a considerable number of
trajectories that do not pass through the same slit as
the atom's track.
The one-bit detectors possess yet another essential
property that is lacking in macroscopic detectors. Be-
fore they are actually read off, nothing irreversible
happens. One can even erase the which-way informa-
tion carefully and recover the two-slit interference pat-
tern in the process. For details consult [9].
There is one more lesson in Bell's article, a psycho-
logical one. Namely that this tentative supporter of
Bohmian mechanics very much wants the Böhm tra-
jectory to pass through the same slit as the observed
track of the atom. Nature, however, does not grant
this favor.
Incomplete Stern-Gerlach Interferometer
The double-slit interferometer for atoms with which-
way detectors is very difficult, if not impossible, to
realize experimentally. We have discussed its features
above mainly for pedagogical reasons which include
the contact made with [6] and [8]. We now turn to the
much more realistic experiments in which atoms tra-
Stern-Gerlach magnets
.edge,», edge w
S N
/ \
A
z **
t/
-Mg—
\
\ /
\
X
A v*
N
flat
>o
>o
flat
<
o
<
o
•y.t
screen
1 quarter 2r'and3 quarter
Fig. 4. Three quarters of a Stern-Gerlach interferometer. The
dashed curves represent ±Az of (15) along which the centers
of the wave function components \jt± propagate. After cross-
ing the z = 0 plane, atoms arrive on the screen either in
region A ("spin up") or in region B ("spin down").
verse three quarters of a Stern-Gerlach interferometer,
see Figure 4. The missing fourth quarter, necessary for
reuniting the partial beams, is absent. Therefore,
entering spin-up atoms are first deflected up, then
down, and finally hit the screen well below the z = 0
plane; conversely, spin-down atoms are first deflected
down, then up, and hit the screen well above the z = 0
plane. We emphasize that both the intermediate and
final separation between the beams are macroscopic
and, in particular, large compared with the individual
beam widths.
The subtleties studied in [10], which are essential for
the coherent reunion of the two partial beams in a
complete (four quarters) Stern-Gerlach interferome-
ter, are irrelevant here. We can, therefore, adopt the
strategy of [11] and disregard the x-motion totally
while regarding the y-motion as classical. Then y is
replaced by vt and the spatial y-dependence of the
magnetic field is thereby effectively turned into a time-
dependence. For the z-motion and the spin evolution
we then have, as in [1], the Hamilton operator
2m oz - F(t) ZGZ , (10)
which has the structure (4) with A standing for the spin
vector operator. The term <?(r) az is the magnetic en-
ergy at the center of the magnets (where z = 0) and
F(t) oz is the force produced by the z-inhomogeneity
of the magnetic field.
The net momentum transferred to a spin-up atom is
Ap(t) = \dt'F(t'), (11)
o
and its accumulated z-displacement becomes
Az(t) = J dt' Ap(t')/m = } dt'F(t'), (12)
o o m
where f = 0 is the instant when the atom enters the
magnetic field. For example, the idealized force
F0 for 0 <t<T0,
F(t)={-F0 for T0<r<3T0,
0 for 3
T0
< t,
(13)
where T0 is the time to traverse one of the three quar-
ters, yields
F„t for 0 < t < T0 ,
Ap(t)={F0(2T0-t) for T0<r<3T0,
-
F0 T0 for 3 T0<t
(14)
1180 B.-G. Englert et al. • Surrealistic Böhm Trajectories
and for 0 < t < T0 ,
Az(t) = ^~{2T2-(t-2T0)2 for T0<t<3T0,
(7
T0 — 21)
T0 for 3 T0 < t (15)
Hence the up and down beams intersect around time
t = ~T0. At time t = 2T0 the (macroscopic) distance
between the two beams is 2 \Az{2 T0)| = 2F0 T02/m,
and if the position of the screen corresponds to
t = 5 T0, the spots produced by the spin-up and spin-
down atoms are 2 \Az(5 T0)| =
3
F0 T^jm apart.
At the initial time
f
= 0 the magnetic atom is sup-
posed to be in a minimum uncertainty state. (This is
an innocuous simplifying assumption; in an atomic
beam minimum uncertainty states are hard to come
by.) To take advantage of the symmetry of the set-up,
we take the atom initially polarized in the x-direction
and choose the spatial wave function symmetric to the
z = 0 plane. Thus the two components referring to the
eigenvalues
az = ±1
of the spin operator
az
are initially
>A+(0, z) = <M0, z)
= (2n)~ 1/4(2<5z0)~1/2
(16)
exp 2 BZR
where
Ö
z0 is the initial spread in z, which uncertainty
is small compared to the maximal separation of the
beams. The corresponding spread in momentum is
dp = jh/ö z0. In a reasonable experiment one must
avoid that the natural spreading happens too fast. In
the present context this requires that the characteristic
time mözjöp is at most of the order of T0.
At later times we get, by solving the Schrödinger
equation,
iP±(t, Z) = (2TT)
-1/4 2 ( <5z0 + i — dp
-1/2
(17)
•
exp 1 (z + zlz(r))2 i / _i
4 öz0(.öz0 + itöplm)±hZAp{t)+2*{t)
up to an irrelevant common z-independent phase fac-
tor. The quantity
<P(t)
= -\dt'<(?(t')
h o
is the accumulated Larmor precession angle. It equals
Qt for 0 <t<T0,
<P(t)={ Q{2 T0— t) for T0<r<3T0, (19)
-ßTn for 3 Tn < t
in the case of
Q for 0 <t<T0,
---Q for T0<t<3T0, (20)
0 for 3 T0 < t.
The two components (17) interchange under the spa-
tial reflection according to
iJ/+{t,z) = il/_(t, -z), (21)
which is the essential symmetry also possessed by the
components ij/K and in (9). Consequently, the
probability density
Q(t,z)=\^ + (t,z)\2+\lP_(t,z)\2
and the z-component
1
j(t, z) = — Re
m
' ,*h 0 , ,* h 8 , "
l OZ l oz
(22)
(t,z) (23)
of the current are symmetric and antisymmetric, re-
spectively:
g(t, -z) = g(t,z), j(t,-z) = -j(t,z). (24)
The resulting Böhm velocity field (z-component again)
is then antisymmetric, too,
.
2
+\ ML-tA,<*(->>-
L m \m öz(t)J J \(öz(t))2J
= - v(t, -z). (25)
Thus it vanishes on the z = 0 plane with the now famil-
iar consequence that the Böhm trajectories do not
cross this plane. In (25), öz(t) = |/(<5z0)2 + (t Sp/m)2 is
the spread in z at time t.
The Bohm-mechanical equation of motion for the
trajectory Z(t), viz.
— Z(t) = v(t,Z(t)),
at
(18) is implicitly solved by
Z(R) Z0
J dz g(t, z) = f dz q (0, z),
(26)
(27)
where Z0= Z(0) is the initial condition. A final con-
dition could be employed analogously and would,
indeed, emphasize the retrodictive nature. In view of
1181 B.-G. Englert et al. • Surrealistic Böhm Trajectories
screen
Fig. 5. Böhm trajectories through the incomplete Stern-Ger-
lach interferometer (magnets not drawn). The dashed curves
represent ±Az of (15); the solid curves are typical Böhm
trajectories, which do not cross the z
—
0 plane. The arrows
indicate the spin vectors of (29).
(22) with (17), (27) is equivalent to
Jz(r) + Z(t)
dz exp 1
2 \&(t)
Az(t)-Z(t)
Z0
r dz
bZf exp
-Z0
L
2\Sz0
(28)
the components up to this cross-over region, beyond
which they follow the other. These statements are
illustrated in Figure 5.
The spin vectors indicated in this figure represent
what is produced by (6) and (17),
COS (p(t) \
fZ(t)Az(t)s
cosh -S(t) = V (&W)2
with the effective Larmor angle
sin
cp(t)
. ufZ(t)Az(t)\
smh —
V J
(29)
cp(t)
= <P(t)-- Z(t)
h (30)
Outside the cross-over region, the argument of the
hyperbolic functions is large, so that, practically
speaking,
S
=
[ 0 j (31)
\(sgn Z0) (sgn Az)/
there. In the cross-over region, the direction of S is
reversed, rapidly though continuously.
We now supplement the incomplete Stern-Gerlach
interferometer of Figure 4 with two one-bit which-
way detectors located where
|
Az
|
is largest, that is just
after two of the three quarters, see Figure 6. When the
detectors are included into the formulation, we have
to replace the two-component wave function
W =
by a four-component one, i.e.
These integrals can, of course, be evaluated in terms of
the error function, but that hardly adds transparence.
Note that both integrands are even functions of z,
which has the immediate consequence that the sign of
Z(t) is that of Z0, quite independent of the sign of
Az(t). So the explicit expression (28) confirms the re-
sult of the symmetry consideration: no Böhm trajec-
tory crosses the z = 0 plane. Further, if Z0 does not
exceed 3z0 by much, as is the most probable situation,
and if, in addition, Az(t) is much larger than öz(t),
signifying a macroscopic separation between and
then \Z(t)\ does not differ markedly from \Az(t)\.
Thus, whereas the spatial wave function components
1\)± intersect where Az(t) changes sign, each Böhm
trajectories first (roughly) follows the center of one of
<A+ lno>
(32)
(33)
at the times before the atom traversed (one of) the
detectors, and by
u/ _
af tor
<A+ I no
>
<A-I"e°s> (34)
after the transition. Since the which-way detectors
have no noticeable influence on the center-of-mass
wave functions tj/±, both the density q and the current
j as well as the velocity field v of
(25)
remain unaltered.
Consequently, the (retrodicted) Böhm trajectories are
identical with the ones found without the which-way
detectors installed. There is only one difference. The
1182 B.-G. Englert et al. • Surrealistic Böhm Trajectories
yes/no
i i
screen
one - bit
detectors
no/yes
B
*y.t
Fig. 6. Incomplete Stern-Gerlach interferometer with one-bit
which-way detectors (magnets not drawn). The dashed
curves represent ±Az(t) of (15). The solid curve shows a
Böhm trajectory through the lower detector, corresponding
to an atom that reached the lower region A through the
upper detector. The arrows indicate the spin vectors of (29)
and (35), respectively.
spin vector "after" is not given by (6) and (29) but by
/ ° \ I ° \
Rafter = 0 = 0
\sgn(|iA+|2- |iA_|2)/ \(sgn Z0)(sgn Az)]
(35)
which now applies in the cross-over region, too. Thus,
spin-flip now appears to happen instantaneously at
the moment when Az changes sign. For an adherent of
Bohmian mechanics this is nothing to worry about,
though, because for him spin is a property of the wave
function and not carried by the atom itself.
With the which-way detectors in place we can again
compare the Böhm trajectory with what is known
about the track of the atom. In the lower region A of
Figure 6. ip_ vanishes effectively and only tj/+ con-
tributes to the density g. Therefore, as implied by (35),
if an atom hits the lower screen in region A, it has left
a trace in the upper detector. Its Böhm trajectory, in
contrast, goes through the lower detector. Likewise,
an atom arriving in the upper region B went through
the lower detector, but its Böhm trajectory through
the upper one. The actual tracks and the Böhm trajec-
tories could not be more at variance than that.
A supporter of Bohmian mechanics woulds insist
that the atom went along its Böhm trajectory through
one of the detectors, but left its mark in the other one.
Thus, he concludes, the which-way detectors do not
deserve their name. The usual which-way detection, he
declares, is an illusion. But, in arguing this way,