Table Of ContentThe Pilot-Wave Perspective on Quantum Scattering and Tunneling
Travis Norsen
Smith College
Northampton, MA 01060
[email protected]
(Dated: October17, 2012)
The de Broglie - Bohm “pilot-wave” theory replaces the paradoxical wave-particle duality of or-
dinary quantum theory with a more mundane and literal kind of duality: each individual photon
orelectron comprises aquantumwave (evolvingin accordance with theusual quantummechanical
waveequation)and aparticlethat,undertheinfluenceofthewave,tracesoutadefinitetrajectory.
The definiteparticle trajectory allows the theory to account for the results of experimentswithout
2 the usual recourse to additional dynamical axioms about measurements. Instead one need simply
1 assume that particle detectors click when particles arrive at them. This alternative understanding
0 ofquantumphenomenaisillustratedherefortwoelementarytextbookexamplesofone-dimensional
2 scattering and tunneling. Weintroduce a novelapproach to reconciling standard textbook calcula-
t tions (made using unphysical plane-wave states) with the need to treat such phenomena in terms
c of normalizable wave packets. This allows for a simple but illuminating analysis of the pilot-wave
O
theory’s particle trajectories, and an explicit demonstration of the equivalence of the pilot-wave
theory predictions with those of ordinary quantumtheory.
6
2
I. INTRODUCTION issimplytheusualquantummechanicalwavefunctionΨ
]
h obeying the usual Schr¨odinger equation:
p
- The pilot-wave version of quantum theory was orig- ∂Ψ ~2 ∂2Ψ
t inated in the 1920s by Louis de Broglie, re-discovered i~ = +V(x)Ψ. (1)
n ∂t −2m ∂x2
a and developedin 1952by David Bohm, and championed
u in more recent decades especially by John Stewart Bell. The particle position X(t) evolves according to
q [1] Usually described as a “hidden variable” theory, the
[ pilot-wave account of quantum phenomena supplements dX j
1 the usual description of quantum systems – in terms of dt = ρ(cid:12) (2)
v wavefunctions–withdefiniteparticlepositionsthatobey (cid:12)x=X(t)
(cid:12)
5 adeterministicevolutionlaw. Thiscanbeunderstoodas (cid:12)
where
6 the simplest possible account of “wave-particleduality”:
2 individual particles (electrons, photons, etc.) manage to ~ ∗ ∂ ∂ ∗
7 j = (Ψ Ψ Ψ Ψ ) (3)
behavesometimeslikewavesandsometimeslikeparticles 2mi ∂x − ∂x
.
0 because each one is literally both. The energy usually
1 associated with the “particle” (e.g., a single electron) is (the usual quantum probability current) and ρ = Ψ2
2 carried by a literal, pointlike particle that follows a def- (the usual quantum probability density) satisfy the c|on|-
1
inite trajectory. This trajectory, however, is not at all tinuity equation
:
v classical;it is instead determined by the structure of the
Xi associated quantum wave which guides or “pilots” the ∂ρ + ∂j =0. (4)
particle. ∂t ∂x
r
a Themainvirtueofthetheory,however,isnotitsdeter- Here we consider the simplest possible case of a single
ministic character, but rather the fact that it eliminates spinless particle moving in one dimension. The general-
theneedforordinaryquantumtheory’s“unprofessionally izationsformotionin3Dandparticleswithspinaretriv-
vagueandambiguous”measurementaxioms. [2]Instead, ial: ∂/∂x and j become vectors, and the wave function
in the pilot-wave picture, measurements are just ordi- becomesamulti-componentspinorobeyingtheappropri-
nary physical processes, obeying the same fundamental ate wave equation. For a system of N particles, labelled
dynamicallawsasotherprocesses. Inparticular,nothing i 1,...,N , the generalization is also straightforward,
like the infamous “collapse postulate” – and the associ- th∈ou{gh it sh}ould be noted that Ψ – and consequently
atedCopenhagennotionthatmeasurementoutcomesare also~j and ρ – are in this case functions on the system’s
i
registered in some separately-postulated classical world configuration space. The velocity of particle i at time
– are needed. The pointers, for example, on laboratory t is given by the ratio ~j /ρ evaluated at the complete
i
measuring devices will end up pointing in definite direc- instantaneous configuration; thus in general the velocity
tions because they are made of particles – and particles, of each particle depends on the instantaneous positions
in the pilot-wave picture, always have definite positions. of all other particles. The theory is thus explicitly non-
Inthe“minimalist”presentationofthepilot-wavethe- local. Bell, upon noticing this surprising feature of the
ory(advocatedespecially byJ.S.Bell), the guiding wave pilot-wave theory, was famously led to prove that such
2
non-locality is a necessary feature of any theory sharing termsoffinite-lengthwave packets –ratherthanunphys-
theempiricalpredictionsofordinaryquantumtheory. [4] icalplane-waves–themethodstobedevelopedprovidea
Althoughthefundamentaldynamicallawsinthepilot- novelperspectiveonordinarytextbookscatteringtheory
wave picture are deterministic, the theory exactly repro- as well. In particular, we describe a certain limit of the
duces the usual stochastic predictions of ordinary quan- usual rigorous approach to scattering [3] in which the
tum mechanics. This arises from the assumption that, specifically conceptual advantages of working with nor-
although the initial wave function can be controlled by malizable wave packets can be had without any compu-
the usualexperimentalstate-preparationtechniques, the tational overhead: the relevant details about the packet
initial particle position is random. In particular, for an shapes can be worked out, in this limit, exclusively via
ensemble of identically-prepared quantum systems hav- intuitive reasoning involving the group velocity.
ing t = 0 wave function Ψ(x,0), it is assumed that the The remainderof the paper is organizedas follows. In
initial particle positions X(0) are distributed according the next section, we review the standard textbook ex-
to ample of reflection and transmission at a step potential,
explaininginparticularwhytheuseofplane-wavesispar-
P[X(0)=x]= Ψ(x,0)2. (5)
| | ticularly problematic in the pilot-wave picture and then
This is called the “quantum equilibrium hypothesis” or indicating how the usual plane-wave calculations can be
QEH.Itisthenapurelymathematicalconsequenceofthe salvaged by thinking about wave packets with a certain
already-postulateddynamicallawsforΨ andX thatthe special shape. The next section explores the pilot-wave
particle positions will be Ψ2 distributed for all times: particle trajectories in detail, showing in particular how
| |
P[X(t)=x]= Ψ(x,t)2, (6) the reflectionandtransmissionprobabilitiescanbe com-
| | puted from the properties of a certain “critical trajec-
a property that has been dubbed the “equivariance” of tory” [11] that divides the possible trajectories into two
the Ψ2 probability distribution. [5] To see this, one classes: thosethattransmitandthosethatreflect. Inthe
| |
need simply note that the probability distribution P for followingsectionwe turnto ananalysisofquantum tun-
anensemble of particles movingin a velocityfield v(x,t) neling through a rectangular barrier from the pilot-wave
will evolve according to perspective. A brief final section summarizes the results
∂P ∂ and provides some further perspective on this still-too-
+ (vP)=0. (7) little-knownalternative to Copenhagenquantum theory.
∂t ∂x
Since j and ρ satisfy the continuity equation, it is then
immediately clear that, for v =j/ρ, P =ρ is a solution. II. SCHRO¨DINGER WAVE SCATTERING AT A
Properlyunderstood,theQEHcanactuallybederived POTENTIAL STEP
from the basic dynamical laws of the theory, much as
the expectation that complex systems should typically
Let us consider the case of a particle of mass m inci-
be found in thermal equilibrium can be derived in clas-
dent, from the left, on the step potential
sical statistical mechanics. [6] For our purposes, though,
it will be sufficient to simply take the QEH as an addi-
0 if x<0
tional assumption, from which it follows that the pilot- V(x)= . (8)
(cid:26)V0 if x>0
wave theory will make the same predictions as ordinary
quantumtheoryforanyexperimentinwhichtheoutcome where V >0. The usual approach is to assume that we
0
is registered by the final position of the particle. That aredealingwithaparticleofdefiniteenergyE (whichwe
the pilot-wave theory makes the same predictions as or- assume here is > V ) in which case we can immediately
0
dinaryQMforarbitrarymeasurementsthenfollowsfrom write down an appropriate general solution to the time-
the fact that, at the end of the day, such measurement independent Schr¨odinger equation:
outcomesarealsoregisteredinthepositionofsomething:
think,forexample,oftheflashonascreensomewherebe- Aeik0x+Be−ik0x if x<0
ψ(x)= (9)
hind a Stern-Gerlach magnet, the position of a pointer (cid:26)Ceiκ0x if x>0
on a laboratory measuring device, or the distribution of
ink droplets in Physical Review. [2] where k = 2mE/~2 and κ = 2m(E V )/~2. The
0 0 0
−
In the present paper, our goal is to illustrate all of A-term reprpesents the incident wpave propagating to the
these ideas by showing in concrete detail how the pilot- right toward the barrier. The B-term represents a re-
wavetheorydealswithsomestandardintroductorytext- flected wave propagating back out to the left. The C-
book examples of one-dimensional quantum scattering termrepresentsatransmittedwave. And thereis, byas-
and tunneling. This alternative perspective should be of sumption, no incoming (i.e., leftward-propagating) wave
interest to students and teachers of this material, since to the right of the barrier.
it provides an illuminating and compelling intuitive pic- The transmission and reflection probabilities depend
ture of these phenomena. In addition, because the pilot- onthe relativeamplitudes(A, B,andC)ofthe incident,
wavetheory(forreasonsweshalldiscuss)forcesustore- reflected,andtransmittedwaves. Byimposingcontinuity
memberthatrealparticlesshouldalwaysbedescribedin of ψ(x) and its derivative at x = 0 (these conditions
3
being required in order that the above ψ(x) satisfy the ignore: These scattering wave functions are
Schr¨odinger equation at x=0) one easily finds that not normalizable, so they don’t actually rep-
resent possible particle states. But we know
B k κ
0 0 what the resolution to this problem is: We
= − (10)
A k0+κ0 must form normalizable linear combinations
of the stationary states just as we did for
and
the free particle – true physical particles are
C 2k represented by the resulting wave packets.
0
= . (11)
A k +κ Thoughstraightforwardinprinciple, this is a
0 0
messy business in practice, and at this point
A typical textbook approach is then to calculate the it is best to turn the problem over to a com-
probability current in each region. Plugging Equation puter.” [8]
(9) into Equation (3) gives
Griffithsgoesontocharacterizethe“peculiar”fact“that
~k0 A2 B 2 if x<0 wewereabletoanalyseaquintessentiallytime-dependent
j =(cid:26) ~mκ0(cid:0)C| 2| −| | (cid:1) if x>0 (12) problem...using stationary states” as a “mathematical
m | | miracle”. Some texts go a little further into this “messy
which can be interpreted as follows. For x < 0 there is business”andtreattheproblemofanincident(typically,
bothanincoming probabilityflux proportionalto k A2 Gaussian) packet in some analytic detail. [9]
0
| |
and an outgoing flux proportional to k0 B 2. The reflec- Andthe needtoexaminescatteringintermsof(finite,
| |
tion probability R can be defined as the ratio of these, normalizable) wave packets has long been recognized in
so thepilot-waveliterature,whichhasincludedforexample
numericalstudies oftrajectoriesforGaussianpacketsin-
j k B 2 B 2 (k κ )2
P = reflected = 0| | = | | = 0− 0 . (13) cident on various barriers. [10–12] The use of Gaussian
R j k A2 A2 (k +κ )2
incident 0 0 0 packets, however, tends to obscure the relationship to
| | | |
the standard textbook plane-wave calculation. There is
Similarly,for x>0, there is anoutgoingprobabilityflux
nowaytoexpresstheprobabilitiesP andP ,foranar-
proportionalto κ C 2. The transmissionprobabilityP R T
0| | T rowGaussianpacket,in anything like the simple formof
can be defined as the ratio of this flux to the incident
Equations(13) and (14). And, in the pilot-wavepicture,
flux, so
thecomplicatedstructureofthewavefunctionduringthe
j κ C 2 2k κ scatteringof the packetgivesrise to equally complicated
transmitted 0 0 0
PT = j = k |A|2 = (k +κ )2. (14) particle trajectories. So although one of course knows
incident 0| | 0 0 basedonEquation(6)thatthe ensemble ofpossiblepar-
ThisapproachtocalculatingP andP ishoweversome- ticle trajectories will “follow” ρ = ψ 2, it is impossible
R T | |
what unintuitive, in so far as the wave function involved toindependentlyverify thiswithoutturningtheproblem
isastationarystate. Thismakesitfarfromobvioushow over to a computer.
to understand the mathematics as describing an actual In the next section, we will develop a simple way to
physical process, unfolding in time, in which a particle, verify that, indeed, just the rightfractionof the possible
initially incident toward the barrier, either transmits or particle trajectories end up in the reflected and trans-
reflects. Thesituationisevenmoreproblematic,though, mitted packets. To lay the ground for this, let us turn
fromthepointofviewofthepilot-wavetheory. Here,the now to setting up a simple approach to reconciling the
particle is supposed to have some definite position at all plane-waveand wave packet approaches. This should be
times, with a velocity given by Equation (2). But, with of pedagogical interest even to those with no particular
thewavefunctionasgivenbyEquation(9),theprobabil- interest in the pilot-wave theory. To be clear, what fol-
ity current j for x<0 is positive (since A > B ). And lowsisinnosenseintendedasareplacementforordinary
of course ρ is positive. So it follows im|m|edia|tel|y that, scattering theory. [3] The point is merely to show how,
in the pilot-wave picture, the particle velocity is neces- by considering incident packets with a particular special
sarily positive: if the particle is in the region x < 0, it shape, the reflection and transmission probabilities can
willbemovingtotheright,towardthebarrier. Itcannot be read off in a trivial way from the packet amplitudes
possibly reflect! and widths.
It is easy to see, however, that this is an artifact of Consider then an incident wave packet
theuseofunphysical(unnormalizable)plane-wavestates.
Manyintroductorytextbooksmentioninpassingthepos- ψ(x)=φ(x)eik0x (15)
sibility of instead using finite wave packets to analyze
with a reasonably sharply-defined wave number k , but
scattering. [7] Griffiths, for example, makes the follow- 0
with a special, non-Gaussian, envelope profile φ(x). In
ing characteristically eloquent remarks:
particular, we imagine φ(x) to be nearly constant over a
“This is all very tidy, but there is a sticky spatial region of length L, and zero outside this region.
matterofprinciplethatwecannotaltogether Then, as long as the (central) wavelength λ = 2π/k is
0 0
4
V(x) In terms of such plane-wave packets, the scattering
V0 process can be understood as shown in Figure 1. Let us
chooset=0 to be the time when the leading edge of the
incidentpacketarrivesatx=0. Theincidentpackethas
x
length L and moves at the group velocity v< = ~k /m.
g 0
ψ(x) Thus, the packet’s trailing edge arrives at the origin at
λ0 t<0 t= T = L/v< =Lm/~k . The whole scattering process
g 0
then naturally breaks up into the following three time
x periods:
L 1. Fort<0theincidentpacketispropagatingtoward
ψ(x) the barrier at x=0.
0<t<T
2. For 0< t < T the wave function in some (initially
small,thenbigger,thensmallagain)regionaround
x
x = 0 is well-approximated by the plane-wave ex-
pressions of Equation (9).
ψ(x) 3. For t > T the incident packet has completely dis-
t>T appeared and there are reflected and transmitted
packets propagating away from the barrier on ei-
x ther side.
It is now possible to understand the usual reflection and
L
transmission probabilities in a remarkably simple way.
LT
To begin with, the incident packet should be properly
FIG. 1: Visualization of a “plane-wave packet” interacting normalized. SinceitgoesasAeik0xoveraregionoflength
with the step potential shown in the top frame. The “t<0” L, it is clear that
frame shows a plane-wave packet of length L and (central)
wavelength λ0 incident from the left. (Note that, strictly 1
speaking, a plane-wave packet by definition has L≫λ0; the |A|= √L. (16)
two length scales are inappropriately similar in the figure so
that several other features will be more readily visible.) At Thetotalprobabilityassociatedwiththereflectedpacket
t=0theleadingedgeoftheincidentpacketarrivesattheori-
canbe foundbymultiplying its probabilitydensityρ =
R
gin andleading edgesforthereflectedandtransmitted pack-
B 2 byitslength. Sincetheleadingandtrailingedgesof
ets are produced. At t = T the trailing edge of the incident | |
the reflected packet are produced respectively when the
packetarrivesattheoriginandtrailingedgesforthereflected
leading and trailing edges of the incident packet arrive
and transmitted packetsare produced. For 0<t<T thein-
cident and reflected packets overlap in some (initially small, atthebarrier–andsincethereflectedpacketpropagates
thenbigger, thensmallagain) region totheleft oftheorigin. in the same region as the incident packet so their group
This is depicted in the “0 < t < T” frame. Note that the velocitiesarethesame–thereflectedpackethasthesame
amplitudes of the reflected and transmitted waves are deter- length, L, as the incident packet. Hence
mined by the usual boundary-matching conditions imposed
at x = 0. Finally, for t > T the reflected and transmitted B 2 (k κ )2
packetspropagate away from the origin. Note that while the PR =|B|2L= |A|2 = (k0−+κ0)2 (17)
wavelength and packet length of the reflected wave matches | | 0 0
thoseof theincidentwave, thewavelength andpacketlength
where in the last step we have used Equation (10) to
of the transmitted wave are respectively greater and smaller
relatethe amplitude B ofthe reflectedpackettothe am-
than those of the incident wave, owing to the different value
of the potential energy tothe right of theorigin. plitudeAoftheincidentone. Theresulthereisofcourse
in agreement with Equation (13).
The transmission probability can be calculated in a
similar way. But here it is crucial to recognize that the
verysmallcomparedtoL(actually,itshouldinaddition group velocity for the x > 0 region, v> = ~κ /m, is
g 0
besmallcomparedtothelengthscaleoverwhichφtran- smaller than the group velocity in the x < 0 region.
sitions to zero at the edges of the packet) the envelope Thus,the positionofthe leadingedgeofthe transmitted
function φ will maintain its shape and simply drift at packet when the trailing edge is created at x = 0, i.e.,
the appropriate group velocity. Let us call this type of the length of the transmitted packet, is only
packeta“plane-wavepacket”;itsconceptualandanalyt-
icalmeritliesin the factthat, whereit doesn’tvanish,it L =v> T =v> L =Lκ0. (18)
is well-approximated by a plane wave. [13] T g · g · v< k
g 0
5
That is, the transmitted packet is shorter, by a factor t
κ /k , than the incident and reflected packets. The to-
0 0
tal probability carried by the transmitted packet is then
easily seen to be
κ κ C 2 4k κ
P =L C 2 =LC 2 0 = 0| | = 0 0 (19)
T T | | | | k k A2 (k +κ )2
0 0 0 0
| |
again in agreement with the earlier result. Note though,
onthisanalysis,howtheperhaps-puzzlingfactorofκ /k
0 0
in Equation(14)admits anintuitively clearoriginin the
relative lengths of the incident and transmitted packets. x
Eveninthecontextofconventional,textbookquantum
theory, the “plane-wave packet” approach has several
pedagogical merits. First, it allows the scattering pro-
cess to be understood and visualized as a genuine, time-
dependent process. Second, the reflection and transmis-
sion probabilities can be calculated without recourse to
thesomewhatcrypticandsomewhathand-wavingdevice
of taking ratios of certain hand-picked terms from the FIG.2: Space-timediagramshowingarepresentativesample
probabilitycurrentsoneachside. Andfinally,theexplicit of possible particle trajectories for the case of a plane-wave
discussion of wave packets helps make clear that the re- packetincidentfromtheleftonasteppotential. Theleading
andtrailing edgesof thevariouspacketsareindicated bythe
sultsofthecalculation–inparticulartheexpressionsfor
dashedgreylines. Particletrajectoriesareshowninblack. In
P and P – can be expected to be accurate only under
R T general,theparticlesimplymovesatthegroupvelocityalong
the conditions (e.g., L λ ) assumed in the derivation.
≫ 0 with the packet that is guiding it. In the (here, triangular)
And of course the over-archingpoint is that all of this is
overlapregion,however,theparticlemovesmoreslowly. This
accomplishedwhilestillusingthemathematicallysimple
givesrisetoabifurcationofthepossibletrajectories,between
plane-wave calculations: there is no particularly “messy those which arrive at the origin before being caught by the
business” and no need “to turn the problem over to a incidentpacket’strailingedge(andthusendupmovingaway
computer.” with thetransmitted packet),and thosewhich are caught by
Inthe followingsection,wewillseetheparticularutil- the incident packet’s trailing edge (and thus end up moving
ity of the “plane-wave packet” approach in the context away with thereflected packet).
of the alternative pilot-wave picture.
is
III. PARTICLE TRAJECTORIES IN THE
PILOT-WAVE THEORY v = jI = ~mk0|A|2 = ~k0. (21)
I ρ A2 m
I
| |
In the pilot-wave theory, the particle velocity is de- Note that this is the same as the group velocity of the
termined by the structure of the wave function in the incident packet. Thus, the particle will approach the
vicinity of the particle, according to Equation (2). By barrierwiththe incidentpacket,indeed keepingits same
considering a plane-wave packet as discussed in the pre- position relative to the front and rear of the packet as
vious section, we can see that there are several possible both things (the wave and the particle) move.
regions in which the particle may find itself. Let us con- At some point – the exact time and place depending
sider these in turn. on its random initial position within the incident packet
To begin with, initially, the particle will be at some – the particle will encounter the leading edge of the re-
(random) location in the incident packet. Since, by as- flected packet. It will then begin to move through the
sumption, the packet length L is very large compared to “overlap region” where both the incident and reflected
the length scaleassociatedwith the packet’s leading and waves are present:
trailing edges,the particle is overwhelminglylikely to be
at a location where the wave function in its immediate ψO(x)=Aeik0x+Be−ik0x. (22)
vicinity is given by
Its velocity in this overlapregion will be given by
ψI(x)=Aeik0x. (20) v = jO = ~mk0 |A|2−|B|2 (23)
O ρ A2+ B 2+(cid:0)2A B cos((cid:1)2k x φ)
(Here and subsequently we omit for simplicity the time- O | | | | | || | 0 −
dependent phase of the wave function, which plays no where φ is the complex phase of B relative to A – zero
role.) It follows immediately that the particle’s velocity in the case at hand. Here the right hand side is to be
6
evaluated at each moment at the instantaneous location
t
of the particle. This first-order differential equation for
X(t)iseasilysolved–moreprecisely,itistrivialtowrite
an exact expression for t(X) – but it is already clear Critical
Trajectory
fromtheaboveexpressionthattheparticle’svelocitywill
oscillate around an average“drift” value given by
~k0 A2 B 2 T−τ
v¯ = | | −| | . (24)
O m A2+ B 2 T
| | | |
Since we are assuming that the packet length L λ =
0
≫
2π/k , the particle’s velocity will (with overwhelming
0
probability)oscillate above and below this averagevalue
τ
many many times while it moves through the overlap
region. It is thus an excellent approximation to simply x
PRL PTL
ignore the oscillations and treat the particle as moving
through the overlapregion with a constant velocity v¯ . L
O
There are two possible ways for the particle to escape
from the overlapregion. If it arrivesat the origin,it will FIG. 3: The critical trajectory, which arrives at the apex of
cross over into the region where only the transmitted the triangular overlap region on this space-time diagram, di-
wave vides trajectories that transmit from those that reflect. The
possible trajectories are distributed with uniform probabil-
ψ (x)=Ceiκ0x (25) ity density throughout theincident packet,so thefraction of
T
the total length L of the packet that is in front of the crit-
ispresent. Itwillthencontinuetomovetotherightwith ical trajectory represents the transmission probability, PT.
a velocity Equivalently, the critical trajectory is a distance PT ·L be-
hind the incident packet’s leading edge. From t = 0, it cov-
jT ~κ0 ers exactly half this distance before encountering the lead-
v = = (26)
T ρ m ing edge of the reflected packet. This occurs after a time
T
τ =(PT·L/2)/(~k0/m). Intraversingtheoverlapregion,the
matching the group velocity of the transmitted packet. criticaltrajectorythenmovesthroughtheremainingdistance
The other possibility is that, while still in the overlap PT ·L/2. Thetimeit takesisT −τ,whereT =L/(~k/m) is
region, the trailing edge of the incident packet catches thetimeneededforthetrailingedgeoftheincidentpacketto
and surpasses the particle. It will then subsequently be arrive at the origin. Equation (29) in the main text follows
immediately by dividingthis distance by this time.
guided exclusively by the reflected wave
ψ (x)=Be−ik0x. (27)
R
Its velocity continue to describe the particles’ probability distribu-
j ~k tion for all t. The pilot-wave theory thus reproduces the
v = R = 0 (28) exactstatisticalpredictionsofordinaryQM,butwithout
R
ρ − m
R anyfurtheraxiomsaboutmeasurement: whereasinordi-
will match that of the reflected wave packet, with which nary QM, for example, the transmission probability PT
it will propagate back out to the left. (equalto theintegralofρ acrossthe transmittedpacket)
It is helpful to visualize the family of possible parti- represents only the probability that the particle will ap-
cle trajectories on a space-time diagram; see Figure 2. pear there if a measurement is made, in the pilot-wave
Notice that a particle which happens to begin near the theoryPT insteadrepresentstheprobabilitythatthepar-
leading edge of the incident packet will definitely trans- ticle really is there in the transmitted packet, ready to
mit, while particles beginning nearer the trailing edge of trigger a “click” in a detector should such a device hap-
the incident packet will definitely reflect. pen to be present.
Although the dynamics here is completely determinis- As a concrete illustration of the equivariance prop-
tic, the theory makes statistical predictions because the erty that guarantees the equivalence between the pilot-
initial position of a particular particle within its guid- wavetheory’sstatisticalpredictionsandthoseofordinary
ing wave is uncontrollable and unpredictable. Recall the quantumtheory,letusderivethereflectionandtransmis-
“quantum equilibrium hypothesis” (QEH) according to sion probabilities directly from the particle trajectories
which, for an ensemble of identically-prepared systems andshowthatweindeedarriveagainatthesameexpres-
withinitial wavefunctionΨ(x,0), the initialparticle po- sions we found before by considering only the quantum
sitions will be random, with distribution given by Equa- wave. Thekeyhereistoexamineinparticularthe“criti-
tion (5). It then follows, from the “equivariance” prop- caltrajectory”whichdivides thosetrajectorieswhichre-
erty described in the introduction, that ρ = ψ 2 will sultintransmission,fromthosewhichresultinreflection.
| |
7
This critical trajectory, by definition, arrives just at the than the width a ofthe potential energybarrier. Then –
apex of the triangular overlap region of Figure 2: parti- letting now κ = 2m(V E)/~2 – the wave function
0 0
−
clesontheleading-edgesideofthecriticaltrajectorywill in the vicinity of tphe barrier will be given by
necessarily transmit, while particles on the trailing-edge
side of the critical trajectory will necessarily reflect. Aeik0x+Be−ik0x if x<0
A zoomed-in image of the overlap region from Figure ψ(x)= Ce−κox+Deκ0x if 0<x<a (33)
2 is shown in Figure 3. As explained in the caption, Feik0x if x>a
the critical trajectory moves through the overlap region
for the overwhelming majority of the time when ψ(x)
across a distance P L/2 where P is the transmission
T T
· nearthebarrierisnonzero. (Inparticular,ψ(x)willdiffer
probability. It does this in a time T τ where T =
Lm/~k andτ =(P L/2)/(~k /m). It−followsthatthe substantially from the above expressions just when the
0 T 0
· leading edge of the incident packet first arrives at the
(average)velocity through the overlapregion is
barrier, and again when the trailing edge arrives there.
P L/2 ~k P But this will have negligible effect on our analysis since
T 0 T
v¯O = Lm PTLm = m 2 P . (29) theprobabilityfortheparticletobetooneartheleading
~k0 − 2~k0 − T or trailing edges will be, for very large L, very small.)
Imposing the usual continuity conditions on ψ(x) and
Equating this with the expression for the velocity in the
its first derivative at x=0 and x=a gives a set of four
overlap region worked out in Equation (24) gives
algebraic conditions on the amplitudes A, B, C, D, and
~k P ~k A2 B 2 F. Eliminating C and D allows the amplitudes of the
0 T 0
= | | −| | (30) reflected (B) and transmitted (F) packets to be written
m 2 P m A2+ B 2
− T | | | | in terms of the amplitude A of the incident packet:
which can be solved for P to give
T B (κ2+k2)sinh(κ a)
= − 0 0 0 (34)
P = |A|2−|B|2. (31) A (κ20−k02)sinh(κ0a)−2ik0κ0cosh(κ0a)
T A2
and
| |
Using Equation (10) to put this in terms of the wave F = −2ik0κ0 e−ik0a.
numbers k0 and κ0 of course gives back precisely Equa- A (κ20−k02)sinh(κ0a)−2ik0κ0cosh(κ0a)
tion (14) for the transmission probability. And since (35)
PR = 1 PT, Equation (13) is also implied again by Sincethepacketthatdevelopsonthedownstreamsideof
−
the properties of the critical trajectory. thebarriermoveswiththesamegroupvelocityasthein-
Itisofcoursenosurprisethatwearriveatthesameex- cidentpacket,thetransmittedpacketlengthmatchesthe
pressionsforthetransmissionandreflectionprobabilities incident packet length. The total probability associated
by considering the pilot-wave expression for the particle with the transmitted packet – the “tunneling probabil-
velocity in, especially, the crucial overlap region. But it ity” – is thus
isaclarifyingconfirmationofthesenseinwhichthewave
andparticleevolutionsareconsistent asexpressedinthe P = |F|2 = 4k02κ20 (36)
equivariance property. T A2 (κ2+k2)2sinh2(κ a)+4k2κ2
| | 0 0 0 0 0
with the corresponding reflection probability being
IV. TUNNELING THROUGH A
RECTANGULAR BARRIER P = |B|2 = (κ20+k02)2sinh2(κ0a) . (37)
R A2 (κ2+k2)2sinh2(κ a)+4k2κ2
| | 0 0 0 0 0
To illustrate the more general applicability of the
Asbefore,theseresultscanbeunderstoodintermsofthe
methods developed in the previous sections, let us ana-
particle trajectories as well. In general the trajectories
lyze another standard textbook example – the tunneling
areverysimilartothosefromtheearlierexample. While
ofaparticlethroughaclassicallyforbiddenregion–from
theincidentandreflectedpacketsarebothpresenttothe
the pilot-waveperspective. Letthe potentialbe givenby
leftofthebarrier,anoverlapregionissetupinwhichthe
motion of the incoming particle is slowed. The particle
V if 0<x<a
V(x)= 0 (32) velocity in this region is again described by Equation
(cid:26)0 otherwise
(23), although now there is a nontrivial complex phase
between the amplitudes B and A:
and let the particle, with a reasonably sharply defined
energy E < V , be incident from the left. As before,
we take the ini0tial wave function Ψ(x,0) to be a plane- φ=tan−1 2k0κ0cosh(κ0a) . (38)
(cid:18)(κ2 k2)sinh(κ a)(cid:19)
wavepacketwith (central)wavelengthλ0 =2π/k0 (with 0− 0 0
k = 2mE/~2) and length L λ . In addition, we The average drift velocity through the overlap region,
0 0
≫
assumpe here that the packet length L is much greater however, remains as in Equation (24), so the analysis
8
t The fact that the relative complex phase of D and C
is not zero is crucial: if this were zero, the probability
current j and hence the particle velocity would vanish,
and it would be impossible for the particles to tunnel
across the barrier. Instead, though, we have that
2~κ
j (x)= 0 C 2sin(θ)e−2κ0a (43)
CFR
m | |
and
ρ (x)
CFR =e−2κ0x+e−4κ0ae2κ0x+2e−2κ0acos(θ) (44)
C 2
| |
x
so that the particle velocity is given by
~κ sin(θ)
0
v (x)= . (45)
CFR
m cos(θ)+cosh[2κ (a x)]
FIG.4: Space-timediagramshowingarepresentativesample 0
−
of possible particle trajectories for the case of a plane-wave
Thus, particles speed up as they cross (from x = 0 to
packet incident from the left on a rectangular potential bar-
x=a) over the CFR.
rier. Particlesbeginningneartheleadingedgeoftheincident
Figure4displaysthebehaviorofarepresentativesam-
packet will tunnel through the barrier and emerge on thefar
side. Theunusualacceleratingcharacterofthetrajectoriesin ple of particle trajectories. The overallpattern is similar
the (here, grey shaded) classically-forbidden region – indeed tothecaseofscatteringfromthesteppotential: particles
themerepresenceoftrajectorieshere–reflectsthehighlynon- that begin near the trailing edge of the incident packet
classical nature of the law of motion for the particle. (Note will be swept up by the reflected packet before reaching
that the analysis in the main text assumes that the incident x=0,whilethosewhichbeginnearertheleadingedgeof
packetlengthLisverylargecomparedtothebarrierwidtha. the incident packet will reach the barrier, tunnel across
This separation of length scales is not accurately depicted in
it, and emerge with the transmitted packet.
theFigure, inorderthat thequalitativenatureof thetrajec-
toriesinallrelevantregionscanbevisualizedsimultaneously.)
V. DISCUSSION
Wehaveanalyzedtwostandardtextbookcasesofone-
surroundingFigure3stillappliesandwehaveagainthat dimensional quantum mechanical scattering and tunnel-
the transmission (or here, tunneling) probability as de- ingfromthepointofviewofthedeBroglie-Bohmpilot-
termined by the critical trajectory is wavetheory. Inparticular,we haveshownhow the stan-
dardtextbookexpressionsforthereflectionandtransmis-
A2 B 2
sion/tunnelingprobabilities–calculatedusing infinitely-
P = | | −| | (39)
T A2 extended plane-wave states – can instead be understood
| |
as arising from a certain type of idealized, non-Gaussian
in agreement with the result arrived at by considering incident wave packet. We then took advantage of this
justthe waves. This againconfirmsthatthedistribution “plane-wave packet” approach to generate a tractable,
of possible particle trajectories evolves in concert with indeed quite simple, picture of how the particle trajecto-
the waveintensityρsuchthatEquation(6)remainstrue ries in the pilot-wave theory develop.
at all times.
Itishopedthattheplane-wavepacketapproachmight
The nature of the pilot-wave theory particle trajecto- prove clarifying for students learning standard textbook
ries in the classically-forbidden region (CFR) is of some quantum mechanics. It is also hoped that the simplifi-
interest. The wave function in the CFR goes as cation (compared to numerical simulations using Gaus-
sianpackets)affordedbytheplane-wavepacketapproach
ψ (x)=Ce−κ0x+Deκ0x (40)
CFR might – so to speak – lower the energy barrier that cur-
rentlypreventsthepilot-wavetheoryfromtunnelinginto
where the four algebraic conditions mentioned just prior
the undergraduate physics curriculum.
to Equation (34) imply that
By way of summarizing the conceptual, pedagogical,
historical, and above all physical reasons for taking the
D (κ +ik )2
= 0 0 e−2κ0a =eiθe−2κ0a (41) pilot-wavetheoryseriously,one candono better thanto
C κ20+k02 quotefromJ.S.Bell’sdiscussionofthedouble-slitexper-
iment:
where the relative phase θ is given by
“Whilethefoundingfathersagonizedoverthe
θ =2tan−1(k /κ ). (42) question
0 0
9
‘particle’ or ‘wave’ vidualparticles(theiractualpositions)andso
also for macroscopic assemblies of particles.
de Broglie in 1925 proposed the obvious an-
“The de Broglie - Bohm synthesis, of parti-
swer
cle and wave, could be regarded as a precise
illustration of Bohr’s complementarity ... if
‘particle’ and ‘wave’.
Bohrhadbeenusingthiswordintheordinary
way. This picture combines quite naturally
Isitnotclearfromthesmallnessofthescintil-
both the waviness of electron diffraction and
lationonthescreenthatwehavetodowitha
interferencepatterns,andthesmallnessofin-
particle? Andisitnotclear,fromthediffrac-
dividual scintillations, or more generally the
tion and interference patterns, that the mo-
definitenatureoflargescalehappenings. The
tionofthe particle is directedby a wave? De
deB-Bpictureisalso,bytheway,quitedeter-
Broglie showed in detail how the motion of
ministic. Theinitialconfigurationofthecom-
a particle, passing through just one of two
bined wave-particle system completely fixes
holes in [the] screen, could be influenced by
the subsequent development. That we can-
waves propagating through both holes. And
not predict just where a particular electron
so influenced that the particle does not go
will scintillate on the screen is just because
where the waves cancel out, but is attracted
we cannot know everything. That we cannot
to where they cooperate.
arrange for impact at a chosen place is just
“Thisideaseemstomesonaturalandsimple, because we cannot control everything.” [14]
to resolve the wave-particle dilemma in such
a clear and ordinary way, that it is a great
Or, as Bell pointedly asked on another occasion:
mystery to me that it was so generally ig-
“Whyisthepilotwavepictureignoredintext
nored. Ofthefounding fathers,onlyEinstein
books? Should it not be taught, not as the
thoughtthatdeBrogliewasontherightlines.
onlyway,butasanantidotetotheprevailing
Discouraged, de Broglie abandoned his pic-
complacency? To show that vagueness, sub-
tureformanyyears. Hetookitupagainonly
jectivity,andindeterminismarenotforcedon
when it was rediscovered, and more system-
us by experimental facts, but by deliberate
atically presented, in 1952, by David Bohm.
theoretical choice?” [15]
Inparticular Bohmdevelopedthe picture for
manyparticlesinsteadofjustone. Thegener-
alizationis straightforward. Thereis no need Needless to say, the present author agrees entirely with
inthispicturetodividetheworldinto‘quan- Bell. It is hoped that the present paper will provide a
tum’ and ‘classical’ parts. For the necessary simpleconcretewayforthealternativepilot-wavepicture
‘classicalterms’areavailablealreadyforindi- of quantum phenomena to be introduced to students.
[1] An English translation of Louis de Broglie’s 1927 pilot- D. Tausk, and N. Zanghi, “Bell’s Theorem” at
wave theory can be found in G. Bacciagalluppi and scholarpedia.org/article/Bell%27s_theorem
A. Valentini, Quantum Theory at the Crossroads, Cam- [5] D. Du¨rr, S. Goldstein, and N. Zanghi, “Quantum Equi-
bridge,2009.DavidBohm’s1952re-discoveryofthethe- libriumandtheOriginofAbsoluteUncertainty”,Journal
ory is presented in “A Suggested Interpretation of the of Statistical Physics, 67, pp.843-907
QuantumTheoryintermsofHiddenVariables,IandII”, [6] Ibid.SeealsoM.D.Towler,N.J.Russell,andA.Valentini,
PhysicalReview,85(1952),pp.166-193.Amorecontem- “Timescales fordynamicalrelaxation totheBornrule”,
poraryoverview,withfurtherreferences,canbefoundat Proceedings of the Royal Society A, 468 (2012), pp. 990-
plato.stanford.edu/entries/qm-bohm. 1013
[2] J.S.Bell,“BeablesforQuantumFieldTheory”(1984)in [7] J.S. Townsend, Quantum Physics: A Fundamental Ap-
Speakable and Unspeakable in Quantum Mechanics, 2nd proach to Modern Physics, University Science Books,
ed., Cambridge, 2004. Sausalito, California, 2010
[3] See for example R.G. Newton, Scattering Theory of [8] D.J. Griffiths, Introduction to Quantum Mechanics,
Waves andParticles, 2ndEd.,Springer,Berlin/Heidel- Prentice Hall, New Jersey, 1995, p.58
berg/NewYork,1982; M.ReedandB.Simon,Methods [9] R. Shankar, Principles of Quantum Mechanics, Second
of Modern Mathematical Physics III: Scattering Theory, Edition, Springer, 1994
San Diego, Academic Press, 1979. [10] C. Dewdney and B.J. Hiley, “A Quantum Potential De-
[4] J.S. Bell, “On the Einstein-Podolsky-Rosen Para- scription of One-Dimensional Time-Dependent Scatter-
dox”, Physics 1 (1964) 195-200. Reprinted in Bell, ingFromSquareBarriersandSquareWells”Foundations
2004, op cit. For a contemporary systematic re- of Physics, Vol.12, 1982, pp.27-48.
view of Bell’s Theorem, see S. Goldstein, T. Norsen, [11] D.BohmandB.J.Hiley,The Undivided Universe,Rout-
10
ledge, London and New York,1993, pp. 73-8 69, 042702 (2004)
[12] P. Holland, The Quantum Theory of Motion, Cam- [14] J.S. Bell, “Six Possible Worlds of QuantumMechanics,”
bridge University Press, 1993, pp. 198-203 (and refer- 1986, reprinted in Bell, 2004, op cit. (Note: ellipsis in
ences therein) original.)
[13] For a rather different (alternative) approach to scatter- [15] J.S.Bell,“Ontheimpossiblepilotwave,”1982,reprinted
ing that also uses the terminology “plane-wave packet” in Bell, 2004, op cit.
seeStuartC.Althorpe,“Generaltime-dependentformu-
lationofquantumscatteringtheory”,PhysicalReviewA,
