Lorentz Covariant Quantum 4-Potential and Orbital Angular Momentum for the Transverse Confinement of Matter Waves

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by  Robert J. Ducharme

Lorentz Covariant Quantum 4-Potential and Orbital Angular Momentum for the
Transverse Confinement of Matter Waves
R. Ducharme1 and I. G. da Paz2
1 2112 Oakmeadow Pl., Bedford, TX 76021, USA and
2 Departamento de F´ısica, Universidade Federal do Piau´ı,
Campus Ministro Petrˆonio Portela, CEP 64049-550, Teresina, PI, Brazil
In two recent papers exact Hermite-Gaussian solutions to relativistic wave equations have been
obtained for both electromagnetic and particle beams that include Gouy phase. The solutions for
particle beams correspond to those of the Schr¨odinger equation in the non-relativistic limit. Here,
6 distinctcanonicalandkinetic4-momentumoperatorswillbedefinedforquantumparticlesinmatter
1 wave beams. The kinetic momentum is equal to the canonical momentum minus the fluctuating
0 terms resulting from the transverse localization of the beam. Three results are obtained. First,
2 the total energy of a particle for each beam mode is calculated. Second, the localization terms
couple into the canonical 4-momentum of the beam particles as a Lorentz covariant quantum 4-
y
a potential originating at the waist. The quantum 4-potential plays an analogous role in relativistic
M Hamiltonian quantummechanics to the Bohm potential in the non-relativistic quantum Hamilton-
Jacobiequation. Third,theorbitalangularmomentum(OAM)operatormustbedefinedintermsof
2 canonical momentum operators. It is further shown that kinetic 4-momentum does not contribute
1 to OAM indicating that OAM can therefore be regarded as a pure manifestation of quantum 4-
potential.

]
h PACSnumbers: 41.85.-p,03.65.Pm,03.65.Vf,42.50.Tx
p
-
t
n I. INTRODUCTION theformofthistermdependsonthespecificformulation
a of quantum mechanics under consideration but that all
u
the variants interrelate and have two distinct common
q Experiment shows that beams of particles still behave
[ properties;theyvanishinthefreeparticlelimitandhave
like beams even if only one particle is traveling through
null expectation values.
2 the apparatus at a time [1]. The converse of this argu-
External devices are responsible for collimating and
v ment is that isolated particles can behave like beams.
focusing the particles in a beam. Once a particle has
4 Specifically, it is understood that the wavefunction Ψ
4 passed through these devices, it remains localized but is
for the particle must take account of a full compliment
5 no longer confined. Our solutions describe the localized
of wave beam features such as mode numbers [2], Gouy
7 state of the particles but not the passage of the particle
0 phase [3, 4] and orbital angular momentum [5].
through the devices responsible for confining the beam.
.
1 The purpose of this paper is to explain the localiz- Thebasicstructureofawavebeamcanbeunderstood
0 ing affect of transverse confinement on a beam particle
usingthe Heisenberguncertaintyprinciple[7]thatstates
6 using a quantum 4-potential. The concept of a quan-
uncertaintyinmomentumisinverselyproportionaltoun-
1
tum 4-potential as it is introduced here is similar to the
: certaintyinposition. Inacontinuouswavebeamthereis
v Bohm potential [6] in the the sense that it is a concept
no localizationof the particle along the axis of the beam
Xi extracted from Ψ rather than a representation of a field meaning that each particle can be assumed to have a
separatefromΨ. Thepointofdepartureisthatthequan-
r preciseaxialmomentumandthereforeapreciseaxialve-
a tum 4-potential couples into each individual component
locity v3. The uncertainty in the position of the particle
of the 4-momentum operators for the particle whereas
along the transverse axis is smallest at the beam waist.
thescalarBohmpotentialisacomponentinaHamilton-
It is therefore the size of the waist that determines the
Jacobiequationbelonging to analternate formulationof
uncertainty in the transverse momentum of the particle.
quantum mechanics. The quantum 4-potential therefore
The presence of transverse momentum explains the fact
requires a distinction to be made between the canoni-
beamsspread. Italsoaccountsfortheexistenceoforbital
cal pˆ (µ=1,2,3,4)and kinetic Pˆ 4-momentum for the
µ µ angular momentum in beams.
particle. The canonical (total) 4-momentum is the sum
Linear wave equations have both plane-wave and lo-
ofthe kinetic 4-momentumandthe quantum4-potential
calized solutions [8] often called wave packets [9]. The
term.
wave packet is smallest at the time of an event that lo-
Freeparticleshavenoquantumpotentialbutlocalized calizes the particle then continuously growsin size after-
particles do have it. The signature of a quantum po- wards. One distinguishing characteristic of plane-wave
tential is therefore the appearance a term in a quantum and localized wave functions is the number of 4-position
mechanical equation that generates localization and has dependencies in them. Plane-waves are local functions
no association to external source. It will be shown that that only depend on the position x (i = 1,2,3) of the
i
2
particle at time t. By contrast, localized wave solutions tionofOAMwithamagneticfieldwasalsostudiedinthe
arebilocalfunctions sincetheymustdependonboththe non relativistic context [22]. More recently the effect of
current4-positionoftheparticleaswellasthe4-position the interaction of relativistic electron vortex beam with
X =(X ,cT)of the preceding confinement eventwhere a laser field was studied showing that the beam center is
µ i
andwhenthe sizeofthe wavepacketwasataminimum. shifted and that the shift in the paraxialbeams is larger
It is the bilocal nature of wave packets that permits the than that in the nonparaxialbeams [23, 24]. The results
probability density of finding free particles to have spa- that we are obtaining in this paper could be useful to
tial extension as well as a 4-position. It is also the de- explore the relativistic effects in the properties such as
pendence of Ψ onX aswellas x thatwillenable us to OAM Hall and Zeeman effect resulting, respectively, of
µ µ
define distinct kinetic P and canonicalp 4-momentum the interaction of relativistic scalar (without spin) elec-
µ µ
vectors. tron vortex beam with an electric and magnetic field.
Bateman-Hillion functions [10, 11] are exact local- Further we can similarly solve the Dirac equation to in-
ized solutions of relativistic wave equations that trace clude the effects of the interaction of spin angular mo-
back to early work of Bateman on conformal transfor- mentum (SAM) with a magnetic field.
mations [12]. In two recent papers, exact Bateman- ItwillbeshowninthispaperthattheSchr¨odingerand
HillionsolutionswereobtainedfortheHermite-Gaussian Klein-Gordon equations give the same orbital angular
modes of both electromagnetic [13] and quantum parti- momentum for each scalarmode of a Laguerre-Gaussian
cle [14] beams. These are detailed solutions for parti- beam. To find relativistic corrections to orbital angular
cle beams that include Gouy phase [15–17]. The parax- momentum it is therefore necessary to investigate solu-
ial wave equation [2] for electromagnetic beams and the tions that mix multiple modes. For example, in the case
Schr¨odinger equation for non-relativistic particle beams ofBesselbeamsolutionstotheDiracequationithasbeen
have both been demonstrated as limiting cases of the found [21] the corrective amplitude coefficients take the
Bateman-Hillion method. forma= 1 E0/E)sinθ0 whereE denotesthe energy
−
OnemethodofobtainingBateman-Hillionsolutionsto of eachparticle, E0 is the restenergyand θ0 is the polar
p
a wave equation is to start from an ansatz. In the case angle indicating the divergence of the beam. This re-
of the Klein-Gordon equation the ansatz eliminates the sults in a relativistic correction sa2 to the total angular
second order time derivative reducing the wave equation momentum of each particle with spin s. The correction
to a parabolic form. This resolves problems of negative clearly vanishes in both the non-relativistic (E E0)
→
energiesandnegativeprobabilitydensitiesthatafflictthe andparaxial(θ0 0)limits butcanotherwiseaffectthe
→
unconstrained Klein-Gordon equation. It will be further energies of beam particles in external electric and mag-
showninthispaperthattheprobabilitydensityoffinding netic fields. Another source for relativistic corrections
a particle in a Bateman-Hillionbeam is just Ψ2 similar that may affect OAM is the repulsion between charged
| |
to the Schr¨odinger equation except that the probability particles. This can be a stronger effect than the spin-
density for Bateman-Hillion solutions is also form pre- orbit interaction that could be studied using either the
serving under Lorentz transformations. Klein-GordonorDiracequations. Therepulsionbetween
In this paper a transformation will be made to the charged particles is also known to have a greater affect
Bateman-Hillion solutions of the Klein-Gordon equation on the beam for lower energy particles.
for particle beams to account for an earlier finding [14] ThefactΨ(x ,t,X ,T)depends ontwo4-positionvec-
i i
that the components of the 4-momentum of the parti- tors requires the introduction of a constraint condition
cles must have a shift in them related to the complex [25, 26] to eliminate one of the independent time coor-
shift in the 4-position coordinates needed for the accu- dinates in the calculation of the physical properties for
rate description of any wave beam. This will be shown the beam. As in an earlier paper [14] the solution to be
to facilitate a calculation for the total energy of each applied here is to use Dirac delta function notation to
particle in terms of the rest mass of the particle, the ki- impose a relationship ξ3 v3τ = 0 between the relative
−
netic energy of the propagation of the particle along the position ξ = x X and relative time τ = t T. This
i i i
− −
axis of the beam and the kinetic energy locked up in relatesbacktotheideathatparticlesincontinuouswave
the transverse mass flows. Results will be presented for beams can be assigned a precise axial velocity v3.
both Hermite-Gaussian and Laguerre-Gaussian beams. In Sec. II, we use the Bateman-Hillion ansatz to solve
Laguerre-Gaussian beams are useful to describe the or- the Klein-Gordon equation for a particle that passes
bital angular momentum states of the particle. through a beam waist . In Sec. III, we determine the
After the seminal paper by Bliokh at al introducing Lorentz invariant probability density of finding a parti-
vortex beams carrying OAM for free quantum electrons cle in a Bateman-Hillion beam. In Sec. IV, we calcu-
[18] several experimental [19] and theoretical [20, 21] re- late the kinetic 4-momentum in terms of the canonical
sultswereobtained. PropertiesoftheinteractionofOAM 4-momentum and the localization terms. In Sec. V, we
withanelectricfieldsuchasOAMHalleffectwasstudied calculate the quantum 4-potential. In Sec. VI, we con-
in the non relativistic context [18]. Further the interac- clude our results in a summary.
3
II. BATEMAN-HILLION BEAMS where
Nmn
Consider a beam of particles each having a rest mass κmn = (k3+k4)w02, (9)
m0, a 4-position xµ = (xi,ct) and a 4-momentum pµ =
(p ,E/c). Let us assume each particle passes through a andNmnisaconstant. ThegeneralformofNmn istobe
i
beamwaistwithapositionX atthetimeT. TheKlein- determinedbutitcanbeseenfromcomparisonofeqs. (7)
i
Gordon equation for the wave function Ψ(x ,t,X ,T) and (9) that N00 = 1. It is also readily verified that eq.
i i
representingeachoftheparticlesinMinkowskispacecan (8) is form invariant under the Lorentz transformation
be expressed as equations:
pˆµpˆµΨ= c12(Eˆ2−c2pˆ2i)Ψ=m20c2Ψ, (1) x′3 =(x3−v3τ)γ, τ′ =(τ − vc23x3)γ (10)
where ~ ∂ ~ ∂ k3′ =(k3− vc3k4)γ, k4′ =(k4− vc3k3)γ (11)
pˆ = , Eˆ = , (2)
i
ı ∂xi −ı ∂t where γ = 1/ 1 v2/c2. Applying the transformation
− 3
are the canonical 4-momentum operators, ~ is Planck’s (8) to eq. (3) gives
p
constant divided by 2π and c is the velocity of light.
One approach to solving eq. (1) for a beam is to use Ψmn =Φmn(ξ1,ξ2,ξ3+cτ)
a Bateman inspired ansatz. In an earlier paper [14], the exp[ı(k3+κmn)x3 ıc(k4 κmn)t], (12)
× − −
following trial form was taken as the starting point for
the derivation of the positive-energy Hermite-Gaussian equivalent to making the replacements k3 k3 +κmn
→
beam solutions and k4 k4 κmn. These replacements can be used, in
→ −
turn to reduce eq. (5) to the simplified to the form
ΨOmn =Φmn(ξ1,ξ2,ξ3+cτ)exp[ı(k3x3−k4ct)], (3) ~
Ψ00 ˆjµ Ψ00 = (0,0,k3,k4), (13)
where h | | i m0
whereitcanbeseenκ00hasbeeneliminated. Oneimpor-
ξ =x X , τ =t T, (4)
i i i
− − tant goal of this paper will be to show that there exists
gives the position of each point x relative to the 4- Nmn such that the condition
µ
positionofthe beamwaist, kµ =(0,0,k3,k4) is the wave ~
vector and Φmn are scalar functions. The positive inte- hΨmn|ˆjµ|Ψmni= m0 (0,0,k3,k4), (14)
gers m and n indicate the mode of the beam.
A curious feature of eq. (3) derived in [14] is that it issatisfied. Ifthishypothesisistrue,itimplies ~ k can
m0 µ
leads to the following expression for the particle current be interpreted as the expectation value for the particle
in a Gaussian beam currentinarelativisticbeamthusgivingaclearphysical
~ meaning to kµ.
hΨO00|ˆjµ|ΨO00i= m0(kµ−κ0µ0), (5) Inserting eq. (12) into the Klein-Gordon equation (1)
gives
where
∂2Φ ∂2Φ ∂Φ
Ψ∗ˆjµΨ= 2m10(Ψ∗pˆµΨ−ΨpˆµΨ∗), (6) ∂xm21n + ∂xm22n +2ı(k3+κmn) ∂xm3n
2ı ∂Φ
and κmn =(0,0,κmn, κmn). Here, the axial parameter + c (k4−κmn) ∂mtn =0, (15)
µ −
κmn takes the form
where
1
κ00 = , (7) m2c2
(k3+k4)w02 k42 =k32+2κmn(k3+k4)+ ~02 . (16)
where w0 is the radius of the beam at the waist. It can be seen the unitary transformation (8) has intro-
Eq. (5) suggests that k is related to the expectation
µ duced the term
value of the axial current for a particle in a beam. In
seeking an intuitive definition for kµ we shall now make KTmn =2κmn(k3+k4), (17)
use of the unitary transformation
into this dispersion relationship. The physical interpre-
Ψmn =ΨOmnexp[ıκmn(x3+ct)], (8) tationofKTmn willbediscussedlateroncetherelativistic
4
energy formula for each particle in the beam has been where L|l| are the generalized Laguerre polynomials and
p
derived.
ξ3+cτ
It is instructive to observe that glp(ξ3,τ)=(1+ l +2p)arctan , (27)
| | 2b
(cid:18) (cid:19)
∂ 1 ∂
Φmn = Φmn, (18) is the Gouy phase in terms of the radial Laguerre in-
∂x3 c∂t
dex p and the azimuthal index l that may be positive or
and equivalently negative.
The operator for the axial component of canonical
∂ 1 ∂
Ψ 2 = Ψ 2, (19) OAM can be expressed as
mn mn
∂x3| | c∂t| | ~ ∂
owing the fact Φmn only depends on ξ3 and τ in the Lˆ3 =ξρ×pˆφ = ı ∂ξ (28)
φ
linear combination ξ3+τ. Eqs. (15) and (18) can now
The Laguerre-Gaussianbeam functions (25) can thus be
be combined to obtain the operator relationships
seen to give
pˆ2+pˆ2
pˆ3Φmn =−pˆ4Φmn =−2~(1k3+2k4)Φmn, (20) Lˆ3Ψlp =l~Ψlp, (29)
showingL3 =l~arethe possibleeigenvaluesofOAMfor
These results will prove useful later.
a Laguerre-Gaussianbeam.
Equation(15) canbe solvedanalogouslyto the parax-
ial equation [2] to give
III. PROBABILISTIC INTERPRETATION
Φ = CmHnGw0H √2ξ1 H √2ξ2
mn w m w ! n w !× In this section, the correspondence between the parti-
clecurrent(6)forBateman-Hillionbeamsandthatofthe
ı2b(ξ2+ξ2)
exp 1 2 ıg , (21) Schr¨odinger equation for particle beams will be investi-
(cid:20)w02(ξ3+cτ −ı2b) − mn(cid:21) gatedas means of determining the probabilitydensity of
findingaparticleinaBateman-Hillionbeam. Asastart-
where H and H are Hermite polynomials,
m n
ing point it will be useful to evaluate each component of
w2 the Bateman-Hillion particle current
b= 0 (k3+k4), (22)
4
jmn =Ψ∗ ˆj Ψ . (30)
µ mn µ mn
This leads to
2
w(ξ3,τ)=w0s1+(cid:18)ξ32+bcτ(cid:19) , (23) j1mn = w02[(4ξb3(ξ+3+cτc)2τ)+ξ14b2]m~0|Ψmn|2, (31)
is the beam radius such that w0 =w(0,0) and
jmn = 4b(ξ3+cτ)ξ2 ~ Ψ 2, (32)
gmn(ξ3,τ)=(1+m+n)arctan ξ3+cτ , (24) 2 w02[(ξ3+cτ)2+4b2]m0| mn|
2b
(cid:18) (cid:19)
is tItheisGnooutyabplheatsheaotftahereKlalteiivni-sGticorqduoannteuqmuatpiaornti(c1le).can j3mn = k3+κmn− (ξ23b(+1+cτm)2++n4b)2 m~0|Ψmn|2
(cid:20) (cid:21)
also be usefully solvedincylindricalcoordinatesstarting 2b(ξ12+ξ22)[(ξ3+cτ)2−4b2] ~ Ψ 2, (33)
from the expression − w02[(ξ3+cτ)2+4b2]2 m0| mn|
Ψlp =Φlp(ξρ,ξφ,ξ3+cτ) 2b(1+m+n) ~
×exp ı(k3+κlp)x3−ıc(k4−κlp)t (25) j4mn =(cid:20)k4−κmn+ (ξ3+cτ)2+4b2(cid:21)m0|Ψmn|2
ξeφqu=ivaatlaenn2t(ξt2o,ξe1q).. T(cid:2)h(i1s2g)ivweshere ξρ = pξ12+(cid:3)ξ22 and + 2b(ξw1202+[(ξξ322)+[(ξc3τ)+2c+τ)42b2−]24b2]m~0|Ψmn|2, (34)
where
Φlp = ClLpwGw0 √w2ξρ!|l|L|pl| 2wξ2ρ2!× |Ψmn|2 =(cid:18)CmHwnGw0(cid:19)2Hm2 √w2ξ1!Hn2 √w2ξ2!
ı2bξρ2 8b2(ξ2+ξ2)
exp"w02(ξ3+cτ −ı2b) +ılξφ−ıglp#, (26) ×exp(cid:20)−w02[(ξ3+1cτ)2+2 4b2](cid:21). (35)
5
ThecontinuityequationfortheKlein-Gordonequation istherelativisticprobabilitydensityforfindingaparticle
(1) is in a Bateman-Hillionbeam. This differs from the widely
cited [27] Klein-Gordon probability density
∂j1 ∂j2 ∂j3 1∂j4
∂x1 + ∂x2 + ∂x3 + c ∂t =0. (36) P = j4 (46)
KG
c
Eqs. (33)and(34)enablethisexpressiontoberewritten
due to the fact Ψ is further constrained under the
in the form mn
parabolic equation (15). It is also of interest to notice
∂j1 + ∂j2 + 1 k3 ∂ +k4 ∂ Ψmn 2 =0, (37) ttihoantsPwBhHereiassfoPrm iinsvnaroitanats aunndiseorlaLtoerdenctozmtproannesnfotromfaa-
∂x1 ∂x2 m0 ∂x3 ∂t | | KG
(cid:18) (cid:19) 4-vector.
or equivalently Bateman-Hillionfunctionscanbenormalizedusingthe
integral expression
∂∂xj11 + ∂∂xj22 + m10 (k3+k4)∂∂t|Ψmn|2 =0, (38) +∞ +∞ +∞ Ψ2δ(ξ3 v3τ)dξ1dξ2dτ = 1, (47)
| | − L
havingusedeq. (19). Thisresultreducestothesimplified Z−∞ Z−∞ Z−∞
expression having set the probability of finding the particle in a
beam of length L is 1. This evaluates to
∂j1 + ∂j2 + ∂ ΨS 2 =0, (39)
∂x1 ∂x2 ∂t| mn| CHG = 2 (48)
in the non-relativistic limit where k3 k4 and m0c2 mn sπw02L2m+nm!n!
c~k4. ≪ ≃ for Hermite-Gaussian beams; and
In an earlierpaper [14] it was shownthat eqs. (1) and
(3) reduce to the Schr¨odinger equation 4p!
CLG = (49)
∂2ΨS ∂2ΨS ∂2ΨS m∂ΨS lp sw02L(p+|l|)!
mn + mn + mn +2ı mn =0, (40)
∂x2 ∂x2 ∂x2 ~ ∂t for Laguerre-Gaussianbeams.
1 2 3
Expectation values for the measurable properties of
and the non-relativistic form of the Bateman-Hillion each particle in the beam can be calculated as
ansatz
ΨOmSn =ΦSmn(ξ1,ξ2,τ)exp ~ı(P3x3−Est) , (41) hΨ|O+ˆ∞|ΨiP+=∞ +∞
h i (Ψ∗OˆΨ)δ(ξ3 v3τ)dξ1dξ2dτ, (50)
−
whereES isthenon-relativisticenergyoftheparticleand Z−∞ Z−∞ Z−∞
where Oˆ is the quantum mechanical operator for each
ΦSmn = Φmnδ(ξ3−vτ)dξ3. (42) observable quantity. Here, the subscript P has been in-
Z cluded as a reminder that the integration is performed
For comparison to results in the present context ΨOS over a planar cross-section perpendicular to the axis of
mn
must be further subject to the unitary transformation the beam but not along the axis itself.
(8) that simplifies to
ıNmn~t IV. CALCULATION OF 4-MOMENTUM
ΨS =ΨOS exp (43)
mn mn (cid:18) m0w02 (cid:19)
The canonical 4-momentum operator pˆ is defined in
µ
in the the non-relativistic limit c . eq. (2) in terms of the 4-position vector x . We next
→∞ µ
Itisreadilyshownthateq. (39)isthecontinuityequa- seek to use the fact Ψ depends on X as well as x
mn µ µ
tion for the Schr¨odinger equation (40) since to define a distinct kinetic 4-momentum operator Pˆ to
µ
satisfy the eigenvalue equation
∂j3 = Ps ∂ ΨS 2 =0. (44)
∂x3 ~ ∂x3| mn| PˆµΨmn =~kµΨmn (51)
Itisthusconcludedfromadirectcomparisonofeqs. (38) The first step is to write
and (39) that
Φmn(ξ1,ξ2,ξ3+cτ)=
PBH =m0kj33++jk44 =|Ψmn|2 (45) Φmn(x1−X1,x2−X2,x3−X3+ct−cT) (52)
6
having used eq. (4). This indicates Equations (57) and (62) enable the total energy Emn
HG
foreachparticleinaHermite-Gaussianmode tobe writ-
∂Φ ∂Φ
mn mn ten as
= (53)
∂x − ∂X
µ µ
2~2
and therefore EHmGn =cs~2k32+ w02 (1+m+n)+m20c2. (64)
∂ ∂
ı~ + Ψ =~ k +κmn Ψ (54) Comparing this result to the energy of a free particle
∂xµ ∂Xµ mn µ µ mn
(cid:18) (cid:19)
(cid:0) (cid:1)
E =c ~2k2+m2c2 (65)
From comparisonof this expressionto eq. (51) it can be FP 3 0
seen that q
of identical mass m0 and axial wave number k3 shows
∂ ∂ thatthebeamparticlepicksupanadditionalenergycon-
Pˆ Ψ = ı~ +ı~ ~κmn Ψ =~k Ψ
µ mn ∂xµ ∂Xµ − µ mn µ mn tribution
(cid:18) (cid:19)
(55) 2~2
or equivalently ~2Kmn = (1+m+n) (66)
T w2
0
Pˆ1Ψmn =Pˆ2Ψmn =0, Pˆ3Ψmn =~k3Ψmn, (56) where Kmn is defined in eq. (17), as a result of being
T
localized. The remaining task is therefore is to assign a
physical interpretation to this term.
Pˆ4Ψmn = ~2k32+2κmn(k3+k4)+m20c2Ψmn, (57) It can be inferred from inspection of eq. (6) that the
q expectation values of canonical 4-momentum and mass
having used eq. (16). These results are the eigenvalue
current must be related through the expression
equationsforthe kinetic4-momentumofeachparticlein
arelativisticHermite-Gaussianbeam. Incompletingthis Ψmn m0ˆjµ Ψmn P = Ψmn pˆµ Ψmn P (67)
argument,itisnecessarytofindtheexplicitformofNmn h | | i ℜh | | i
from eq. (14). wheretheoperator takestherealpartoftheargument.
ℜ
InsertingtheBateman-Hillionansatz(12)intoeq. (14) Equations(20),(58),(61)and(67)canthereforebeused
gives together to give
hΨmn|m0ˆjµ|ΨmniP =~(kµ+κmµn)+hΦmn|m0ˆjµ|ΦmniP. ℜhΨmn|pˆ21+pˆ22|ΨmniP = 2w~22(1+m+n) (68)
(58) 0
Here,theterm Φmn m0ˆjµ Φmn P canbeevaluatedusing This shows that the middle term under the square root
h | | i
the integrals
signin eq. (64)representsthe contributionof the fluctu-
+∞ ating transverse components of momentum to the total
xH2(√αx)e−αx2dx=0, (59) energy of each particle.
m
Z−∞
V. QUANTUM POTENTIAL
+∞x2H2(√αx)e−αx2dx= π 1 +m . (60)
m α3 2
Z−∞ r (cid:18) (cid:19) The concept of distinguishing between canonical and
kinetic4-momentumhasfamiliarityfromthedescription
The result is
[27] of a particle of charge e moving in an electromag-
Φmn m0ˆjµ Φmn P = ~κmn (61) netic 4-potential Aµ. The kinetic 4-momentum for this
h | | i − problem is
having set
πˆ =pˆ eA . (69)
µ µ µ
−
Nmn =1+m+n. (62)
For the purposes ofcomparisonthe relationshipbetween
the kinetic and the canonical 4-momentum of a beam
Putting eq. (61) into (58) gives
particle given in eq. (55) can be written as
hΨmn|m0ˆjµ|ΨmniP =~kµ (63) Pˆµ =pˆµ m0Uˆµ, (70)
−
Itisthusestablishedthattheeigenvaluesofthekinetic4-
where
momentumoperatorPˆ areequaltotheexpectationsval-
µ
ues for the mass current for all Hermite-Gaussian beam ~ 1 ∂
Uˆ = +κmn . (71)
modes. µ m0 ı ∂Xµ µ
(cid:18) (cid:19)
7
Eqs. (69) and (70) are similar in form but A is an where j2 =m2j jµ. It is clear that if we now add back
µ M 0 µ
external 4-potential whereas Uˆ is an operator. The un- the quantum 4-potential into both eqs. (78) and (79)
µ
derstanding here is that wave equations are constructed then eq. (79) must pick up an additional scalarterm V2
using kinetic 4-momentum to take account of external such that
potentials and canonical 4-momentum if no external po-
j2 m2c2 Ψ2+V2 =0 (80)
tential is present. The Hermite-Gaussian function Ψmn M − 0 | |
was derived from a wave equation that contains only
where
canonical 4-momentum operators but it is still possible
to identify a 4-potential like term Uˆµ in the definition V2 = ~kµ m0Uµ 2 m20c2 Ψ2 (81)
−| − | − | |
of the kinetic 4-momentum P analogous to the role of
µ
the external 4-potential A in π . Equation (71) will be Expanding this expressiongives
µ µ
referred to as the 4-potential operator.
Kinetic 4-momentum was defined in eq. (51) to be a V2 =−m20|Uµmn|2−2~kµm0Uµmn−~2KTmn =
real quantity. It follows from eq. (70) that the particle 1+m+n ξ2+ξ2
current can be written in the form 4~2 1 2 (82)
w2 − w4
(cid:20) (cid:21)
~k
j = µ +U Ψ2 (72) having used
µ µ
m0 | |
(cid:18) (cid:19)
where Umn 2 = ~2 16b2(ξ3+cτ)2(ξ12+ξ22), (83)
| µ | −m20 w04[(ξ3+cτ)2+4b2]2
~ ı ∂Ψ
U = +κmn (73)
µ m0ℜ Ψ∂Xµ µ
(cid:18) (cid:19) ~ Kmn 8b2(1+m+n)
kµUmn = T +
is a real quantum 4-potential field. Comparing eq. (72) µ m0 (cid:20)− 2 w02[(ξ3+cτ)2+4b2](cid:21)
to the component eqs. (31) through (34) gives + ~ 8b2(ξ12+ξ22)[(ξ3+cτ)2−4b2], (84)
Umn = ~ 4b(ξ3+cτ)ξ1 , (74) m0 w04[(ξ3+cτ)2+4b2]2
1 m0w02[(ξ3+cτ)2+4b2] alongside eq. (16). It is concluded from this argument
that V2 is itself a quantum potential appearing in eq.
Umn = ~ 4b(ξ3+cτ)ξ2 , (75) (81) as the scalar analog of the quantum 4-potential Uµ
2 m0w02[(ξ3+cτ)2+4b2] in eq. (72).
The OAMoperator(28) canbe rewrittenin Cartesian
coordinates to give
~ 2b(1+m+n)
U3mn = m0 (cid:20)κmn− (ξ3+cτ)2+4b2(cid:21) Lˆ3 =ξ1pˆ2−ξ2pˆ1 (85)
~ 2b(ξ12+ξ22)[(ξ3+cτ)2−4b2], (76) or equivalently
− m0 w02[(ξ3+cτ)2+4b2]2
Lˆ3 =ξ1(Pˆ2+m0Uˆ2) ξ2(Pˆ1+m0Uˆ1) (86)
−
~ 2b(1+m+n) having used eq. (70). This last result simplifies to
Umn = κmn+
4 m0 (cid:20)+− ~ 2b(ξ(12ξ3++ξ2c2τ)[)(2ξ3++4bc2τ(cid:21))2−4b2], (77) Lˆ3 =ξ1m0Uˆ2−ξ2m0Uˆ1 (87)
m0 w02[(ξ3+cτ)2+4b2]2 since Pµ =(0,0,~k3,~k4). It is therefore concluded that
thequantum4-potentialoperatorUˆ andnotthekinetic
µ
to be the explicit form of the quantum 4-potential for a
4-momentum operator Pˆ is the source of the mass flow
Hermite-Gaussian beam. µ
resulting in OAM.
Expression(72) is a quantum mechanicalequationde-
Calculating the expectation value of each component
scribing a particle in a localized state. In the absence of
Uˆ of the quantum 4-potential and the scalar analog V2
localization (w0 ) it reduces to the form µ
→∞ we obtain
~k
jµ = m0µ|Ψ|2 (78) hΨmn|Uµ|ΨmniP =hΨmn|Vµ2|ΨmniP =0, (88)
showingthatthequantum4-potentialtermhasvanished. Thisresultshowsthatquantum4-potentialisafluctuat-
Squaring eq. (78) gives ing phenomenon. Specifically, the presence of quantum
4-potentialcancausethe canonical4-momentumofa lo-
j2 m2c2 Ψ2 =0 (79) calizedparticleinabeamtoinstantaneouslydeviatefrom
M − 0 | |
8
thekinetic4-momentumbutithasnoaffectatallonthe It is clear from eqs. (82) and (95) that
expected 4-momentum of the particle.
V2
Theoriginalconceptofaquantumpotentialwasintro- Q= lim . (96)
duced by David Bohm[6] who startedfrom an ansatz to c→∞2m0
solve the Schr¨odinger equation. This takes the form ThisresultshowsthattheBohmpotentialforaHermite-
Gaussian beam is the non-relativistic limit of the scalar
Ψ=Rexp ıS , (89) form V2 of the relativistic quantum potential defined in
~
(cid:18) (cid:19) eq. (82).
wheretheamplitudeRandS/~arerealvaluedfunctions.
On inserting eq. (89) into the Schr¨odinger equation VI. SUMMARY
(40),theimaginarypartoftheequationcanbeidentified
as the continuity equation (39) and the real part as the
A relativistic solution for Hermite-Gaussian particle
Hamilton-Jacobi equation
beams presented in an earlier paper [14] has been used
to calculate the properties of the particles in the beam.
∂S S 2
mn mn
= |∇ | +Q. (90) In the original paper, the solutions were obtained using
− ∂t 2m0 aBateman-HillionansatzthatreducestheKlein-Gordon
equationto aparabolicformthus enabling Ψ2 to be in-
where
| |
terpretedasthe probabilitydensityforfinding the parti-
~2 2Rmn cle. Itwasshownthesolutionsareformpreservingunder
Q= ∇ , (91)
−2m0 Rmn Lorentz transformations and correspond to those of the
Schr¨odingerequationin the non-relativisticlimit. It was
istheBohmpotential. Itisofinterestnexttoinvestigate alsoshownthe solutions take accountof the Gouy phase
howthequantum4-potentialandtheBohmpotentialare
in the beam.
related to each other.
Inthispaper,aLorentzcovariantkinetic4-momentum
The solution to the Schr¨odingerequationfor Hermite- operator has been introduced equal to canonical 4-
Gaussian beams is given in eqs. (41) and (42). On com- momentum operator minus a quantum 4-potentialterm.
paring eq. (41) and (89) the explicit form of the am- The quantum 4-potential originates at the beam waist
plitude Rmn and phase function Smn can be read off to where it introduces fluctuating terms into the canoni-
be cal 4-momentum of transversely localized particles. All
theeigenvaluesofthekinetic4-momentumoperatorhave
R = CmHnGw0H √2ξ1 H √2ξ2 in fact been shown to equal the expectation values of
mn w m w n w therealpartsofthecanonical4-momentumcomponents.
S S ! S !
The total energy of a particle for each beam mode has
ξ2+ξ2
exp 1 2 , (92) also been calculated. It has been found, in particular,
× − w2
(cid:18) S (cid:19) that the energy of a particle in a beam differs from the
energy of a free particle as a result of fluctuating trans-
and
verse momentum components in the spatial plane per-
pendicular to the axis of the beam.
Smn =P3x3−Et−(1+m+n)~ω0t Transverse momentum is needed to explain both the
+ 2(ξ12+wξ222)~ω0τ −~(1+m+n)arctan(2ω0τ), (93) dwiaviesrtgeanscweeolfl tahseObAeaMm. aHfteerre,psaoslsuintigonthsrhoauvgehbtheeenbeparme-
S
sented for Laguerre-Gaussianmodes to demonstrate the
where possibility for OAM in the Bateman-Hillion formalism.
~ It has also been found that in our proposed partition-
wS =w0 1+4ω0τ2, ω0 = m0w02. (94) i4n-gpootfenctainaolnpiacratls4t-hmatomtheenktuinmetiicntpoarktinmeatikcesanndocqounatnrtibuum-
p
tiontoOAMmeaningthatOAMisapuremanifestation
Insertingeq. (92)intoeq. (91)showstheBohmquantum
of quantum 4-potential. A clear indicator to this is that
potentialforanon-relativisticHermite-Gaussianbeamto
particles must be localized to exhibit OAM. Free parti-
be
clescannothaveOAMintheabsenceoflocalizationsince
2~2 1+m+n ξ2+ξ2 they have no quantum 4-potential.
Q= + 1 2 . (95)
m0 (cid:20) wS2 wS4 (cid:21) Thequantum4-potentialhasbeendiscussedinrelation
to the electromagnetic 4-potential and the Bohm poten-
Eq. (95)caninturnbeinsertedintotheHamilton-Jacobi tial. Quantum 4-potential acts on mass in an analogous
equation (90) giving eq. (93) as a solution. manner to how electromagnetic 4-potential operates on
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