Identification of Geometric Potential from Quantum Conditions for a Particle on a Curved Surface

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by  D. K. Lian

Identification of Geometric Potential from Quantum Conditions for a Particle on a
Curved Surface
D. K. Lian, L. D. Hu, and Q. H. Liu1
1School for Theoretical Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China
(Dated: February 15, 2017)
Combination of a construction of unambiguous quantum conditions out of the conventional one
and a simultaneous quantization of the positions, momenta, angular momenta and Hamiltonian
leads to thegeometric potential given by theso-called thin-lay quantization.
PACS numbers: 03.65.Ca Formalism; 04.60.Ds Canonical quantization; 02.40.-k Geometry, differential ge-
7 ometry,andtopology;68.65.-kLow-dimensional,mesoscopic,andnanoscalesystems: structureandnonelec-
1 tronicproperties;
0
2

b I. INTRODUCTION
e
F
In quantum mechanics for a system, the construction of a proper quantum Hamiltonian operator takes the central

4 position. For a free particle constrained to live on a curved surface or a curved space, DeWitt in 1957 used a
1 specific generalization of Feynman’s time-sliced formula in Cartesian coordinates and found a surprising result that

hisamplitudeturnedouttosatisfyaSchr¨odingerequationdifferentfromwhathadpreviouslyassumedbySchr¨odinger
]
h [2]andPodolsky[3]. InadditiontothekinetictermwhichisLaplace-Beltramioperatordividedbytwotimesofmass,
p his Hamilton operator contained an extra effective potential proportional to the intrinsic curvature scalar.
- JensenandKoppein1972[4]andsubsequentlydaCosta[5]in1981developedathin-layerquantization(alsoknown
t
n asconfiningpotentialformalism)todealwiththefreemotiononthecurvedsurfaceanddemonstratedthattheparticle
a experiencesaquantumpotentialthatisafunctionoftheintrinsicandextrinsiccurvaturesofthecurvedsurface,which
u
waslatercalledthegeometricpotential [6]. Bythethin-layerquantizationwemeanatreatmentof(n−1)-dimensional
q smooth surface Sn−1 in flat space Rn (n (cid:23) 1) and two infinitely high potential walls at the distance δ →0 from the
[
surface. Since the excitationenergiesofthe particle inthe directionnormalto the surfacearemuchlargerthan those

2 in the tangential direction so that the degree of freedom along the normal direction is actually frozen to the ground
v
state,aneffectivedynamicsfortheconstrainedsystemonthesurfaceisthusestablished. Thisthin-layerquantization
0
has a distinct feature for no presence ofoperator-orderingdifficulty orother ambiguities. It is thus a powerfultoolto
7
examine various curvature-induced consequences in low-dimensional curved nanostructures, for instance, spin-orbit
3
8 interaction of electrons on a curved surface [7], the mechanical-quantum-bit states [6], the geometry-induced charge
0 separationonhelicoidalribbon[8],the curvature-inducedp-njunctionsinbilayergraphene[9],theperiodiccurvature
. dependent electrical resistivity of corrugated semiconductor films [10] as well as the geometry-driven shift in the
1
0 Tomonaga-Luttingerliquid [11], electronicband-gapopening incorrugatedgraphene[12], low-temperatureresistivity
7 anomalies in periodic curved surfaces [13], curvature effects in thin magnetic shells [14], and the induced magnetic
1 moment for a spinless charged particle on a curved wire [15], etc. [16–20] Experimental confirmations include: an
: optical realization of the geometric potential [21] in 2010 and the geometric potential in a one-dimensional metallic
v
i C60 polymer with an uneven periodic peanut-shaped structure in 2012 [22]. Applying the thin-layer quantization
X
to momentum operators which are fundamentally defined as generators of a space translation, we have geometric
r momenta [23] which depends on the extrinsic curvatures of the curved surface.
a
It is generally accepted that the canonical quantization offers a fundamental framework to directly construct the
quantum operators, and the fundamental quantum conditions are commutators between components of position and
momentum [24, 25]. Many explorations have been devoted to searchingfor the geometric potential within the frame-
work[26–38]. Itiscuriousthatnoattemptissuccessfulforevensimplesttwo-dimensionalcurvedsurfaceS2 embedded
inR3. Thebestresultwastostartfromsurfaceequationdf(x)/dt=0,thetimetderivativeofthedirectonef(x)=0
ratherthan f(x)=0 itself, to obtaina potential depending onthe the intrinsic andextrinsic curvaturesvia twoarbi-
traryrealcoefficients[30]. Someresultsarecontradictorywitheachother,forinstance,forafreeparticleona(n−1)
dimensionalsphereKleinertandShabanovpredictednoexistenceofanyquantumpotential[33],butHongandRothe
gave a quantum potential whose n-dependent multiple is (n +1)(n−3) [35], whereas the thin-layer quantization
presented (n−1)(n−3) for such a multiple [28, 30, 39]. We revisited all these attempts, and concluded that the
canonical quantization together with Schr¨odinger-Podolsky-DeWitt approach of Hamiltonian operator construction
was dubious, for the kinetic energy in it takes some presumed forms that are primarily a sum of the Cartesian mo-
mentasquared. Since2011,wehavetriedtoenlargethecanonicalquantizationschemetosimultaneouslyquantizethe
Hamiltoniantogetherwithpositionsandmomenta[40],ratherthansubstitutedthepositionandmomentumoperators
intosomepresumedformsofHamiltonian. Yetthesuccessislimited. i)Weobtainedthe geometricmomentumwhich
2
is identical to that given by the thin-layer quantization [23], and ii) we got the correct form of geometric potential
for the (n−1) dimensional sphere [39], but iii) there are ambiguities associated with geometric potential for other
curved surfaces [41].
In this Letter, we report that for a particle on a two-dimensional curved surface, simultaneous quantization of
the Hamiltonian together with the fundamental quantities as position, momentum and the angular momentum, the
Hamiltonian includes correct form of the geometric potential. It is the first time to achieve this result within the
canonical quantization scheme.
II. DIRAC’S THEORY OF THE CONSTRAINED SYSTEMS: CLASSICAL AND QUANTUM
MECHANICS
Letusconsideranon-relativisticallyfreeparticlethatisconstrainedtoremainonasurfacedescribedbyaconstraint
in configurational space f(x) = 0, where f(x) is some smooth function of position x, whose normal vector is n ≡
∇f(x)/|∇f(x)|. We can always choose the equation of the surface such that |∇f(x)| = 1, so that n ≡ ∇f(x).
In classical mechanics, the Hamiltonian is simply H = p2/2µ where p denotes the momentum, and µ denotes the
mass. However,inquantummechanics,wecannotimpose the usualcanonicalcommutationrelations[x ,p ]=i¯hδ ,
i j ij
(i,j = 1,2,3). Dirac was aware of the fact the presence of this constraint which needed to be eliminated before
quantization could very well cause the remaining classical phase to not admit Cartesian coordinates.
A. Dirac brackets formulation of the classical motion
DiracgaveageneraltheoryforalargeclassofconstrainedHamiltoniansystemsincludingthemotiononthesurface
[24]. He introducedabracketinsteadofthe Poissonone[f(x,p),g(x,p)] betweenanypairofquantitatesf(x,p)and
P
g(x,p) in the following,
−1
[f(x,p),g(x,p)] ≡[f(x,p),g(x,p)] −[f(x,p),χ (x,p)] C [χ (x,p),g(x,p)] , (1)
D P α P αβ β P
where repeated indices are summed over in whole of this Letter, and C ≡ [χ (x,p),χ (x,p)] are the matrix
αβ α β P
elements in the constraint matrix (C ) and the functions χ (q,p) (α,β =1,2) are two constraints [38],
αβ α
χ (x,p)≡f(x)(=0),and χ (x,p)≡n·p(=0). (2)
1 2
The bracket [f(x,p),g(x,p)] is called the Dirac bracket. We have elementary Dirac brackets in the following, with
D
use of symbol n ≡∂n /∂x [38],
i,j i j
[x ,x ] =0, (3)
i j D
[x ,p ] =δ −n n , (4)
i j D ij i j
[p ,p ] =(n n −n n )p . (5)
i j D j i,k i j,k k
These brackets were in general taken the fundamental set which after quantization forms the set of the so-called
fundamental quantum conditions [24, 25]. In classical mechanizes, the motion particle follows the geodesic whose
curvature is κ and torsion is τ. Now we define the orbital angular momentum G≡ x×p for a particle on a curved
surface. We do not use the conventional symbol L that is usually used to denote the orbital angular momentum
wherethreecomponentsformaso(3)algebra[L ,L ] =ε L . Forthe particleonthecurvedsurface,wehaveafter
i j P ijk k
calculations,
[G ,G ] =ε {G −x τx·p+(x κ−n )n·G}. (6)
i j D ijk k k k k
It reduces to the [G ,G ] = ε G for both relations τx·p = 0 and (x κ−n )n·G = 0 come true. Clearly, for
i j D ijk k k k
particle moves in the free space, or in central force potential, or on the sphere, G is identical to L.
Next, we have following equations of motion for x,p and G, respectively,
dx p
≡[x,H] = , (7)
D
dt µ
dp n
≡[p,H] =− (p·∇n·p), (8)
D
dt µ
dG p·∇n·p
≡[G,H] =−(x×n) ≡T. (9)
D
dt µ
3
A important property of vector T=dG/dt is that it lies on the tangential surface, for we have n·T = 0. Relations
(7)-(9) are revealing but somewhat trivial. In contrast, the consequence of these relations is significant in quantum
mechanics, as we see shortly.
B. Quantum conditions of the constrained system
The scheme of the canonicalquantizationhypothesizes that in generalthe definition of a quantum commutator for
any variables f and g is given by [25],
[f,g]=i¯hO{[f,g] } (10)
D
in which O{f} stands for the quantum operator corresponding to the classical quantity f. The fundamental
quantum conditions are [x ,x ],[x ,p ] and [p ,p ]. For a particle moves in the free space, we have two funda-
i j i j i j
mental operators in quantum mechanics, which in the configuration representation are position x and momentum
p = −i¯h∇. For a particle moves on a surface, there is in general a great difficulty in getting the momentum op-
erator [28–30], because we run into the notorious operator-ordering difficulty of momentum and position operators
in O{(n n −n n )p }(=[p ,p ]/(i¯h)) from (5). Even worse is not the ambiguities in defining the Hamiltonian
j i,k i j,k k i j
operator, but the Schr¨odinger-Podolsky-DeWitt approach is not able to give the correct form of the Hamiltonian
operator no matter what form of the momentum operator is obtained [30]. Thus the commutators [p ,p ] must be
i j
excluded from the so-called fundamental quantum conditions for they contain severe vagueness. Thus, we should
search for quantum conditions beyond the usual fundamental ones. A straightforward enlargement of the quantum
conditions is to simply follow the hypothesis given by (10) to include all [f,H] as f =x, p and G to simultaneously
determinetheoperatorspandH. Itisfruitlessatall,becausetherearemuchdepressingoperator-orderingdifficulties
inO{n(p·∇n·p)}/µ(=−[p,H]/(i¯h))from(9)andO{(x×n)p·∇n·p}/µ(=−[G,H]/(i¯h)). Tosurmountthese
difficulties, we note following vanishing relations,
p
n·[x,H] =n· =0,n·[G,H] =n·F=0 and n×[p,H] =0. (11)
D D D
µ
The resultant quantum conditions free from the operator-ordering difficulty are given by,
[x ,x ]=0, (12)
i j
[x ,p ]=i¯h(δ −n n ), (13)
i j ij i j
p
[x,H]=i¯h , (14)
µ
i¯h
n·[x,H]+[x,H]·n= (n·p+p·n)=0, (15)
µ
n×[p,H]+[p,H]×n=0, (16)
n·[G,H]+[G,H]·n=i¯h(n·F+F·n)=0. (17)
The Hamiltonian operator must take the following form for it in classical limit reduces to the classicalHamiltonian
H =p2/2µ,
2
¯h
2
H =− ∇ +α(x)·∇ +V , (18)
2µ LB s G
where α(x) and V are functions that go overto zero not only in classicallimit but also for free motion in flat space,
G
2
whichingeneraldoes nothaveananaloginclassicalmechanics,and∇ =∇ ·∇ is the Laplace-Beltramioperator
LB s s
which is the dot product of the gradient operator ∇ on the surface.
s
The first condition (12) sets the configuration representation with Cartesian coordinates. The second condition
(13) gives the essential part of the momentum p is the gradient ∇ on the surface,
s
p=−i¯h(∇ +β(x)), (19)
s
where β(x) is anundetermined vectorfunction. Substituting (19)and(18)into the thirdcondition(14), the momen-
tum operator p and Hamiltonian operator H becomes, respectively [39, 40],
M
p=−i¯h ∇ + n (20)
s
2
(cid:18) (cid:19)
4
−1 −1 −1 −1
where M is a sum of two principal curvatures R and R , (R +R ) the mean curvature at point x on the
1 2 1 2
surface S2
p2 ¯h2M2
H = +V − , (21)
G
2µ 8µ
where α(x)=0 in Eq. (18). It is easily to verify that the fourth and fifth conditions (15) and (16) are automatically
satisfied whatever form of potential V is. Lastly, let us calculate the n·[G,H]+[G,H]·n, and after a lengthy but
G
straightforwardmanipulation, we arrive at,
¯h2M2 ¯h2 ¯h2 M 2
2
V = − (M −2K)+ϕ=− −K +ϕ, (22)
G
8µ 4µ 2µ 2
((cid:18) (cid:19) )
−1
where K is the gaussiancurvature which is the product of the two principal curvatures as (R1R2) , and function ϕ
satisfies following differential equation,
n×∇ϕ=0. (23)
It means that ∇ϕ is parallel to the normal direction n≡∇f(x), and we have ∇ϕ=Φ(x)∇f(x) with Φ(x) being the
magnitude of gradient of function ϕ. Since |∇f(x)| = 1, we have thus Φ(x) = ±|∇ϕ|. So, the function ϕ defines a
surface whose normal vector is identical to the surface f(x) = 0. So, the new surface is identical to f(x) = 0, but
takes another form ϕ[f(x)]=0, i.e., we have ϕ=0. The quantum Hamiltonian operator turns out to be,
p2 ¯h2 ¯h2 M 2
2 2
H = − M −2K =− ∇ + −K . (24)
2µ 2µ 2µ LB 2
(cid:18) (cid:19) !
(cid:0) (cid:1)
2 −2 −2
Inthefirstexpression,thecurvature-inducedpotentialisnegativedefiniteforwehave−¯h /(2µ) R +R ,whereas
1 2
2 −1 −1 2
inthesecondexpression,thecurvature-inducedpotentialissemi-negativedefiniteforwehave−¯h /(8µ) R −R ,
(cid:0) 1 (cid:1) 2
which is the so-called geometric potential,
(cid:0) (cid:1)
2 2
¯h M
V =− −K . (25)
G
2µ 2
((cid:18) (cid:19) )
C. Further comments on the commutators [pi,pj]
The naive utilization of the relation (5), i.e., O{(n n −n n )p }=[p ,p ]/(i¯h), is highly controversialtopic to
j i,k i j,k k i j
constructmomentum. Forinstance,oncecanassumeO{(njni,k−ninj,k)pk}≡c1njni,kpk+c2pknjni,k+c3njpkni,k+
c4ni,kpknj−(i↔j)wherec1+c2+c3+c4 =0[28]. But,becausenj andni,k containvariousfunctionsofcoordinates
of x, y and z, e.g., we can have n = x x2+y2+z2 −1/2, then operators p n in the 4th term n p n in above
j j k j i,k k j
decomposition of O{(njni,k−ninj,k)pk}(cid:0)can at least b(cid:1)e further decomposed as pknj =d1 x2+y2+z2 −1/2pkxj +
d2xjpk x2+y2+z2 −1/2whered1+d2 =1. Noprinciplefromeitherphysicsormathematic(cid:0)scanbeused(cid:1)toterminate
thisprocedure. So,thecommutators[p ,p ]canhardlybemembersofthefundamentalquantumconditions. However,
i j
(cid:0) (cid:1)
our geometric momentum turns out to satisfy the following relation,
1
O{(n n −n n )p }= ((n n −n n )p +p (n n −n n )). (26)
j i,k i j,k k j i,k i j,k k k j i,k i j,k
2
Instead,ourproposalistoreversethequantizationconditions[p ,p ]/(i¯h)=O{(n n −n n )p }andtoconstruct
i j j i,k i j,k k
a function containing momentum operators such that we have,
n·P+P·n=0, (27)
where P ≡n·[p,p ]+[p,p ]·n. Since this relation (27) alone is not yet able to give the momentum operators, we
j j j
see that the commutators [p ,p ] are not fundamental quantum conditions either.
i j
5
III. CONCLUSIONS AND DISCUSSIONS
Thequantumconditionsgivenbythestraightforwardapplicationsoftheequation(10)arenotalwaysfruitful, even
misleading. For the particle on the curved surface, in order to obtain the geometric potential predicted by the so-
called thin-lay quantization, a proper enlargement of the quantum conditions turns out to be compulsory to contain
positions, momenta, orbital angular momentum and Hamiltonian. What is more, a construction of unambiguous
quantum conditions out of the equation (10) proves inevitable. Combining the enlargement and the construction,
we obtain the geometric potential. Since momentum in the orbital angular momentum is the geometric one, which
only in some special casesreduces to the usualone, we cancall the orbitalangular momentum the geometric angular
momentum. Even the present paper deals with only the two-dimensional curved surface, we conjecture that our
method can be used for particle on an arbitrarily dimensional curved surface, which is still under investigation.
Finally, wewouldliketo pointoutthatthere areotherforms ofthe enlargementandthe constructioninliterature,
for instance Refs. [42, 43], but they were devised to serve entirely different purposes.
Acknowledgments
This work is financially supported by National Natural Science Foundation of China under Grant No. 11675051.
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