The Classical and Quantum Mechanics of a Particle on a Knot

0.13 MB·English

by  V. V. Sreedhar

The Classical and Quantum Mechanics of a Particle on a Knot
V. V. Sreedhar∗
Chennai Mathematical Institute, Plot H1, SIPCOT IT Park,
Siruseri, Kelambakkam Post, Chennai 603103, India
A free particle is constrained to move on a knot obtained by winding around a putative torus.
The classical equations of motion for this system are solved in a closed form. The exact energy
eigenspectrum, in the thin torus limit, is obtained by mapping the time-independent Schr¨odinger
equation to the Mathieu equation. In the general case, the eigenvalue problem is described by
the Hill equation. Finite-thickness corrections are incorporated perturbatively by truncating the
Hill equation. Comparisons and contrasts between this problem and the well-studied problem of a
particle on a circle (planarrigid rotor) are performed throughout.
5
1
PACSnumbers: 03.65.Ta,02.10.Kn
0
2
n Keywords: Classical; Quantum; Particle; Knot
a
J

6 INTRODUCTION

]
h Theexampleofaparticleconstrainedtomovealongacircle–theso-calledplanarrigidrotor–isoneofthesimplest
p problems that is discussed in text-books of quantum mechanics. The beguiling simplicity of this problem is at the
- heart of many non-trivial ideas that pervade modern physics. For understanding many issues like the existence of
t
n inequivalent quantizations of a given classicalsystem [1], the role of topology in the definition of the vacuum state in
a
gaugetheories[2], bandstructureofsolids [3],generalisedspinandstatistics ofthe anyonictype [4], andthe study of
u
mathematically interesting algebrasof quantum observableson spaces with non-trivialtopology [5], the problem of a
q
[ particle on a circle serves as a toy model.

In this paper, we consider the problem of a particle constrained to move on a torus knot. Besides adding a new
1
twist to the aforementioned problems, the present system can be thought of as a double-rotor (analogous to the
v
8 double-pendulum, but without the gravitational field) which is a genuine non-planar generalization of the planar
9 rotor.
0 The paper is organised as follows: In the next section we introduce toroidal coordinates in terms of which the
1
constraintswhichrestrictthemotionoftheparticletothetorusknotaremostnaturallyincorporated. Asawarm-up,
0
we then analyse the particle on a circle in toroidal coordinates. This prelude allows us to compare and contrast the
.
1 results of the subsequent sections with the well-known results for the particle on a circle. The following two sections
0
deal with the classical and quantum mechanics of a particle on a torus knot. In the penultimate section we briefly
5
touch upon the possibility of inequivalent quantizations of the particle on a knot. These will be labelled by two
1
: parameters, in contrast to the particle on a circle. The concluding section summarises and presents an outlook.
v
i
X
TOROIDAL COORDINATES
r
a
The toroidal coordinates [6] are denoted by 0 η < , π < θ π, 0 φ < 2π. Given a toroidal surface
≤ ∞ − ≤ ≤
of major radius R and minor radius d, we introduce a dimensional parameter a, defined by a2 = R2 d2, and a
dimensionless parameter η , defined by η = cosh−1(R/d). The equation η = constant, say η , define−s a toroidal
0 0 0
surface. The combination R/d is called the aspect ratio. Clearly, larger η corresponds to smaller thickness of the
0
torus. In the limit η , the torus degenerates into a limit circle.
0
→∞
The toroidal coordinates are related to the usual Cartesian coordinates by the equations
asinhηcosφ asinhηsinφ asinθ
x= , y = , z = . (1)
(coshη cosθ) (coshη cosθ) (coshη cosθ)
− − −
∗ [email protected]
2
The metric coefficients are given by the equations
a
h =h = , h =h sinhη (2)
1 2 3 1
(coshη cosθ)
−
and the volume element is
a3sinhη
dV = (3)
(coshη cosθ)3
−
With the help of these basic relations, it is straightforward to rewrite well-known Cartesian expressions in toroidal
coordinates.
A Particle Constrained to Move on a Circle
The Lagrangianfor a free particle of mass m in Cartesian coordinates (x,y,z) is
m
L= (x˙2+y˙2+z˙2) (4)
2
In the above expression, and henceforth, an overdot refers to a time derivative. After some algebra, this expression
can be rewritten in toroidal coordinates as
m (η˙2+θ˙2+sinh2η φ˙2)
L= a2 (5)
2 (coshη cosθ)2
−
To restrict the motion of the particle to lie on a circle in the xy plane, we impose the constraints η =η , a constant,
0
and θ =θ , another constant. The Lagrangianthen takes the form
0
ma2 sinh2η φ˙2
0
L= (6)
2 (coshη cosθ )2
0 0
−
The Euler-Lagrangeequation
d ∂L ∂L
( )= (7)
dt ∂φ˙ ∂φ
then yields, as expected,
φ¨=0 φ(t)=ωt+φ (8)
0
⇒
where ω is a constant and has the physical interpretation of frequency, and φ is a constant of integration which
0
specifies the position of the particle on the circle at time t=0 – similar to plane polar coordinates.
DefiningarescaledmassM =m sinh2η0 ,wegettheHamiltonianH = p2φ withthemomentumcanonically
(coshη0−cosθ0)2 2Ma2
conjugate to φ being given by p =Ma2φ˙ as usual. Using this to set up the Schr¨odinger equation and solving it, we
φ
get, for the eigenvalues and the normalised eigenfunctions respectively,
n2~2 1
E = , ψ (φ)= e±inφ n=0,1,2, (9)
n 2Ma2 n √2π ···
For large η , the thickness of the putative torus decreases and M m: we approach the well-known expressions in
0
→
plane polar coordinates.
Interestingly,it is also possible to get a particle to moveon a circle by imposing the constraintsη =η , a constant,
0
and φ=φ , another constant. This however results in a more complicated Lagrangianviz.
0
ma2 θ˙2
L= (10)
2 (coshη cosθ)2
0
−
3
The resulting Euler-Lagrangeequation is
θ¨(coshη cosθ)= sinθ θ˙2 (11)
0
− −
which can be re-written as
d
[θ˙(coshη cosθ)]=0
0
dt −
and readily integrated to yield
θ˙(coshη cosθ)=κ (12)
0
−
κbeingaconstant. Thusthesolutionisreducedtoquadratures. Thankstothepresenceofthefactor(coshη cosθ),the
−
solutionis notassimpleas the oneinplanepolarcoordinates. The Hamiltoniancanbe obtainedinastraightforward
manner and is given by
p2
H = θ (coshη cosθ)2 (13)
2ma2 0−
The presence of the θ-dependent multiplicative factor is portentous of additional complications that arise when we
makeatransitiontoquantummechanics. Inparticular,thefactthattheconjugateoperatorsp andθdonotcommute
θ
requires us to perform an operator-orderingof the classical Hamiltonian.
The above analysis shows that while toroidal coordinates are ideally suited to consider the motion of a particle on
a circle in the xy-plane, they are more cumbersome when it comes to handling paths which stray from the xy-plane.
Since a knot is intrinsically non-planar,we should be prepared to confrontthe attendant complications. It should be
mentioned, however, that these complications would also be present in other coordinate systems. We choose to work
with toroidal coordinates because of their suitability in imposing the constraints that define a torus knot.
CLASSICAL MECHANICS OF A PARTICLE ON A KNOT
As already mentioned, the constraint η = η defines a toroidal surface. A (p,q) torus knot can be obtained by
0
considering a closed path that loops p times around one of the cycles of a torus while looping around the other
cycle q times, p,q being relatively prime integers. The desired property can be enforced by imposing the constraint:
pθ+qφ=0. It is easy to check that θ θ+2πq φ φ 2πp i.e. as we complete q cycles in the θ direction, we
→ ⇒ → −
are forced to complete p cycles in the φ direction – as required. Imposing the above two constraints on equation (5),
we get the Lagrangianfor a particle on a torus knot to be
M
L= f(φ)φ˙2 (14)
2
where
a2
f(φ)= and M =m(α2+β2) (15)
(γ cosαφ)2
−
with
α= q/p, β =sinhη , γ =coshη (16)
0 0
−
The maindifference betweenthe Lagrangianin (14)andthe one fora particleonacircle viz. equation(6), lies inthe
appearance of the φ-dependent factor f(φ) in the Lagrangian which contains the information about the non-trivial
embedding of the knot in three dimensions.
The Euler-Lagrangeequation is given by
f(φ)φ¨+ 1f′(φ)φ˙2 =0 (17)
2
wheretheprimedenotesaderivativeofthe functionf withrespecttoits argumentφ. Now,usingthe aboveequation
4
of motion, it is straightforwardto show that
d
[ fφ˙]=0 fφ˙ = φ˙ = A(γ cosαφ) (18)
dt ⇒ A⇒ a −
p p
where is a constant. Noting that (1 γ2)<0, the latter equation can be integrated to get
A −
1 γ 1 αβt
φ(t)= tan−1 − tan(A ) (19)
α rγ+1 2a
(cid:2) (cid:3)
In the limit η , the right hand side is linear in t, as expected for a particle on a circle.
0
→∞
The momentum p canonically conjugate to φ and the Hamiltonian H can be easily worked out and are given by
φ
the following expressions
p2
p =Mf(φ)φ˙, H = φ (20)
φ
2Mf(φ)
QUANTUM MECHANICS OF A PARTICLE ON A KNOT
Inprinciple,onceaHamiltonianisgiven,itisastraightforwardexercisetowritedowntheSchr¨odingerequation. In
the presentcase, the classicalHamiltonian involves a term which mixes the coordinate and the canonically conjugate
momentum. Since these canonical pairs will be elevated to the level of operators in the quantum theory, we need
to prescribe an ordering for the operator products. We choose the so-called Weyl ordering which symmetrises the
product as follows:
1 1 1 1
H = [ p2 +p p +p2 ] (21)
6M f φ φf φ φf
In the above equation, and in what follows, we refrain from putting hats, but it should be remembered that both φ
and pφ are operators which obey the canonical commutation relations viz. [φ,pφ]− = i~. Pulling all the momentum
terms to the extreme right in preparation to make them act on a wavefunction, we get
H = 1 [(1)p2 (i~)(1)′p ~2(1)′′] (22)
2M f − f − 3 f
In the above form, the Hamiltonian is tailor-made for constructing the time-independent Schr¨odinger equation,
with the usual prescription for replacing the momentum by the corresponding differential operator. The resulting
Schr¨odinger equation is
~2 1 d2 ~2 1 ′ d ~2 1 ′′
[ ( ) ( ) ( ) ]ψ =Eψ (23)
−2M f dφ2 − 2M f dφ − 6M f
The first derivative in φ can be eliminated by the well-knowntrick of substituting ψ =χΣ in the above equationand
getting rid of terms proportionalto dΣ/dφ by choosing χ appropriately. This yields for χ,
χ f (24)
∝
p
Substitutingthisin(23)andcollectingtheremainingterms,theSchr¨odingerequationreducestothefollowingequation
for Σ
d2
[ +V(φ)]Σ=0 (25)
dφ2
where the ‘potential’ V is defined by
2f′′f f′2 2MEf
V(φ)=[ − + ] (26)
12f2 ~2
5
V is an even function of φ. Substituting for f from (15), we get after some algebra,
2MEa2/~2+α2/2 α2γcosαφ/3 α2cos2αφ/6
V = − − (27)
(γ cosαφ)2
−
Since V(φ) is a periodic function, the above potential can be expanded in a Fourier series and equation (25) gets
identified with the Hill differential equation [7].
The Thin-Torus Approximation
As already mentioned, large values of η and hence large values of γ, correspond to a thin torus around which the
0
particle’s trajectory winds. In this limit, we can restrict to terms of the order of 1/γ. Then β2 γ2, hence M m,
∼ →
and equation (27) simplifies to
α2 α2
V = λ cosαφ (28)
4 − 3γ
where
8mEa2
λ= (29)
~2α2
Equation (25) now takes the form
d2 α2 α2
[ + λ cosαφ]Σ=0 (30)
dφ2 4 − 3γ
Changing variables such that αφ=2z, the above equation becomes
d2 4
[ +λ cos2z]Σ=0 (31)
dz2 − 3γ
which is immediately recognised to be the well-known Mathieu equation [8].
The solutions Σ of the Mathieu equation with the required periodicity are given by the Mathieu functions of
fractional order ν viz.
1 cos(ν +2)z cos(ν 2)z
ce (z,γ)=cosνz − (32)
ν
− 6γ (ν+1) − (ν 1) ···
(cid:2) − (cid:3)
1 sin(ν+2)z sin(ν 2)z
se (z,γ)=sinνz − (33)
ν
− 6γ (ν+1) − (ν 1) ···
(cid:2) − (cid:3)
The complete solution with two arbitrary coefficients A and B is given by
Σ=Ase (z,γ)+Bce (z,γ) (34)
ν ν
Setting ν = 2n where n is an integer, we see that the above functions have a periodicity qπ in z, which translates
q
into the required periodicity 2pπ in φ.
The condition relating λ to ν is given by
λ=ν2+ 2 (35)
9γ2(ν2−1) ···
Since we are restricting our attention to 1/γ order,the boxed terms can be neglected. The allowedvalues of λ follow
by setting ν = 2n. These values of λ, in conjunction with equation (29), determine the energy eigenvalues to be
q
n2~2α2
E = (36)
n 2ma2q2
6
Beforeproceeding further,it is worthrecallingthatthe complete solutionofequation(23)thatwe are tryingto solve
is given by ψ = χΣ with χ √f. The complete solutions for the (unnormalised) eigenfunctions ψ with the correct
∝
boundary conditions are therefore given by
a 1 cos((n+q)αφ/q) cos((n q)αφ/q
ψ(n)(φ)= cos(nαφ/q) − (37)
+ (γ cosαφ ) ×{ −6γ (2n/q+1) − (2n/q 1) ··· }
− (cid:2) − (cid:3)
a 1 sin((n+q)αφ/q) sin((n q)αφ/q
ψ−(n)(φ)= sin(nαφ/q) − (38)
(γ sinαφ ) ×{ −6γ (2n/q+1) − (2n/q 1) ··· }
− (cid:2) − (cid:3)
wherewehaveused2z =αφ. Further, sinceweretainonlyterms oforder1/γ,the boxedterms inequations(37)and
(38) can be neglected.
In passing, we mention that the two independent solutions (32) and (33) can be combined into a single equation
given by [8]
Σ=eiνzu (39)
where
u=sin(z σ)+a cos(3z σ)+b sin(3z σ)+a cos(5z σ)+b sin(5z σ)+ (40)
3 3 5 5
− − − − − ···
whereσ isanewparametersuchthatσ =π/2yieldsthesolution(32)andσ =0yieldsthesolution(33). Intheabove
the coefficients a,b are determined in terms of γ and σ. To order 1/γ that we are interested in, only b = 1 , is
3 −12γ
non-zero. This succinctwayofwritingthegeneralsolutionwillbe particularlyusefulinincorporatingfinite-thickness
corrections.
It may be noted that for q = 1, p = 1, and hence α = 1, the above results (35-38) reduce to the well-known
− −
results for a particle on a circle. For a (2,3) torus knot, namely, the trefoil, p = 2, and the eigenfunctions have a
period 4π. The general solutions of Mathieu equations with a period 4π were first worked out by Lars Onsager in
1935 in his dissertation for a PhD at Yale [9]. While it is gratifying to note this, it is also a little disappointing. The
energy levels and energy eigenfunctions are the same as that of a particle on a circle, except for the factor of α. This
is a consequence of the fact that, in the weak coupling limit (large γ), the putative torus degenerates into a limit
circle, with the attendant vagueness associated with the winding in the θ direction. Correspondingly, the Mathieu
functionsdegenerateintotrigonometricfunctions. Itmaybetemptingtothinkthatthegeneralsolution(forarbitrary
γ) will be givenby Mathieu functions, with the boxed expressions in equations (35) and (37-38)being the next order
corrections. The story, however, is slightly more complicated. Our penchant for boxing negligible pieces relates to
this fact.
The Slightly-Thick-Torus Correction
Tothenextorderin1/γ,thecorrectexpressionforthepotentialisobtainedbystartingwithequation(27),making
a binomial expansion of the denominator, and collecting terms up to order 1/γ2. This straightforward exercise,
followed by the steps that lead up to equation (31), yields the so-called Hill-Whittaker equation [10]
d2
[ +Θ +2Θ cos2z+2Θ cos4z]Σ=0 (41)
dz2 0 1 2
with
2 2 1
Θ =λ+ , Θ = , Θ = (42)
0 3γ2 1 −3γ 2 −γ2
where now
8MEa2
λ= (43)
~2α2γ2
7
Following Ince [10], the most general solution of the Hill-Whittaker equation can be obtained along the same lines
adopted for solving Mathieu’s equation and yields the following energy eigenvalues
~2α2γ2 4 2 7
E = 4ν2+(1 cos2σ sin2σ ) (44)
n 8Ma2 − − 3γ − 3γ − 9γ2
(cid:8) (cid:9)
The corresponding solutions are
2γ 1 4γ2 2γ 4γ2
Σ(n) =e2inz/q sin(z σ)+ sin(3z σ)+( )sin(5z σ)+ cos(3z σ) cos(5z σ) (45)
− 3 − 108 − 9 − 3 − − 9 −
(cid:8) (cid:9)
where σ is the parameter introduced earlier. Once again multiplying by the factor χ = N√f, expanding the de-
nominator, retaining terms up to order 1/γ2 and, rewriting everything in terms of φ using αφ = 2z, gives the final
expression for the eigenstate to be
ψ(n)(φ)=Neinαφ/q
×
4γ 5αφ 5αφ
[sin( σ)+cos( σ)]
− 9 2 − 2 −
(cid:8)
2 3αφ 3αφ 2 5αφ 5αφ
+ [sin( σ)+cos( σ) cosαφ sin( σ) cos( σ) ]
3 2 − 2 − − 3 { 2 − − 2 − }
1 αφ 1 5αφ 2cosαφ 3αφ 3αφ
+ [sin( σ)+ sin( σ)+ sin( σ)+cos( σ)
γ 2 − 108 2 − 3 { 2 − 2 − } (46)
4cos2αφ 5αφ 5αφ
sin( σ)+cos( σ) ]
− 9 { 2 − 2 − }
1 αφ 1 5αφ 2cos2αφ 3αφ 3αφ
+ [cosαφ sin( σ)+ sin( σ) + sin( σ)+cos( σ)
γ2 { 2 − 108 2 − } 3 { 2 − 2 − }
4cos3αφ 5αφ 5αφ
sin( σ)+cos( σ) ]
− 9 { 2 − 2 − }
(cid:9)
where N is a normalization constant.
The Result For An Arbitrarily Thick Torus
For the sake of completion, we mention that this method can be systematically continued to arbitrary orders in
1/γ. The corresponding equation satisfied by Σ is the Hill equation given by
∞
d2
[ +Θ +2 Θ cos2rz]Σ=0 (47)
dz2 0 2r
Xr=1
As in the earlier section, we follow Ince [10], and try a general solution of the form
Σ=eiνzu (48)
where
ν =p (σ)Θ +p (σ)Θ + +q (σ)Θ2+q (σ)Θ2+ +q Θ Θ +q Θ Θ +q Θ Θ + +r (σ)Θ3+ (49)
1 1 2 2 ··· 1 1 2 2 ··· 12 1 2 13 1 3 23 2 3 ··· 1 1 ···
and
u=sin(z σ)+A (z,σ)Θ +A (z,σ)Θ +B (z,σ)Θ2+B (z,σ)Θ2+ +B (z,σ)Θ Θ + (50)
− 1 1 2 2··· 1 1 2 2 ··· 12 1 2 ···
with σ being determined by the relation
Θ =1+λ (σ)Θ +λ (σ)Θ +µ (σ)Θ2+µ (σ)Θ2+ +µ (σ)Θ Θ +ν (σ)Θ3 (51)
0 1 1 2 2··· 1 1 2 2 ··· 12 1 2··· 1 1···
8
Substituting theseexpressionsinequation(47)wecansolveforthecoefficientstoanydesiredorder,andhenceobtain
the corresponding eigenvalues and eigenvectors. We don’t pursue this exercise further since it does not shed any
further light on the solution to the problem.
INEQUIVALENT QUANTIZATIONS
Let us briefly recapitulate the interesting consequences that arise if the particle which is constrained to move on a
circleischarged,andifthecircleenclosesaninfinitely long,infinitesimallythin, andimpenetrablesolenoidcarryinga
uniform current. As is well-known,the wavefunction of the particle picks up a nontrivialphase factor which depends
on the net flux enclosed by the trajectory of the particle as it goes around the circle. Thus the wavefunction is
multi-valued, which is a manifestation of the nontrivial topology of the circle which, in turn, is a consequence of
the fact that the path cannot be shrunk to a point in the presence of the impenetrable solenoid. Redefining the
wavefunction such that it is single-valued modifies the Hamiltonian in such a way that the energy spectrum depends
on the enclosed flux. Given that the corresponding Lagrangians, with and without the flux, differ only by a total
derivative term, the classical theory is unaltered; although different values of the flux yield different energy spectra,
and hence inequivalent quantum theories. It is reasonable to expect similar features in the case of a charged particle
constrained to move along a knot.
For the torus knot of interest, two independent magnetic fluxes can be introduced. The first is the usual magnetic
fieldobtainedbyplacingauniformcurrentcarrying,long,thinsolenoidparalleltothez-axisandpassingthroughthe
centre of the putative torus around which the knot winds. Let us denote the corresponding flux by Φ . The second
S
flux is obtained by a uniform poloidal currentwinding aroundthe torus which produces a magnetic field which has a
support only inside the torus, the so-called toroidal magnetic field. Let us denote this flux by Φ .
T
A particle constrained to move on a (p,q) torus knot, starts at a point on the surface of the putative torus and
returns to the initial point after completing one circuit of the knot; in the process winding aroundthe solenoidalflux
ptimesandthetoroidalfluxq times. Thetotalfluxenclosedistherefore: Φ=pΦ +qΦ . Thuswehavetheequation
S T
which highlights the multi-valued nature of the wavefunction viz.
ψ(η ,θ+2πq,φ 2πp)=exp(iΦ)ψ(η ,θ,φ) (52)
0 0
−
Defining the single-valued wavefunction
Φ
ψ˜(η ,θ,φ)=exp( i φ)ψ(η ,θ,φ) (53)
0 0
− 2pπ
and the corresponding Hamiltonian obtained by the transformation
Φ Φ
H˜ =exp( i φ)Hexp(i φ) (54)
− 2pπ 2pπ
we see that the momentum operator in the Hamiltonian is shifted by iΦ , which leads to a corresponding shift in
2pπ
ν and hence the energy spectrum defined in equations (36) and (44). It is noteworthy that the phase picked up by
the wavefunction of the particle, for a given (p,q) knot, is a sum of two independent fluxes. Thus the inequivalent
quantizations are labelled by two parameters.
CONCLUSIONS AND OUTLOOK
The classical and quantum mechanics of a particle constrained to move on a torus knot were studied. The results
were compared and contrasted with the well-known results for a particle constrained to move on a circle. Defining
the knotasa trajectorywhichwinds aroundaputativetorusinawell-definedfashion,andusingtoroidalcoordinates
to parametrise the knot, makes it possible to rewrite the time-independent Schr¨odinger equation as a Hill equation
which can then be studied perturbatively in the thickness of the putative torus.
Attributing a chargeto the particle and introducing two independent magnetic fields having supports in physically
disconnected, but topologically linked, regions, leads to a two-parameter family of inequivalent quantizations of the
particle moving on a knot.
9
The model discussed in this paper has several features which are worth discussing further. First, it would be
natural to study the model non-perturbatively i.e. using instanton methods made popular in [2][3]. Second, it would
be interesting to generalise the treatment to more than one particle moving on the knot. The non-trivial phase
factor can then be related to exotic quantum statistics of the anyonic type. It would also be interesting to construct
coherent states and study algebras of quantum observables associated with a particle on a knot. All these problems
havenaturalanaloguesforthecorresponding,butmuchsimpler,exampleofaparticleconstrainedtomoveonacircle.
I thank G. Krishnaswami and A. Laddha for discussions. This work is partially funded by a grant from Infosys
Foundation.
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