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Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum
Accepted Manuscript Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum H. Panahi, L. Jahangiri PII: DOI: Reference:
S0003-4916(16)30030-6 http://dx.doi.org/10.1016/j.aop.2016.04.013 YAPHY 67092
To appear in:
Annals of Physics
Received date: 20 November 2015 Accepted date: 19 April 2016 Please cite this article as: H. Panahi, L. Jahangiri, Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum, Annals of Physics (2016), http://dx.doi.org/10.1016/j.aop.2016.04.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum H. Panahi ∗, L. Jahangiri †, Department of Physics , University of Guilan, Rasht 41635-1914, Iran.
February 22, 2016
Abstract In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using of Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one. Keywords: Quantum mechanics, Ellipsoid, Dirac method, Bein formalism, Geometric momentum. PACS numbers: 03.65.-w, 04.60.Ds ∗
Corresponding author (E-mail: [email protected])
†
( E-mail: [email protected])
1
1
Introduction
The quantum mechanics in curved space has attracted much attention during the last decades and different authors have considered this problem via various methods [1-5]. In Ref. [6], the classical and quantum dynamics of a particle has been investigated in 2D space-time with constant curvature. They have considered the motion of a relativistic particle in the cases of one-sheet hyperboloid, stripe and half-plane. Also the classical and quantum free particle on one-sheeted hyperboloid have been discussed in [7]. They have solved the classical equations of motion and identified the Hilbert space and observables for this particle as well. As another effort in Ref. [8], these authors have performed a detailed study of the classical and quantum mechanics for a particle on the double cone in two cases: free and bounded by the harmonic oscillator potential. In Ref. [9], the classical and quantum motion of a particle on Riemannian manifolds have been studied by considering the curved space as a hypersurface in higher dimensional Euclidean space via Dirac method. The Dirac method has been introduced as a powerful method in order to study the classical and quantum mechanical systems using the Hamiltonian method specially when the Lagrangian is singular [10]. In this approach the Poisson brackets are replaced by Dirac brackets and passing from classical to quantum mechanics is attainable through canonical quantization procedure. According to Dirac approach for a constrained motion, there are commutators between position and momentum operators which are not sufficient to lead to unique forms of the momentum and Hamiltonian operators for the system. For instance, for a spherical surface, there are more than five different forms of momentum and Hamiltonian which have been discussed in [11]. In order to remove this problem there are two formalisms to study the quantum motion on two dimensional curved surface including Schrodinger formalism and canonical quantization scheme which lead to geometric potential and geometric momentum. These formalisms have been applied in Ref. [11,12,13] and the authors have discussed the quantum motion of a constrained particle on some two dimensional curved surfaces like torus, sphere and 2
etc. Their calculations are based on gaussian and mean curvatures which are intrinsic and extrinsic curvatures respectively. Now in this paper we apply the Dirac method in order to study the classical and quantum mechanical behaviour of a particle on an ellipsoid and then determine the momentum operator representations by using of bein formalism. We show that irrespective of the geometrical point of view, our momentum operator representations are in agreement with geometric momentum. The paper is organized as follows: In section 2, we review the basic concepts of Dirac approach briefly. In section 3, we apply the results of the previous section for calculating the constraints and the Dirac brackets and then by quantizing, we calculate the commutators between position and momentum operators for an ellipsoid. In section 4, using of the bein formalism, we try to obtain the momentum operator representations that satisfy the obtained commutators. In section 5, we discuss the special cases like sphere, spheroid ad etc which shows that our calculations are in agreement with the obtained former results in other papers. In section 6, we end our work with a brief conclusion.
2
A brief review of Dirac approach
For a system with a finite number of degrees of freedom, the Lagrangian is a function of coordinates and velocities as L({qi }, {q˙i }). In order to get the equations of motion one should construct the Hamiltonian. For this purpose, the momenta are obtained as follows pi =
∂L . ∂ q˙i
(1)
For a non-singular Lagrangian, the velocities q˙i can be expressed in terms of coordinates and momenta [14] q˙i = fi (q, p).
(2)
The Hamiltonian is then obtained as a function of momenta and coordinates as H(q, p) =
X i
pi fi (q, p) − L(q, f (q, p)). 3
(3)
But when the Lagrangian is singular, there is not an explicit dependence of velocities on momenta and so it is impossible to write the Hamiltonian only in terms of qi and pi similar to (3). In such a case, according to Dirac method, those momenta that cause difficulty are considered as a set of primary constraints ϕj which are added to Hamiltonian as follows HT = H +
X
uj ϕ j .
(4)
j
Then the consistency conditions, time evolution of primary constraints, should be calculated by using of the following equation ϕ˙ i = {ϕi , HT }P ,
(5)
where the {, }p is the Poisson bracket. The consistency conditions may lead to new constraints which are called secondary constraints. It should be pointed out that the primary and secondary constraints are weakly equal to zero which in the next section we explain about it clearly. Another important part of the Dirac method is replacing the Poisson bracket with the Dirac bracket which is defined as {A, B}D = {A, B}P −
X k,l
−1 {A, ϕk }P σkl {ϕl , B}P ,
(6)
where σkl = {ϕk , ϕl }P .
(7)
One of the applications of Dirac method that yields interesting results is in studying the motion of a particle on curved hypersurfaces. In this situation the hypersurface is considered in higher dimensional Euclidean space which means the motion with constraint. As we mentioned before, in Ref. [9] the authors have discussed the classical and quantum mechanical treatments on the hypersurface VN −1 embedded in N dimensional Euclidean space RN from the view point of Dirac method. In the next section we apply this method in order to study the classical and quantum mechanics on an ellipsoid.
4
3
Classical ad quantum mechanics on an ellipsoid as a 2-dimensional surface embedded in 3-dimensional Euclidean space
In this section we consider an ellipsoid in 3-dimensional Euclidean space which is specified by cartesian coordinates (x, y, z). The surface equation in cartesian coordinates is give by x2 y 2 z 2 + 2 + 2 = 1. a2 b c
(8)
The Lagrangian for this system is
1 x2 y 2 z 2 L = m x˙ 2 + y˙ 2 + z˙ 2 + λ 2 + 2 + 2 − 1 , 2 a b c
(9)
where according to Dirac method the equation of ellipsoid has been added to Lagrangian as a constraint and λ is a dynamical variable. In order to write the Hamiltonian let us first calculate the conjugate mumenta px =
∂L = mx, ˙ ∂ x˙
py =
∂L = my, ˙ ∂ y˙
pλ =
pz =
∂L = mz, ˙ ∂ z˙
∂L ≈ 0. ∂ λ˙
(10)
(11)
As we mentioned before the second equation is considered as the primary constraint φ1 = pλ ≈ 0,
(12)
so the total Hamiltonian of the system becomes
x2 y 2 z 2 1 p2x + p2y + p2z − λ 2 + 2 + 2 − 1 + upλ , HT = 2m a b c
(13)
where u is a multiplier. For this constrained motion, the consistency condition should be examined so we calculate the time evolution of primary constraint (12) as follows:
x2 y 2 z 2 φ2 = φ˙1 = {φ1 , HT }P = + 2 + 2 − 1 ≈ 0, a2 b c 5
(14)
2 xpx ypy zpz φ3 = φ˙2 = {φ2 , HT }P = + 2 + 2 m a2 b c
≈ 0,
(15)
2 2 1 p2x p2y p2z x y2 z2 ˙ φ4 = φ3 = {φ3 , HT }P = + 2 + 2 + 2λ 4 + 4 + 4 ≈ 0, m m a2 b c a b c
4 4λ xpx ypy zpz φ5 = φ˙4 = {φ4 , HT }P = + 4 + 4 m m a4 b c
x2 y 2 z 2 + 4 + 4 + 4 u ≈ 0. a b c
(16)
(17)
At this stage the process of finding new constraints terminates. In above relations the expression ≈ means weak equality since we need these constraints for the purpose of calculating Dirac brackets and after that they become strong equations. As we see φ2 is the surface equation and φ3 shows that for a confining particle on an ellipsoid, the momentum has no component in the normal direction of surface by supposing that the unit normal vector is as follows
x ~ 2 i + ~n = a q 2 x a4
z ~ k c2 . 2 + zc4
~j +
y b2
y2 b4
+
(18)
Also (16) and (17) determine the variables λ and u as:
λ=
−1 m
2
u=
−4λ m
p2x a2
x2 a4
x2 a4
y2 b4
+
xpx a4
p2y b2
+
+
+
+
ypy b4
y2 b4
p2z c2
+
+
z2 c4
+ z2 c4
,
zpz c4
(19)
.
(20)
Now we want to calculate the Dirac brackets between the variables of phase space. But first −1 let us determine the antisymmetric matrix σkl and its inverse σkl according to (7) as
σkl
=
0
0
0
0
0
σ14
0
−σ14 −σ24 −σ34 6
σ14
−σ14 σ24 0
σ34 0
,
(21)
−1 σkl
where
=
0
σ34 2 σ14
−σ24 2 σ14
− σ114
−σ34 2 σ14
0
1 σ14
0
σ24 2 σ14
− σ114
0
0
1 σ14
0
0
0
σ14 = − σ24 =
σ34
,
(22)
4 x2 y 2 z 2 + 4 + 4 , m a4 b c
(23)
(24)
8 xpx ypy zpz + 4 + 4 , m 2 a4 b c
2 8 1 p2x p2y p2z x y2 z2 = 2 + 4 + 4 − 2λ 6 + 6 + 6 . m m a4 b c a b c
(25)
Using (6) and (22) we get the following Dirac brackets: {xi , xj }D = 0, {xi , pj }D = δij − {pi , pj }D =
(26)
ri rj , r2
(27)
1 r j Ki − r i Kj , r2
(28)
where ~r =
x ~ y ~ z i + 2 j + 2 ~k, 2 a b c
~ = px ~i + py ~j + pz ~k. K a2 b2 c2
(29)
(30)
As we mentioned before, after calculating Dirac brackets, constraints are treated as strong equations. So we have only two necessary constraints which are: x2 y 2 z 2 + 2 + 2 − 1 = 0, a2 b c 7
(31)
xpx ypy zpz + 2 + 2 = 0. a2 b c
(32)
We consider (32) as ~n · p~ = 0 even though both are equivalent in classical mechanics but different in quantum mechanics. Also the Hamiltonian (13) becomes
1 H= p2 + p2y + p2z . 2m x
(33)
From the classical point of view the particle is characterized by the set equations (26)-(28) and constraints (31), (32). As usual, the quantization problem is attainable through the Dirac quantization. The only difference is that the Poisson brackets should be replaced by Dirac brackets so [A, B] = i¯ h{A, B}D .
(34)
As a result the classical relations (26)-(28) become [ˆ xi , xˆj ] = 0,
[ˆ xi , pˆj ] = i¯ h δij − [ˆ pi , pˆj ] =
(35)
rˆi rˆj , r2
(36)
i¯ h rˆj ˆ ˆ i rˆj − rˆi K ˆj − K ˆ j rˆi . Ki + K 2 2 2 2 r r r r2
(37)
Also the constraint (31) and (32) are given by x2 y 2 z 2 + 2 + 2 − 1 = 0, a2 b c
(38)
1 1 1 (ˆ xpˆx + pˆx xˆ) + 2 (ˆ y pˆy + pˆy yˆ) + 2 (ˆ z pˆz + pˆz zˆ) = 0. 2 2a 2b 2c
(39)
Taking into consideration the hermiticity condition, in the right hand side of (37) and left hand side of (39), the ordering problem has been imposed by letting the anticommutator between the momentum operator and those which depend on position. In the next section we try to find the momentum operators pˆx , pˆy , pˆz that satisfy (36), (37) and (39).
8
4
The momentum operator representations for an ellipsoid: bein formalism
So far all results are obtained in Euclidean space. By considering (8), we change the coordinate as x = a sin θ cos ϕ
y = b sin θ sin ϕ
z = c cos θ.
(40)
As we see instead of three variables (x, y, z) in cartesian coordinate, we have only two coordinates (θ, ϕ) on ellipsoid because of (8). As we know by changing the coordinate, the beins ei
µ
and their inverse eµ
i
are used to transform the physical quantities from flat to curved
frames and vice versa. As a convention we put the Latin and Greek indices as global and local indices which take the values 1,2,3 and θ, ϕ respectively. Since a vector is independent of the basis chosen, so we expect that under change of coordinate frame, the components of a vector transform as [15] ∂xa µ p = e µp = p , ∂xµ
(41)
∂xµ a p . ∂xa
(42)
a
a
µ
pµ = eµ a pa =
So using of the bein formalism and knowing the momentum in cartesian coordinate we can obtain the momentum in curved space and vice versa. The only important point is that from the quantum mechanical point of view, (41) and (42) should be rewritten as
where {ˆ a, ˆb} =
1 ˆ (ˆ ab 2
pˆa = {ˆ ea µ , pˆµ },
(43)
pˆµ = {ˆ eµ a , pˆa },
(44)
+ ˆbˆ a). Since beins are operators that depend on position and one
should consider their symmetric form with momentum operator. By letting θ = cos−1 9
z c
and
ϕ = tan−1
ay , bx
according to (40) in (35)-(37) we get ˆ ϕ] ˆ θ] ˆ = [ϕ, [θ, ˆ = [θ, ˆ ϕ] ˆ = 0,
ˆ pˆθ ] = [ϕ, [θ, ˆ pˆϕ ] = i¯ h,
ˆ pˆϕ ] = [ϕ, [θ, ˆ pˆθ ] = 0,
[ˆ pθ , pˆϕ ] = [ˆ pθ , pˆθ ] = [ˆ pϕ , pˆϕ ] = 0,
(45)
(46)
(47)
a
∂x ˆa } which can be obtained from (42) and also the fundamental here we have used pˆν = { ∂x ν,p
quantum mechanical relation [9] ˆ = [F (x), A]
∂F ˆ [ˆ x, A]. ∂x
(48)
From equation (46) and (47) we find that pˆθ and pˆϕ can be represented as pˆθ = −i¯ h
∂ + i¯ h f (θ, ϕ), ∂θ
(49)
pˆϕ = −i¯ h
∂ + i¯ h g(θ, ϕ). ∂ϕ
(50)
The functions f (θ, ϕ) and g(θ, ϕ) should be chosen so that pθ and pϕ be hermitian operators. Taking into consideration the surface element of the ellipsoid, we can construct the Hilbert space which is specified by the following scalar product hψ1 |ψ2 i =
Z
2π
0
Z
0
π
q
sin θ a2 b2 cos2 θ + a2 c2 sin2 θ sin2 ϕ + b2 c2 sin2 θ cos2 ϕ ψ1∗ ψ2 dθdϕ.
(51)
So we propose the following representations for pˆθ and pˆϕ which are hermitian with respect to scalar product (51) i¯ h i¯ h sin θ cos θ(−a2 b2 + a2 c2 sin2 ϕ + b2 c2 cos2 ϕ) ∂ − cot θ − , ∂θ 2 2 a2 b2 cos2 θ + a2 c2 sin2 θ sin2 ϕ + b2 c2 sin2 θ cos2 ϕ
(52)
∂ i¯ h sin2 θ sin ϕ cos ϕc2 (a2 − b2 ) . − ∂ϕ 2 a2 b2 cos2 θ + a2 c2 sin2 θ sin2 ϕ + b2 c2 sin2 θ cos2 ϕ
(53)
pˆθ = −i¯ h pˆϕ = −i¯ h
10
We should notice that these hermitian representations satisfy (46) and (47). Now by knowing pˆθ and pˆϕ we can determine the operators pˆx , pˆy , pˆz from the quantum mechanical form of equation (41) as follows pˆb = {
∂xa δab g µν , pˆν }, µ ∂x
(54)
where g µν and δab are the metrics of curved and Euclidean space respectively. This relation makes us sure that the obtained pˆa are obviously hermitian. Let us calculate the metric gµν and its inverse g µν related to ellipsoid as
2 2 2 2 2 2 2 2 a cos θ cos ϕ + b cos θ sin ϕ + c sin θ
gµν = g µν =
(b2 − a2 ) sin θ cos θ sin ϕ cos ϕ
1 a2 b 2 c 2
2
sin θ(
sin2
θ cos2
ϕ
a2
+
sin2
θ sin2
ϕ
b2
+
cos2 c2
θ
)
(b2 − a2 ) sin θ cos θ sin ϕ cos ϕ
a2 sin2 θ sin2 ϕ + b2 sin2 θ cos2 ϕ a2 sin2 θ sin2 ϕ +b2 sin2 θ cos2 ϕ (a2 − b2 ) sin θ cos θ sin ϕ cos ϕ
,
(55)
(a2 − b2 ) sin θ cos θ sin ϕ cos ϕ a2 cos2 θ cos2 ϕ + b2 cos2 θ sin2 ϕ +c2 sin2 θ
After long calculations we get the momentum operator representations from (54) as 2
cos θ sin ϕ θ sin ϕ + sin ab ∂ ∂ 2 ac2 sin θ + i¯ h pˆx = −i¯ h 2 2 2 2 2 2 2 2 2 2 ϕ sin θ cos ϕ ac2 ( sin θacos + sin θb2sin ϕ + cosc2 θ ) ∂θ + sin θb2sin ϕ + cosc2 θ ∂ϕ 2 a2 b2 i¯ h sin θ cos ϕ 2 2 2 sin ϕ + cos θ cos ϕ + (cos2 ϕ + + 2ϕ 2 sin2 θ sin2 ϕ cos2 θ 2 a 2 ab2 c2 ( sin2 θ cos + + 2 ) 2 2
cos θ cos ϕ
a
b
c
c2 cos2 θ sin2 ϕ) + 2 sin2 θ , a
(57) 2
cos θ cos ϕ ϕ + sin aθ 2cos ∂ ∂ bc2 sin θ b pˆy = −i¯ h − i¯ h 2 2 2 2 2 2 2 2 2 2 sin θ cos ϕ ϕ + sin θb2sin ϕ + cosc2 θ ∂ϕ bc2 ( sin θacos + sin θb2sin ϕ + cosc2 θ ) ∂θ 2 a2 sin θ sin ϕ i¯ h + cos2 ϕ + cos2 θ sin2 ϕ + 2 θ cos2 ϕ sin2 θ sin2 ϕ sin cos2 θ 2 2 a2 bc2 ( + + c2 ) a2 b2 2 2 a c 2 2 2 2 (sin ϕ + cos θ cos ϕ) + 2 sin θ , b2 b
cos θ sin ϕ
pˆz = i¯ h
sin θ(a2 sin2 ϕ + b2 cos2 ϕ) a2 b2 c(
sin2
+
θ cos2 a2
ϕ
+
sin2
θ sin2 b2
ϕ
+ cos θ
cos2 c2
(58)
∂ ∂ (a2 − b2 ) cos θ sin ϕ cos ϕ + i¯ h sin2 θ sin2 ϕ sin2 θ cos2 ϕ θ ∂θ cos2 θ ∂ϕ 2 2 ) a b c( + + 2 ) 2 2
a
b
2
c
i¯ h a sin2 θ + 2 (sin2 ϕ + cos2 θ cos2 ϕ) + 2 θ cos2 ϕ 2 θ sin2 ϕ 2θ sin sin cos 2 2 2 2 a b c( c + + c2 ) a2 b2 b2 (cos2 ϕ + cos2 θ sin2 ϕ) , (59) 2 c 11
.(56)
which satisfy the commutation relations (36) and (37) and also the third constraint (39). Our momentum representations (57)-(59) satisfy geometric momentum p~ = −i¯ h(rµ ∂µ + H~n) which has been introduced as a proper momentum for a constrained particle on two dimensional curved surfaces in terms of mean curvature H [11-13]. In follow we discuss four special cases of this curved surface.
5
Special cases
1) Spheroid we let a = b. In this case the commutators between position and momentum operators are the same as (35)-(37) where z 1 ~ (xi + y~j) + 2 ~k, 2 a c ~ = 1 (px~i + py~j) + pz ~k. K a2 c2 ~r =
(60) (61)
Also the metric can be rewritten as gµν
a2 cos2 θ + c2 sin2 θ 0 = , 2 2 0 a sin θ
g µν =
a2
cos2
1 θ+c2 sin2 θ
0
0 1 a2 sin2 θ
,
and the momentum operator representations are
c2 −i¯ h cos θ cos ϕ ∂ sin ϕ ∂ sin θ cos ϕ 2 2 − − pˆx = 2 (1 + cos θ + 2 sin θ) ,(62) 2 2 2 a c2 sin θ + cos θ ∂θ sin θ ∂ϕ a 2 2a2 c2 sina2 θ + cosc2 θ a2 c2 pˆy =
−i¯ h cos θ sin ϕ ∂ cos ϕ ∂ sin θ sin ϕ c2 2 + − sin2 θ) ,(63) (1 + cos θ + 2 2 a c2 sin2 θ + cos2 θ ∂θ sin θ ∂ϕ a 2 2 θ 2a2 c2 sin + cosc2 θ a2 c2 a2
2
cos θ(sin2 θ + ac2 (1 + cos2 θ)) i¯ h ∂ sin θ pz = + , 2 a ac sin2 θ + cos2 θ ∂θ sin2 θ cos2 θ 3 2a c a2 + c2 a2 c2 which in agreement with previous results in Ref. [13]. 2) Sphere 12
(64)
By letting a = b = c, Eq. (8) is reduced to equation on a sphere. The commutators between the position and momentum operators can be obtained from (35)- (37) as [ˆ xi , xˆj ] = 0, [ˆ xi , pˆj ] = i¯ h(δij − [ˆ pi , pˆj ] =
(65) xˆi xˆj ), a2
(66)
i¯ h (ˆ xj pˆi + pˆi xˆj − xˆi pˆj − pˆj xˆi ). 2a2
(67)
Also from (55), (56) we see that the metric gµν and its inverse g µν are reduced to the metric on sphere as gµν
2 0 a = , 0 a2 sin2 θ
g µν =
1 a2
0
0
1 a2 sin2 θ
.
(68)
According to (57)-(59), the momentum operator representations that satisfy (66) and (67) are as follows
i¯ h ∂ sin ϕ ∂ cos θ cos ϕ − − sin θ cos ϕ , a ∂θ sin θ ∂ϕ i¯ h ∂ cos ϕ ∂ pˆy = − cos θ sin ϕ + − sin θ sin ϕ , a ∂θ sinθ ∂ϕ i¯ h ∂ pˆz = sin θ + cos θ . a ∂θ
pˆx = −
(69) (70) (71)
These results are in agreement with the former representations obtained in [11].
3)Elliptical cylinder For an ellipse the two dimensional surface equation in cartesian coordinates is as follows x2 y 2 + 2 = 1. a2 b
(72)
Similar to section 3, the constraints can be calculated which are analogous to (14)-(17) without the terms that depend on z parameter. According to (6), the Dirac brackets between the variables of phase space for an ellipse can be calculated and after quantizing we have [ˆ x, yˆ] = 0, 13
(73)
[ˆ x, pˆx ] = i¯ h 1−
xˆ2 2 a4 ( xa4 +
[ˆ x, pˆy ] = [ˆ y , pˆx ] = −i¯ h
[ˆ y , pˆy ] = i¯ h 1− i¯ h [ˆ px , pˆy ] = 2 2 2a b
x2 a4
yˆ +
y2 b4
pˆx + pˆx x2 a4
xˆyˆ +
2 a2 b2 ( xa4
yˆ2 2 b4 ( xa4 + yˆ +
y2 ) b4
y2 b4
−
y2 ) b4 x2 a4
(74)
y2 ) b4
xˆ +
, ,
(75)
,
y2 b4
(76) pˆy − pˆy x2 a4
xˆ +
y2 b4
.
(77)
By considering the parametric equation as x = a cos ϕ,
y = b sin ϕ,
(78)
and rewriting the proposed formalism in section 4, we obtain the following representations for momentum operators px , py pˆx = i¯ h
a sin ϕ ∂ i¯ h ab2 cos ϕ + , 2 (a2 sin2 ϕ + b2 cos2 ϕ)2 a2 sin2 ϕ + b2 cos2 ϕ ∂ϕ
(79)
b cos ϕ ∂ i¯ h a2 b sin ϕ + 2 (a2 sin2 ϕ + b2 cos2 ϕ)2 a2 sin2 ϕ + b2 cos2 ϕ ∂ϕ
(80)
pˆy = −i¯ h
These relations clearly satisfy equations (75)-(77) and also the third constraint. 4)Circular cylinder By letting a = b, (72) is reduced to equation of a circle. So according to (73)-(77), the commutator between position and momentum operators are as follows [ˆ x, yˆ] = 0,
(81)
xˆ2 [ˆ x, pˆx ] = i¯ h 1− 2 , a xˆyˆ [ˆ x, pˆy ] = [ˆ y , pˆx ] = −i¯ h 2, a 2 yˆ [ˆ y , pˆy ] = i¯ h 1− 2 , a i¯ h [ˆ px , pˆy ] = 2 (ˆ y pˆx + pˆx yˆ − xˆpˆy − pˆy xˆ). 2a 14
(82) (83) (84) (85)
The momentum operator representations that satisfy these relations can be obtained from (79), (80) as sin ϕ ∂ i¯ h cos ϕ + , a ∂ϕ 2 a cos ϕ ∂ i¯ h sin ϕ pˆy = −i¯ h + . a ∂ϕ 2 a pˆx = i¯ h
(86) (87)
Let us compare these representations with the previous results obtained in [16] as pˆx = i¯ h
sin ϕ ∂ i¯ h cos ϕ α sin ϕ + + , a ∂ϕ 2 a a
pˆy = −i¯ h In order to explain the extra terms
cos ϕ ∂ i¯ h sin ϕ α cos ϕ + − . a ∂ϕ 2 a a α a
(88) (89)
sin ϕ and − αa cos ϕ in px and py , we should notice that
we let pˆϕ = −i¯ h
∂ , ∂ϕ
(90)
because of [ϕ, ˆ pˆϕ ] = i¯ h and hermiticity condition. But we see that by adding an arbitrary function to (90), the mentioned commutator and hermiticity condition do not change. Our ∂ calculations show that by letting pˆϕ = −i¯ h ∂ϕ − α, the momentum operator representations
(86) and (87) are in agreement with (88) and (89).
6
conclusion
In this work we have obtained the classical Dirac brackets between the variables of phase space for a constrained motion on an ellipsoid. By using these brackets, the commutators between position and momentum operators have been determined via Dirac quantization. Then we have shown that by changing the coordinate, these commutators transform to canonical commutators between position and momentum. So by knowing the momentum representations in curved space pˆθ and pˆϕ , we have determined the momentum representations pˆx , pˆy and pˆz by using of bein formalism which enables us to write the Schrodinger 15
equation. We show that irrespective of geometrical point of view our calculations are in agreement with geometric momentum which has been considered as a proper momentum for constrained particle on curved surfaces. We have also discussed some special cases of this problem and compared them with former results in other papers.
References [1] B. Sakita, K. Kikkawa, Quantum mechanics of many degrees of freedom basing on path integral, Iwanami, Tokyo (1986). [2] N. Ogawa, K.Fuji, N. Chepilko, A. Kobushukin, Prog. Theor. Phys. 85 (1991), 1189. [3] N. M. Chepilko, A. V. Romanenko, Eur. Phys. J. C 21 (2001), 587. [4] P. C. Schuster, R. L. Jaffe, Ann. Phys. 307 (2003), 132. [5] S. Matsutani, J. Phys. A: Math. Gen. 26 (1993), 5133. [6] G. Jorjadze, W. Piechoki, Phys. Lett. B. 461 (1999), 183. [7] K. Kowalski, J. Rembielinski, Ann. Phys. 336 (2013), 167. [8] K. Kowalski, J. Rembielinski, Ann. Phys. 329 (2013), 146. [9] N.Ogawa, K. Fuji, A. Kobushukin, Prog. Theor. Phys. 83 (1990), 894. [10] P. A. M. Dirac, Lectures on quantum mechanics, Yeshiva university, New York (1964). [11] Q. H. Liu, L. H. Tang, D. M. Xun, Phys. Rev. A. 84 (2011), 042101. [12] D. M. Xun, Q. H. Liu, X. M. Zhu, Ann. Phys. 338 (2013), 123. [13] Q. H. Liu, C. L. Tong, M. M. Lai, J. Phys. A: Math. Theor. 40 (2007), 4161.
16
[14] H. J. Rothe, K. D. Rothe, Classical and quantum dynamics of constrained Hamiltonian systems, World Scientific Lecture Notes in Physics- Vol. 81. [15] M. Nakahara, Geometry, Topology and physics, Institute of Physisics publishing Bristol and Philadelphia, 2nd edition (2003). [16] A. Scardicchio, Phys. Lett. A. 300 (2002), 7.
17
S0003-4916(16)30030-6 http://dx.doi.org/10.1016/j.aop.2016.04.013 YAPHY 67092
To appear in:
Annals of Physics
Received date: 20 November 2015 Accepted date: 19 April 2016 Please cite this article as: H. Panahi, L. Jahangiri, Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum, Annals of Physics (2016), http://dx.doi.org/10.1016/j.aop.2016.04.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum H. Panahi ∗, L. Jahangiri †, Department of Physics , University of Guilan, Rasht 41635-1914, Iran.
February 22, 2016
Abstract In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using of Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one. Keywords: Quantum mechanics, Ellipsoid, Dirac method, Bein formalism, Geometric momentum. PACS numbers: 03.65.-w, 04.60.Ds ∗
Corresponding author (E-mail: [email protected])
†
( E-mail: [email protected])
1
1
Introduction
The quantum mechanics in curved space has attracted much attention during the last decades and different authors have considered this problem via various methods [1-5]. In Ref. [6], the classical and quantum dynamics of a particle has been investigated in 2D space-time with constant curvature. They have considered the motion of a relativistic particle in the cases of one-sheet hyperboloid, stripe and half-plane. Also the classical and quantum free particle on one-sheeted hyperboloid have been discussed in [7]. They have solved the classical equations of motion and identified the Hilbert space and observables for this particle as well. As another effort in Ref. [8], these authors have performed a detailed study of the classical and quantum mechanics for a particle on the double cone in two cases: free and bounded by the harmonic oscillator potential. In Ref. [9], the classical and quantum motion of a particle on Riemannian manifolds have been studied by considering the curved space as a hypersurface in higher dimensional Euclidean space via Dirac method. The Dirac method has been introduced as a powerful method in order to study the classical and quantum mechanical systems using the Hamiltonian method specially when the Lagrangian is singular [10]. In this approach the Poisson brackets are replaced by Dirac brackets and passing from classical to quantum mechanics is attainable through canonical quantization procedure. According to Dirac approach for a constrained motion, there are commutators between position and momentum operators which are not sufficient to lead to unique forms of the momentum and Hamiltonian operators for the system. For instance, for a spherical surface, there are more than five different forms of momentum and Hamiltonian which have been discussed in [11]. In order to remove this problem there are two formalisms to study the quantum motion on two dimensional curved surface including Schrodinger formalism and canonical quantization scheme which lead to geometric potential and geometric momentum. These formalisms have been applied in Ref. [11,12,13] and the authors have discussed the quantum motion of a constrained particle on some two dimensional curved surfaces like torus, sphere and 2
etc. Their calculations are based on gaussian and mean curvatures which are intrinsic and extrinsic curvatures respectively. Now in this paper we apply the Dirac method in order to study the classical and quantum mechanical behaviour of a particle on an ellipsoid and then determine the momentum operator representations by using of bein formalism. We show that irrespective of the geometrical point of view, our momentum operator representations are in agreement with geometric momentum. The paper is organized as follows: In section 2, we review the basic concepts of Dirac approach briefly. In section 3, we apply the results of the previous section for calculating the constraints and the Dirac brackets and then by quantizing, we calculate the commutators between position and momentum operators for an ellipsoid. In section 4, using of the bein formalism, we try to obtain the momentum operator representations that satisfy the obtained commutators. In section 5, we discuss the special cases like sphere, spheroid ad etc which shows that our calculations are in agreement with the obtained former results in other papers. In section 6, we end our work with a brief conclusion.
2
A brief review of Dirac approach
For a system with a finite number of degrees of freedom, the Lagrangian is a function of coordinates and velocities as L({qi }, {q˙i }). In order to get the equations of motion one should construct the Hamiltonian. For this purpose, the momenta are obtained as follows pi =
∂L . ∂ q˙i
(1)
For a non-singular Lagrangian, the velocities q˙i can be expressed in terms of coordinates and momenta [14] q˙i = fi (q, p).
(2)
The Hamiltonian is then obtained as a function of momenta and coordinates as H(q, p) =
X i
pi fi (q, p) − L(q, f (q, p)). 3
(3)
But when the Lagrangian is singular, there is not an explicit dependence of velocities on momenta and so it is impossible to write the Hamiltonian only in terms of qi and pi similar to (3). In such a case, according to Dirac method, those momenta that cause difficulty are considered as a set of primary constraints ϕj which are added to Hamiltonian as follows HT = H +
X
uj ϕ j .
(4)
j
Then the consistency conditions, time evolution of primary constraints, should be calculated by using of the following equation ϕ˙ i = {ϕi , HT }P ,
(5)
where the {, }p is the Poisson bracket. The consistency conditions may lead to new constraints which are called secondary constraints. It should be pointed out that the primary and secondary constraints are weakly equal to zero which in the next section we explain about it clearly. Another important part of the Dirac method is replacing the Poisson bracket with the Dirac bracket which is defined as {A, B}D = {A, B}P −
X k,l
−1 {A, ϕk }P σkl {ϕl , B}P ,
(6)
where σkl = {ϕk , ϕl }P .
(7)
One of the applications of Dirac method that yields interesting results is in studying the motion of a particle on curved hypersurfaces. In this situation the hypersurface is considered in higher dimensional Euclidean space which means the motion with constraint. As we mentioned before, in Ref. [9] the authors have discussed the classical and quantum mechanical treatments on the hypersurface VN −1 embedded in N dimensional Euclidean space RN from the view point of Dirac method. In the next section we apply this method in order to study the classical and quantum mechanics on an ellipsoid.
4
3
Classical ad quantum mechanics on an ellipsoid as a 2-dimensional surface embedded in 3-dimensional Euclidean space
In this section we consider an ellipsoid in 3-dimensional Euclidean space which is specified by cartesian coordinates (x, y, z). The surface equation in cartesian coordinates is give by x2 y 2 z 2 + 2 + 2 = 1. a2 b c
(8)
The Lagrangian for this system is
1 x2 y 2 z 2 L = m x˙ 2 + y˙ 2 + z˙ 2 + λ 2 + 2 + 2 − 1 , 2 a b c
(9)
where according to Dirac method the equation of ellipsoid has been added to Lagrangian as a constraint and λ is a dynamical variable. In order to write the Hamiltonian let us first calculate the conjugate mumenta px =
∂L = mx, ˙ ∂ x˙
py =
∂L = my, ˙ ∂ y˙
pλ =
pz =
∂L = mz, ˙ ∂ z˙
∂L ≈ 0. ∂ λ˙
(10)
(11)
As we mentioned before the second equation is considered as the primary constraint φ1 = pλ ≈ 0,
(12)
so the total Hamiltonian of the system becomes
x2 y 2 z 2 1 p2x + p2y + p2z − λ 2 + 2 + 2 − 1 + upλ , HT = 2m a b c
(13)
where u is a multiplier. For this constrained motion, the consistency condition should be examined so we calculate the time evolution of primary constraint (12) as follows:
x2 y 2 z 2 φ2 = φ˙1 = {φ1 , HT }P = + 2 + 2 − 1 ≈ 0, a2 b c 5
(14)
2 xpx ypy zpz φ3 = φ˙2 = {φ2 , HT }P = + 2 + 2 m a2 b c
≈ 0,
(15)
2 2 1 p2x p2y p2z x y2 z2 ˙ φ4 = φ3 = {φ3 , HT }P = + 2 + 2 + 2λ 4 + 4 + 4 ≈ 0, m m a2 b c a b c
4 4λ xpx ypy zpz φ5 = φ˙4 = {φ4 , HT }P = + 4 + 4 m m a4 b c
x2 y 2 z 2 + 4 + 4 + 4 u ≈ 0. a b c
(16)
(17)
At this stage the process of finding new constraints terminates. In above relations the expression ≈ means weak equality since we need these constraints for the purpose of calculating Dirac brackets and after that they become strong equations. As we see φ2 is the surface equation and φ3 shows that for a confining particle on an ellipsoid, the momentum has no component in the normal direction of surface by supposing that the unit normal vector is as follows
x ~ 2 i + ~n = a q 2 x a4
z ~ k c2 . 2 + zc4
~j +
y b2
y2 b4
+
(18)
Also (16) and (17) determine the variables λ and u as:
λ=
−1 m
2
u=
−4λ m
p2x a2
x2 a4
x2 a4
y2 b4
+
xpx a4
p2y b2
+
+
+
+
ypy b4
y2 b4
p2z c2
+
+
z2 c4
+ z2 c4
,
zpz c4
(19)
.
(20)
Now we want to calculate the Dirac brackets between the variables of phase space. But first −1 let us determine the antisymmetric matrix σkl and its inverse σkl according to (7) as
σkl
=
0
0
0
0
0
σ14
0
−σ14 −σ24 −σ34 6
σ14
−σ14 σ24 0
σ34 0
,
(21)
−1 σkl
where
=
0
σ34 2 σ14
−σ24 2 σ14
− σ114
−σ34 2 σ14
0
1 σ14
0
σ24 2 σ14
− σ114
0
0
1 σ14
0
0
0
σ14 = − σ24 =
σ34
,
(22)
4 x2 y 2 z 2 + 4 + 4 , m a4 b c
(23)
(24)
8 xpx ypy zpz + 4 + 4 , m 2 a4 b c
2 8 1 p2x p2y p2z x y2 z2 = 2 + 4 + 4 − 2λ 6 + 6 + 6 . m m a4 b c a b c
(25)
Using (6) and (22) we get the following Dirac brackets: {xi , xj }D = 0, {xi , pj }D = δij − {pi , pj }D =
(26)
ri rj , r2
(27)
1 r j Ki − r i Kj , r2
(28)
where ~r =
x ~ y ~ z i + 2 j + 2 ~k, 2 a b c
~ = px ~i + py ~j + pz ~k. K a2 b2 c2
(29)
(30)
As we mentioned before, after calculating Dirac brackets, constraints are treated as strong equations. So we have only two necessary constraints which are: x2 y 2 z 2 + 2 + 2 − 1 = 0, a2 b c 7
(31)
xpx ypy zpz + 2 + 2 = 0. a2 b c
(32)
We consider (32) as ~n · p~ = 0 even though both are equivalent in classical mechanics but different in quantum mechanics. Also the Hamiltonian (13) becomes
1 H= p2 + p2y + p2z . 2m x
(33)
From the classical point of view the particle is characterized by the set equations (26)-(28) and constraints (31), (32). As usual, the quantization problem is attainable through the Dirac quantization. The only difference is that the Poisson brackets should be replaced by Dirac brackets so [A, B] = i¯ h{A, B}D .
(34)
As a result the classical relations (26)-(28) become [ˆ xi , xˆj ] = 0,
[ˆ xi , pˆj ] = i¯ h δij − [ˆ pi , pˆj ] =
(35)
rˆi rˆj , r2
(36)
i¯ h rˆj ˆ ˆ i rˆj − rˆi K ˆj − K ˆ j rˆi . Ki + K 2 2 2 2 r r r r2
(37)
Also the constraint (31) and (32) are given by x2 y 2 z 2 + 2 + 2 − 1 = 0, a2 b c
(38)
1 1 1 (ˆ xpˆx + pˆx xˆ) + 2 (ˆ y pˆy + pˆy yˆ) + 2 (ˆ z pˆz + pˆz zˆ) = 0. 2 2a 2b 2c
(39)
Taking into consideration the hermiticity condition, in the right hand side of (37) and left hand side of (39), the ordering problem has been imposed by letting the anticommutator between the momentum operator and those which depend on position. In the next section we try to find the momentum operators pˆx , pˆy , pˆz that satisfy (36), (37) and (39).
8
4
The momentum operator representations for an ellipsoid: bein formalism
So far all results are obtained in Euclidean space. By considering (8), we change the coordinate as x = a sin θ cos ϕ
y = b sin θ sin ϕ
z = c cos θ.
(40)
As we see instead of three variables (x, y, z) in cartesian coordinate, we have only two coordinates (θ, ϕ) on ellipsoid because of (8). As we know by changing the coordinate, the beins ei
µ
and their inverse eµ
i
are used to transform the physical quantities from flat to curved
frames and vice versa. As a convention we put the Latin and Greek indices as global and local indices which take the values 1,2,3 and θ, ϕ respectively. Since a vector is independent of the basis chosen, so we expect that under change of coordinate frame, the components of a vector transform as [15] ∂xa µ p = e µp = p , ∂xµ
(41)
∂xµ a p . ∂xa
(42)
a
a
µ
pµ = eµ a pa =
So using of the bein formalism and knowing the momentum in cartesian coordinate we can obtain the momentum in curved space and vice versa. The only important point is that from the quantum mechanical point of view, (41) and (42) should be rewritten as
where {ˆ a, ˆb} =
1 ˆ (ˆ ab 2
pˆa = {ˆ ea µ , pˆµ },
(43)
pˆµ = {ˆ eµ a , pˆa },
(44)
+ ˆbˆ a). Since beins are operators that depend on position and one
should consider their symmetric form with momentum operator. By letting θ = cos−1 9
z c
and
ϕ = tan−1
ay , bx
according to (40) in (35)-(37) we get ˆ ϕ] ˆ θ] ˆ = [ϕ, [θ, ˆ = [θ, ˆ ϕ] ˆ = 0,
ˆ pˆθ ] = [ϕ, [θ, ˆ pˆϕ ] = i¯ h,
ˆ pˆϕ ] = [ϕ, [θ, ˆ pˆθ ] = 0,
[ˆ pθ , pˆϕ ] = [ˆ pθ , pˆθ ] = [ˆ pϕ , pˆϕ ] = 0,
(45)
(46)
(47)
a
∂x ˆa } which can be obtained from (42) and also the fundamental here we have used pˆν = { ∂x ν,p
quantum mechanical relation [9] ˆ = [F (x), A]
∂F ˆ [ˆ x, A]. ∂x
(48)
From equation (46) and (47) we find that pˆθ and pˆϕ can be represented as pˆθ = −i¯ h
∂ + i¯ h f (θ, ϕ), ∂θ
(49)
pˆϕ = −i¯ h
∂ + i¯ h g(θ, ϕ). ∂ϕ
(50)
The functions f (θ, ϕ) and g(θ, ϕ) should be chosen so that pθ and pϕ be hermitian operators. Taking into consideration the surface element of the ellipsoid, we can construct the Hilbert space which is specified by the following scalar product hψ1 |ψ2 i =
Z
2π
0
Z
0
π
q
sin θ a2 b2 cos2 θ + a2 c2 sin2 θ sin2 ϕ + b2 c2 sin2 θ cos2 ϕ ψ1∗ ψ2 dθdϕ.
(51)
So we propose the following representations for pˆθ and pˆϕ which are hermitian with respect to scalar product (51) i¯ h i¯ h sin θ cos θ(−a2 b2 + a2 c2 sin2 ϕ + b2 c2 cos2 ϕ) ∂ − cot θ − , ∂θ 2 2 a2 b2 cos2 θ + a2 c2 sin2 θ sin2 ϕ + b2 c2 sin2 θ cos2 ϕ
(52)
∂ i¯ h sin2 θ sin ϕ cos ϕc2 (a2 − b2 ) . − ∂ϕ 2 a2 b2 cos2 θ + a2 c2 sin2 θ sin2 ϕ + b2 c2 sin2 θ cos2 ϕ
(53)
pˆθ = −i¯ h pˆϕ = −i¯ h
10
We should notice that these hermitian representations satisfy (46) and (47). Now by knowing pˆθ and pˆϕ we can determine the operators pˆx , pˆy , pˆz from the quantum mechanical form of equation (41) as follows pˆb = {
∂xa δab g µν , pˆν }, µ ∂x
(54)
where g µν and δab are the metrics of curved and Euclidean space respectively. This relation makes us sure that the obtained pˆa are obviously hermitian. Let us calculate the metric gµν and its inverse g µν related to ellipsoid as
2 2 2 2 2 2 2 2 a cos θ cos ϕ + b cos θ sin ϕ + c sin θ
gµν = g µν =
(b2 − a2 ) sin θ cos θ sin ϕ cos ϕ
1 a2 b 2 c 2
2
sin θ(
sin2
θ cos2
ϕ
a2
+
sin2
θ sin2
ϕ
b2
+
cos2 c2
θ
)
(b2 − a2 ) sin θ cos θ sin ϕ cos ϕ
a2 sin2 θ sin2 ϕ + b2 sin2 θ cos2 ϕ a2 sin2 θ sin2 ϕ +b2 sin2 θ cos2 ϕ (a2 − b2 ) sin θ cos θ sin ϕ cos ϕ
,
(55)
(a2 − b2 ) sin θ cos θ sin ϕ cos ϕ a2 cos2 θ cos2 ϕ + b2 cos2 θ sin2 ϕ +c2 sin2 θ
After long calculations we get the momentum operator representations from (54) as 2
cos θ sin ϕ θ sin ϕ + sin ab ∂ ∂ 2 ac2 sin θ + i¯ h pˆx = −i¯ h 2 2 2 2 2 2 2 2 2 2 ϕ sin θ cos ϕ ac2 ( sin θacos + sin θb2sin ϕ + cosc2 θ ) ∂θ + sin θb2sin ϕ + cosc2 θ ∂ϕ 2 a2 b2 i¯ h sin θ cos ϕ 2 2 2 sin ϕ + cos θ cos ϕ + (cos2 ϕ + + 2ϕ 2 sin2 θ sin2 ϕ cos2 θ 2 a 2 ab2 c2 ( sin2 θ cos + + 2 ) 2 2
cos θ cos ϕ
a
b
c
c2 cos2 θ sin2 ϕ) + 2 sin2 θ , a
(57) 2
cos θ cos ϕ ϕ + sin aθ 2cos ∂ ∂ bc2 sin θ b pˆy = −i¯ h − i¯ h 2 2 2 2 2 2 2 2 2 2 sin θ cos ϕ ϕ + sin θb2sin ϕ + cosc2 θ ∂ϕ bc2 ( sin θacos + sin θb2sin ϕ + cosc2 θ ) ∂θ 2 a2 sin θ sin ϕ i¯ h + cos2 ϕ + cos2 θ sin2 ϕ + 2 θ cos2 ϕ sin2 θ sin2 ϕ sin cos2 θ 2 2 a2 bc2 ( + + c2 ) a2 b2 2 2 a c 2 2 2 2 (sin ϕ + cos θ cos ϕ) + 2 sin θ , b2 b
cos θ sin ϕ
pˆz = i¯ h
sin θ(a2 sin2 ϕ + b2 cos2 ϕ) a2 b2 c(
sin2
+
θ cos2 a2
ϕ
+
sin2
θ sin2 b2
ϕ
+ cos θ
cos2 c2
(58)
∂ ∂ (a2 − b2 ) cos θ sin ϕ cos ϕ + i¯ h sin2 θ sin2 ϕ sin2 θ cos2 ϕ θ ∂θ cos2 θ ∂ϕ 2 2 ) a b c( + + 2 ) 2 2
a
b
2
c
i¯ h a sin2 θ + 2 (sin2 ϕ + cos2 θ cos2 ϕ) + 2 θ cos2 ϕ 2 θ sin2 ϕ 2θ sin sin cos 2 2 2 2 a b c( c + + c2 ) a2 b2 b2 (cos2 ϕ + cos2 θ sin2 ϕ) , (59) 2 c 11
.(56)
which satisfy the commutation relations (36) and (37) and also the third constraint (39). Our momentum representations (57)-(59) satisfy geometric momentum p~ = −i¯ h(rµ ∂µ + H~n) which has been introduced as a proper momentum for a constrained particle on two dimensional curved surfaces in terms of mean curvature H [11-13]. In follow we discuss four special cases of this curved surface.
5
Special cases
1) Spheroid we let a = b. In this case the commutators between position and momentum operators are the same as (35)-(37) where z 1 ~ (xi + y~j) + 2 ~k, 2 a c ~ = 1 (px~i + py~j) + pz ~k. K a2 c2 ~r =
(60) (61)
Also the metric can be rewritten as gµν
a2 cos2 θ + c2 sin2 θ 0 = , 2 2 0 a sin θ
g µν =
a2
cos2
1 θ+c2 sin2 θ
0
0 1 a2 sin2 θ
,
and the momentum operator representations are
c2 −i¯ h cos θ cos ϕ ∂ sin ϕ ∂ sin θ cos ϕ 2 2 − − pˆx = 2 (1 + cos θ + 2 sin θ) ,(62) 2 2 2 a c2 sin θ + cos θ ∂θ sin θ ∂ϕ a 2 2a2 c2 sina2 θ + cosc2 θ a2 c2 pˆy =
−i¯ h cos θ sin ϕ ∂ cos ϕ ∂ sin θ sin ϕ c2 2 + − sin2 θ) ,(63) (1 + cos θ + 2 2 a c2 sin2 θ + cos2 θ ∂θ sin θ ∂ϕ a 2 2 θ 2a2 c2 sin + cosc2 θ a2 c2 a2
2
cos θ(sin2 θ + ac2 (1 + cos2 θ)) i¯ h ∂ sin θ pz = + , 2 a ac sin2 θ + cos2 θ ∂θ sin2 θ cos2 θ 3 2a c a2 + c2 a2 c2 which in agreement with previous results in Ref. [13]. 2) Sphere 12
(64)
By letting a = b = c, Eq. (8) is reduced to equation on a sphere. The commutators between the position and momentum operators can be obtained from (35)- (37) as [ˆ xi , xˆj ] = 0, [ˆ xi , pˆj ] = i¯ h(δij − [ˆ pi , pˆj ] =
(65) xˆi xˆj ), a2
(66)
i¯ h (ˆ xj pˆi + pˆi xˆj − xˆi pˆj − pˆj xˆi ). 2a2
(67)
Also from (55), (56) we see that the metric gµν and its inverse g µν are reduced to the metric on sphere as gµν
2 0 a = , 0 a2 sin2 θ
g µν =
1 a2
0
0
1 a2 sin2 θ
.
(68)
According to (57)-(59), the momentum operator representations that satisfy (66) and (67) are as follows
i¯ h ∂ sin ϕ ∂ cos θ cos ϕ − − sin θ cos ϕ , a ∂θ sin θ ∂ϕ i¯ h ∂ cos ϕ ∂ pˆy = − cos θ sin ϕ + − sin θ sin ϕ , a ∂θ sinθ ∂ϕ i¯ h ∂ pˆz = sin θ + cos θ . a ∂θ
pˆx = −
(69) (70) (71)
These results are in agreement with the former representations obtained in [11].
3)Elliptical cylinder For an ellipse the two dimensional surface equation in cartesian coordinates is as follows x2 y 2 + 2 = 1. a2 b
(72)
Similar to section 3, the constraints can be calculated which are analogous to (14)-(17) without the terms that depend on z parameter. According to (6), the Dirac brackets between the variables of phase space for an ellipse can be calculated and after quantizing we have [ˆ x, yˆ] = 0, 13
(73)
[ˆ x, pˆx ] = i¯ h 1−
xˆ2 2 a4 ( xa4 +
[ˆ x, pˆy ] = [ˆ y , pˆx ] = −i¯ h
[ˆ y , pˆy ] = i¯ h 1− i¯ h [ˆ px , pˆy ] = 2 2 2a b
x2 a4
yˆ +
y2 b4
pˆx + pˆx x2 a4
xˆyˆ +
2 a2 b2 ( xa4
yˆ2 2 b4 ( xa4 + yˆ +
y2 ) b4
y2 b4
−
y2 ) b4 x2 a4
(74)
y2 ) b4
xˆ +
, ,
(75)
,
y2 b4
(76) pˆy − pˆy x2 a4
xˆ +
y2 b4
.
(77)
By considering the parametric equation as x = a cos ϕ,
y = b sin ϕ,
(78)
and rewriting the proposed formalism in section 4, we obtain the following representations for momentum operators px , py pˆx = i¯ h
a sin ϕ ∂ i¯ h ab2 cos ϕ + , 2 (a2 sin2 ϕ + b2 cos2 ϕ)2 a2 sin2 ϕ + b2 cos2 ϕ ∂ϕ
(79)
b cos ϕ ∂ i¯ h a2 b sin ϕ + 2 (a2 sin2 ϕ + b2 cos2 ϕ)2 a2 sin2 ϕ + b2 cos2 ϕ ∂ϕ
(80)
pˆy = −i¯ h
These relations clearly satisfy equations (75)-(77) and also the third constraint. 4)Circular cylinder By letting a = b, (72) is reduced to equation of a circle. So according to (73)-(77), the commutator between position and momentum operators are as follows [ˆ x, yˆ] = 0,
(81)
xˆ2 [ˆ x, pˆx ] = i¯ h 1− 2 , a xˆyˆ [ˆ x, pˆy ] = [ˆ y , pˆx ] = −i¯ h 2, a 2 yˆ [ˆ y , pˆy ] = i¯ h 1− 2 , a i¯ h [ˆ px , pˆy ] = 2 (ˆ y pˆx + pˆx yˆ − xˆpˆy − pˆy xˆ). 2a 14
(82) (83) (84) (85)
The momentum operator representations that satisfy these relations can be obtained from (79), (80) as sin ϕ ∂ i¯ h cos ϕ + , a ∂ϕ 2 a cos ϕ ∂ i¯ h sin ϕ pˆy = −i¯ h + . a ∂ϕ 2 a pˆx = i¯ h
(86) (87)
Let us compare these representations with the previous results obtained in [16] as pˆx = i¯ h
sin ϕ ∂ i¯ h cos ϕ α sin ϕ + + , a ∂ϕ 2 a a
pˆy = −i¯ h In order to explain the extra terms
cos ϕ ∂ i¯ h sin ϕ α cos ϕ + − . a ∂ϕ 2 a a α a
(88) (89)
sin ϕ and − αa cos ϕ in px and py , we should notice that
we let pˆϕ = −i¯ h
∂ , ∂ϕ
(90)
because of [ϕ, ˆ pˆϕ ] = i¯ h and hermiticity condition. But we see that by adding an arbitrary function to (90), the mentioned commutator and hermiticity condition do not change. Our ∂ calculations show that by letting pˆϕ = −i¯ h ∂ϕ − α, the momentum operator representations
(86) and (87) are in agreement with (88) and (89).
6
conclusion
In this work we have obtained the classical Dirac brackets between the variables of phase space for a constrained motion on an ellipsoid. By using these brackets, the commutators between position and momentum operators have been determined via Dirac quantization. Then we have shown that by changing the coordinate, these commutators transform to canonical commutators between position and momentum. So by knowing the momentum representations in curved space pˆθ and pˆϕ , we have determined the momentum representations pˆx , pˆy and pˆz by using of bein formalism which enables us to write the Schrodinger 15
equation. We show that irrespective of geometrical point of view our calculations are in agreement with geometric momentum which has been considered as a proper momentum for constrained particle on curved surfaces. We have also discussed some special cases of this problem and compared them with former results in other papers.
References [1] B. Sakita, K. Kikkawa, Quantum mechanics of many degrees of freedom basing on path integral, Iwanami, Tokyo (1986). [2] N. Ogawa, K.Fuji, N. Chepilko, A. Kobushukin, Prog. Theor. Phys. 85 (1991), 1189. [3] N. M. Chepilko, A. V. Romanenko, Eur. Phys. J. C 21 (2001), 587. [4] P. C. Schuster, R. L. Jaffe, Ann. Phys. 307 (2003), 132. [5] S. Matsutani, J. Phys. A: Math. Gen. 26 (1993), 5133. [6] G. Jorjadze, W. Piechoki, Phys. Lett. B. 461 (1999), 183. [7] K. Kowalski, J. Rembielinski, Ann. Phys. 336 (2013), 167. [8] K. Kowalski, J. Rembielinski, Ann. Phys. 329 (2013), 146. [9] N.Ogawa, K. Fuji, A. Kobushukin, Prog. Theor. Phys. 83 (1990), 894. [10] P. A. M. Dirac, Lectures on quantum mechanics, Yeshiva university, New York (1964). [11] Q. H. Liu, L. H. Tang, D. M. Xun, Phys. Rev. A. 84 (2011), 042101. [12] D. M. Xun, Q. H. Liu, X. M. Zhu, Ann. Phys. 338 (2013), 123. [13] Q. H. Liu, C. L. Tong, M. M. Lai, J. Phys. A: Math. Theor. 40 (2007), 4161.
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[14] H. J. Rothe, K. D. Rothe, Classical and quantum dynamics of constrained Hamiltonian systems, World Scientific Lecture Notes in Physics- Vol. 81. [15] M. Nakahara, Geometry, Topology and physics, Institute of Physisics publishing Bristol and Philadelphia, 2nd edition (2003). [16] A. Scardicchio, Phys. Lett. A. 300 (2002), 7.
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