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Classical and quantum motion in an inverse square potential
Physics Letters A 373 (2009) 418–421
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Physics Letters A www.elsevier.com/locate/pla
Classical and quantum motion in an inverse square potential M. Ávila-Aoki a , C. Cisneros b , R.P. Martínez-y-Romero c , H.N. Núñez-Yépez d , A.L. Salas-Brito e,∗ a
Centro Universitario Valle de Chalco, Universidad Autónoma del Estado de México, Valle de Chalco, CP 56615, Estado de México, Mexico Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 6-96, CP 62131, Cuernavaca, Morelos, Mexico c Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal 21-267, CP 04000, Coyoacán DF, Mexico d Departamento Física, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-534, CP 09340, Iztapalapa DF, Mexico e Laboratorio de Sistemas Dinámicos, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Apartado Postal 21-267, CP 04000, Coyoacán DF, Mexico b
a r t i c l e
i n f o
Article history: Received 1 August 2008 Received in revised form 22 November 2008 Accepted 25 November 2008 Available online 4 December 2008 Communicated by P.R. Holland Keywords: Inverse square potential Free motion on a hyperbola Superselection rule
a b s t r a c t Classical motion in an inverse square potential is shown to be equivalent to free motion on a hyperbola. The existence of a classical splitting between the q > 0 and q < 0 regions of motion is demonstrated. We show that this last property may be regarded as the classical counterpart of the superselection rule occurring in the corresponding quantum problem. We solve the quantum problem in momentum space finding that there is no way of quantizing its energy but that the eigenfunctions suffice to describe the single renormalized bound state of the system. The dynamical symmetry of the classical problem is found to be O (1, 1). Both this symmetry and the symmetry of inversion through the origin are found to be broken. © 2008 Elsevier B.V. All rights reserved.
The quantum problem of motion under an attractive inverse square potential has been discussed for years [1–12]. Motion in such a potential has been investigated because it is relevant for certain applications [13–21]. The vortex structure of the probability flow of a single particle is related to behaviour under an inverse-square potential [4,10,11]. This sort of potential has also been used for studying oscillators coupled to dipole fields, using the potential β q2 + g /q2 , a case in which tunnelling across the barrier has been found possible [12,22]. A charged particle interacting with an electric dipole can be described through an inverse square potential, so it is useful for describing the scattering of electrons off polar molecules, and to explain the existence of a critical dipole moment for the binding of an electron to a polar molecule. It has been used to argue the existence of an anomaly in molecular physics. The need to having a better understanding of the problem has prompted the use of field theoretic techniques, thus its renormalized solutions have been computed employing cut-off [18] and dimensional regularization [19]. Such schemes have established the existence of a single bound state breaking the scale invariance of the problem and providing an example of dimensional transmutation in a finite system [18,19]. A related problem has been studied
*
Corresponding author. E-mail addresses: [email protected] (M. Ávila-Aoki), carmen@ce.fis.unam.mx (C. Cisneros), [email protected] (R.P. Martínez-y-Romero), [email protected] (H.N. Núñez-Yépez), [email protected] (A.L. Salas-Brito). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.11.054
in [23] but their results are not strictly applicable to the case of a inverse square interaction as they studied potentials of the form |x|− p with p greater or equal than 1 but strictly less than 2. In this Letter we study the one-dimensional problem showing features of its quantum solution that are analogous to properties of the classical motion. We show the existence of a superselection rule which may be regarded as a quantum extension of a classical feature, the independence between the right and the left sides of the singularity at q = 0. Let us recall that classical motion under certain potentials can be made equivalent to geodesic motion on a curved space. An outstanding example is the hydrogen atom which is equivalent to geodesic flow on a sphere [25,26]. We here show that another example is the classical 1D motion under the inverse square potential V (q) = −
k q2
(1)
with k > 0. The motion under (1) can be transformed to free motion on a branch of a hyperbola. These classical results are then used to analyse its quantum behaviour. More features of the classical motion under (1) are analyzed and shown to be related to peculiarities of its quantum solution. In particular we show that the classical splitting that occurs at the singularity at q = 0 is related to both the existence of a superselection rule and to the spontaneous breaking of parity that occur in the quantum system. This result hints towards the possibility of us-
M. Ávila-Aoki et al. / Physics Letters A 373 (2009) 418–421
419
ing this kind of interactions for quantum information processing [28,29]. To show the equivalence of motion under the potential (1) with geodesic motion of a particle on a hyperbola, let us begin with the autonomous classical Hamiltonian H=
p2 2m
−
k q2
(2)
,
so H is a constant of the motion and the energy, E, is clearly conserved. Other conspicuous feature of Hamiltonian (2) is its scale invariance, any change q → λq and p → p /λ2 leaves the equations of motion unchanged if the energy is rescaled as E → λ2 E. Thence the problem has no characteristic dimensional scales. Quantum mechanically this scaling symmetry also holds [24] but such quantum symmetry can be broken at a certain value of the coupling constant [17]. To classically regularize the problem the first step is to perform a π /2 rotation in phase space, p → q and q → − p, to transform (2) to q2 2m
k
−
= E.
p2
(3)
We analyze the case of negative energies so we set q2 /2m ≡ − E > 0 and rewrite (3) as
q2 2m
= k/ p 2 −
2 p 1 +1 = .
(4)
k
Let us define
q2
P = p 1+
4α 2
(5)
,
where we have used α 2 ≡ m (q, p )/2 — where we are making explicit that is merely a shorthand for a function of p and q. We need to choose a Q as to complete a canonical transformation. To this end, we use the generating function F (q, P ) =
2km log 2P q + 2 2km + P 2 q2 ,
√
(6)
from which we get, through p = ∂ F (q, P )/∂ q, the momentum (5), and the coordinate
∂ F (q, P ) p 2 q2 Q = = −q 1 − . ∂P 2km
(7)
The transformed Hamiltonian, H , is then H =
P2 2μ
(8)
,
where μ ≡ k/2. We have thus transformed the Hamiltonian of the problem to a free particle form [26]. Using in (2) the alternate coordinates
q ξ = arcsinh √
2m
,
pξ =
√
2m p cosh(ξ ),
(9)
the original Hamiltonian is also converted to the free form (8), H =
p 2ξ 2μ
.
(10)
These coordinates (ξ, p ξ ) exhibit the phase curve on which the particle freely moves, it is a branch of a hyperbola. The hyperbola branch gets traversed at a constant speed in one of the two possible ways, either from ξ = −∞ to ξ = +∞ or the other way round. Notice that in the process of getting (8) — or (10) — we have regularized the problem, the singularity at q = 0 has disappeared from the Hamiltonian. It is not possible to transform the problem to a
Fig. 1. Phase portrait (p vs q) of the system with Hamiltonian (2) in arbitrary units. Notice how the phase trajectories avoid crossing from negative to positive values of q and vice versa. Classically, the particle remain confined in just one side of the singularity for ever.
free particle moving on a compact region, the motion is necessarily unbounded. The two possible ways of moving on the hyperbola correspond to the bouncing motions that can occur in the original problem, the particle can come towards the origin starting from the right at an initial position q0 = + (k/ ) > 0 with zero speed then move towards the origin and, instead of getting to it, change its velocity from −∞ to +∞, bouncing back towards its starting position in a finite time. This can be explicitly proved just noting that the following Poisson bracket
H , sgn(q) = 0,
(11)
always vanishes. Hence, S = sgn(q) is a classical constant of the motion. The two motions with the same energy are both parityrelated and independent from one another. This feature can be appreciated from the phase portrait, in the original coordinates [used in (2)] shown in Fig. 1, which clearly exhibits how the system avoids crossing through the singularity. For proving the constancy of S we used the original Hamiltonian (2), therefore the constant is valid both for the E 0 case and for the E < 0 case and does not depend on the canonical transformation or the regularization performed. The dynamical invariance group of both the original classical problem (1) and of the free particle on a hyperbola is the same, O (1, 1), with both translations ξ → ξ + ξ0 , p ξ → p ξ and reflections ξ, p ξ → −ξ, − p ξ [30] as can be seen explicitly in Eq. (10). The constancy of the S implies that the classical motion is wholly to the right or wholly to the left of the singularity at q = 0. Parity is already broken at the classical level. As we show below, in the quantum case the operator associated to Sˆ generates a superselection rule. This operator breaks parity. We now pinpoint some quantum properties of the system that are related to the previous classical features. As one may infer from equation (10), the quantum system has eigenfunctions ψ(ξ ) = N exp(±i βξ ), where N a normalization constant and β is any real number. No discrete negative energy levels are found for there is no way to quantize the energy in an unbounded region with no singularities. The energy levels are continuous, given a state with negative energy 0 there are infinitely more with lower energies — as can be also argued from dimensional considerations. This feature implies that there is no ground state in the system
420
M. Ávila-Aoki et al. / Physics Letters A 373 (2009) 418–421
[40]. If the Hamiltonian (2) is interpreted as a quantum operator, it can be factored as
ˆ = H
h¯ 2
2m
d dx
+
b+ x
1 2
b + 12 d − + , dx
(12)
x
where h¯ 2 (b + 1/2)(b − 1/2)/2m = k is the constant in the potential. ˆ cannot be negative if κ 1/4, The mean value of the energy H where κ ≡ 2mk/¯h2 ; hence κ > 1/4 to allow for negative energy eigenstates in Hamiltonian (2) [18]. Thus κ = 1/4 plays the role of a critical coupling and as long as κ is larger than the critical value the theory may admit bound states. However, it still requires of renormalization for producing the bound state that the regularized problem supports. Renormalization breaks the O (1, 1) scale symmetry of the problem and has been shown to produce a simple realization of an anomaly, namely, the binding of an electron to a polar molecule [17,19]. In the classical 1D case these properties are related to the splitting at the origin that breaks the symmetry of the problem. On the other hand, the symmetry under reflections of (1) and the above considerations, mean that the coordinate space eigenfunctions corresponding to the two possible classical bouncing motions, namely |φ+ and |φ− , should vanish, respectively, for q < 0 and for q > 0. Therefore the system has no eigenstates with well defined parity and thus parity is also spontaneously broken. We can prove this property as follows. Assume we were allowed to form parity states,
1 |φ+ + |φ− and |φo = √ |φ+ − |φ− ;
1
|φe = √
2
2
(13)
we should then conclude that such states describe exactly the same physics. To begin with, the probability densities would have the same value everywhere for both the even and the odd states |φe |2 = |φo |2 , and their Wronskian determinant would always vanish, meaning that |φe and |φo could not be regarded as describing two different states. But, if we cannot distinguish between |φe and |φo , it should be clear that the relative phase of the components along the left and right states, |φ− and |φ+ , is irrelevant. Therefore, any quantum state of the form
|Ψ = a− |φ− + a+ |φ+ ,
(14)
where a− and a+ are numbers, is necessarily devoid of meaning as a coherent superposition [31,32]. For (14) to represent a bona fide quantum state, the coefficients should vanish or the state (14) should have to be interpreted as an incoherent superposition [32, 33]. This feature is a manifestation of a superselection rule operating in the system [33–35]. Such rule forbids coherent superpositions of states |φ± on one side of the singularity with states |φ∓ on the other side. We are also forced to acknowledge that parity is broken. The superselection rule may be regarded as the quantum version of the classical property we have shown occurs in the classical problem, the particle is confined completely to the right or completely to the left of q = 0, as proven by the conservation of S = sgn(q). The superselection rule shows that the side of the singularity in which the motion occurs remains a classical property even in the quantum case. Note that we use the term ‘classical property’ meaning a property that can be always sharply and simultaneously measured with any other observable, as the mass in nonrelativistic quantum mechanics [36]. The quantum operator associated with the classically conserved quantity S and generating the superselection rule, is Sˆ =
f f f f φ φ − φ φ , +
+
−
−
(15)
f
where f is a label distinguishing the different states of the system and the sum (or integration if f is continuous) should include
all states of the system both the negative and the positive energy ones. This operator Sˆ commutes with every other observable in the system. However, it does not commute with the operator of inversion through the origin, Pˆ [30,37],
[ Sˆ , Pˆ ] = 0.
(16)
Hence, the associated property does not correspond to a good quantum number and the symmetry under inversion through the origin is broken. The eigenstates of the problem may be regarded as doubly degenerate. This apparent violation of the nondegeneracy theorem for bound states in one dimension, as in the example of the 1D hydrogen atom [31,39,41], is due to fact that the singularity at q = 0 separates the two regions q > 0 and q < 0. Even the quantum flux, j = i (¯h/2m)(ψ∂ψ ∗ /∂ q − ψ ∗ ∂ψ/∂ q), vanishes for all state at q = 0 [31]. There cannot be any relation between the region q > 0 and the region q < 0 as has been also established for the hydrogen atom in one-dimension [27,31,34]. As a quantum analog of the classical space splitting, the Hilbert space of the system H is splitted in two coherent sectors H = H+ ⊕ H− , where H+ (H− ) is the space where the left (right) vanishing eigenfunctions belong [35]. The feature has been termed a “space splitting” by some authors [38]. So, any physically realizable state of the ˆ and its eigenvalues +1, and system must be an eigenstate of S, −1 label the superselection sectors, H+ and H− , in which the Hilbert space splits. The superselection rule we have shown operates, though proved for the one-dimensional case, might nevertheless be useful in some applications [29]. The first step in the classical regularization we carried out, the π /2 rotation in phase space, corresponds to a Fourier transform in the quantum case. This suggest using the momentum representation for analysing the quantum problem. We therefore write, with the help of the correspondence
ˆ −1
q
→
i h¯
p
· · · dp ,
(17)
−∞
the Schrödinger equation of the problem in the p-representation as
p2 2m
+ φ( p ) +
k h¯ 2
p p
φ( p ) dp dp = 0,
(18)
−∞ −∞
where φ( p ) is the momentum eigenfunction. Such expression implies, according to (17) and (18), that the corresponding eigenfunction in the coordinate representation, ψ(q), has to vanish the origin ψ(0) = 0. Selecting (17) corresponds to a sort of quantum version of the regularization performed because it ameliorates the effect of the singularity in q-space. The solutions of (18) can be found by converting it to a differential equation. Every solution to such differential equation is expressible as a linear combination of 2 F1
u − , u + , 1/2, − p 2 /2m
and
2 F1
v − , v + , 3/2, − p 2 /2m p , (19)
where 2 F 1 [a, b, c ; −x ] is a hypergeometric function, and we have introduced u ± ≡ (3 ± χ )/4 and v ± = (5 ± χ )/4, with χ ≡ √ 1/4 − κ . As it should be clear from our discussion, we cannot impose any further limitation on the solutions. Therefore, there also no limitation on the possible energy values. Energy is not quantized as we have already guessed from the classical analysis, it can take therefore any negative value. As the classical analysis has suggested, the system does not have a ground state. Notice that the nature of the solutions must change when κ attains values greater than 1/4 as χ becomes purely imaginary. Thus, κc ≡ 14 is a critical value in the coupling constant. 2
M. Ávila-Aoki et al. / Physics Letters A 373 (2009) 418–421
As we have anticipated the solutions change about κc , which is a threshold separating two regimes, one with no bound states (κ < κc ) and other where bound states may exist (κ κc ). The functions (19) are the negative energy component functions in the p-representation. Such components √ limit the possible √ values the momentum variable can take as − 2m p + 2m in any negative energy state. To finalize, we should check that our component solutions (19) are capable of expressing the single bound state obtained by renormalization [17,19]. Using the expression given for such state given in [19] we get, after Fourier transforming it, that the only bound state of the system in the p-representation is
φbs ( p ) = A 1 (3/4)2 2 F 1 3/4, 3/4, 1/2, − p 2 /2m (20) + A 2 p (5/4)2 2 F 1 5/4, 5/4, 3/2, − p 2 /2m , √ 1/ 2 1/ 4 1/ 2 1/ 4 where √ A 1 ≡ (2/π ) (m ) (1 + −i ), A 2 ≡ i /(2π (m ) ) × (1 + −i ) are constants. Hence, the bound state wave function of the problem can be expressed — as it should be — in terms of the momentum components (19) at precisely the critical case. The existence of the single bound state (20) breaks the scale symmetry of the problem. Hence an anomalous breaking ensues, as has been explicitly proved in [17,19]. We have shown that the classical motion under an inverse square potential is equivalent to free motion on a branch of a hyperbola. We then uncovered the O (1, 1) hidden symmetry of the problem. The impenetrability of the origin also follows since S = sgn(q) is a classically conserved quantity, hence the system exhibits the so-called “space splitting” phenomenon [38]. We have also proved that the quantum operator associated to the classical quantity S induces a superselection rule in the corresponding quantum problem. We solved the quantum problem in the momentum representation exhibiting that its general solution is able to reproduce the previously reported regularized solution in the coordinate representation. We point out that the breaking of the scale symmetry of the problem has been interpreted as a manifestation of a physicochemical anomaly [17,19,24]. Acknowledgements This work has been partially supported by a PAPIIT-UNAM research grant (number IN 115406-3). We dedicate this work to the loving memories of Holda Yolanda Brito-Ancona, Darío Moreno, Oralia Vela-Carolán, M. X’Sac, and P.A. Koshka.
421
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Classical and quantum motion in an inverse square potential M. Ávila-Aoki a , C. Cisneros b , R.P. Martínez-y-Romero c , H.N. Núñez-Yépez d , A.L. Salas-Brito e,∗ a
Centro Universitario Valle de Chalco, Universidad Autónoma del Estado de México, Valle de Chalco, CP 56615, Estado de México, Mexico Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 6-96, CP 62131, Cuernavaca, Morelos, Mexico c Facultad de Ciencias, Universidad Nacional Autónoma de México, Apartado Postal 21-267, CP 04000, Coyoacán DF, Mexico d Departamento Física, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-534, CP 09340, Iztapalapa DF, Mexico e Laboratorio de Sistemas Dinámicos, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Apartado Postal 21-267, CP 04000, Coyoacán DF, Mexico b
a r t i c l e
i n f o
Article history: Received 1 August 2008 Received in revised form 22 November 2008 Accepted 25 November 2008 Available online 4 December 2008 Communicated by P.R. Holland Keywords: Inverse square potential Free motion on a hyperbola Superselection rule
a b s t r a c t Classical motion in an inverse square potential is shown to be equivalent to free motion on a hyperbola. The existence of a classical splitting between the q > 0 and q < 0 regions of motion is demonstrated. We show that this last property may be regarded as the classical counterpart of the superselection rule occurring in the corresponding quantum problem. We solve the quantum problem in momentum space finding that there is no way of quantizing its energy but that the eigenfunctions suffice to describe the single renormalized bound state of the system. The dynamical symmetry of the classical problem is found to be O (1, 1). Both this symmetry and the symmetry of inversion through the origin are found to be broken. © 2008 Elsevier B.V. All rights reserved.
The quantum problem of motion under an attractive inverse square potential has been discussed for years [1–12]. Motion in such a potential has been investigated because it is relevant for certain applications [13–21]. The vortex structure of the probability flow of a single particle is related to behaviour under an inverse-square potential [4,10,11]. This sort of potential has also been used for studying oscillators coupled to dipole fields, using the potential β q2 + g /q2 , a case in which tunnelling across the barrier has been found possible [12,22]. A charged particle interacting with an electric dipole can be described through an inverse square potential, so it is useful for describing the scattering of electrons off polar molecules, and to explain the existence of a critical dipole moment for the binding of an electron to a polar molecule. It has been used to argue the existence of an anomaly in molecular physics. The need to having a better understanding of the problem has prompted the use of field theoretic techniques, thus its renormalized solutions have been computed employing cut-off [18] and dimensional regularization [19]. Such schemes have established the existence of a single bound state breaking the scale invariance of the problem and providing an example of dimensional transmutation in a finite system [18,19]. A related problem has been studied
*
Corresponding author. E-mail addresses: [email protected] (M. Ávila-Aoki), carmen@ce.fis.unam.mx (C. Cisneros), [email protected] (R.P. Martínez-y-Romero), [email protected] (H.N. Núñez-Yépez), [email protected] (A.L. Salas-Brito). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.11.054
in [23] but their results are not strictly applicable to the case of a inverse square interaction as they studied potentials of the form |x|− p with p greater or equal than 1 but strictly less than 2. In this Letter we study the one-dimensional problem showing features of its quantum solution that are analogous to properties of the classical motion. We show the existence of a superselection rule which may be regarded as a quantum extension of a classical feature, the independence between the right and the left sides of the singularity at q = 0. Let us recall that classical motion under certain potentials can be made equivalent to geodesic motion on a curved space. An outstanding example is the hydrogen atom which is equivalent to geodesic flow on a sphere [25,26]. We here show that another example is the classical 1D motion under the inverse square potential V (q) = −
k q2
(1)
with k > 0. The motion under (1) can be transformed to free motion on a branch of a hyperbola. These classical results are then used to analyse its quantum behaviour. More features of the classical motion under (1) are analyzed and shown to be related to peculiarities of its quantum solution. In particular we show that the classical splitting that occurs at the singularity at q = 0 is related to both the existence of a superselection rule and to the spontaneous breaking of parity that occur in the quantum system. This result hints towards the possibility of us-
M. Ávila-Aoki et al. / Physics Letters A 373 (2009) 418–421
419
ing this kind of interactions for quantum information processing [28,29]. To show the equivalence of motion under the potential (1) with geodesic motion of a particle on a hyperbola, let us begin with the autonomous classical Hamiltonian H=
p2 2m
−
k q2
(2)
,
so H is a constant of the motion and the energy, E, is clearly conserved. Other conspicuous feature of Hamiltonian (2) is its scale invariance, any change q → λq and p → p /λ2 leaves the equations of motion unchanged if the energy is rescaled as E → λ2 E. Thence the problem has no characteristic dimensional scales. Quantum mechanically this scaling symmetry also holds [24] but such quantum symmetry can be broken at a certain value of the coupling constant [17]. To classically regularize the problem the first step is to perform a π /2 rotation in phase space, p → q and q → − p, to transform (2) to q2 2m
k
−
= E.
p2
(3)
We analyze the case of negative energies so we set q2 /2m ≡ − E > 0 and rewrite (3) as
q2 2m
= k/ p 2 −
2 p 1 +1 = .
(4)
k
Let us define
q2
P = p 1+
4α 2
(5)
,
where we have used α 2 ≡ m (q, p )/2 — where we are making explicit that is merely a shorthand for a function of p and q. We need to choose a Q as to complete a canonical transformation. To this end, we use the generating function F (q, P ) =
2km log 2P q + 2 2km + P 2 q2 ,
√
(6)
from which we get, through p = ∂ F (q, P )/∂ q, the momentum (5), and the coordinate
∂ F (q, P ) p 2 q2 Q = = −q 1 − . ∂P 2km
(7)
The transformed Hamiltonian, H , is then H =
P2 2μ
(8)
,
where μ ≡ k/2. We have thus transformed the Hamiltonian of the problem to a free particle form [26]. Using in (2) the alternate coordinates
q ξ = arcsinh √
2m
,
pξ =
√
2m p cosh(ξ ),
(9)
the original Hamiltonian is also converted to the free form (8), H =
p 2ξ 2μ
.
(10)
These coordinates (ξ, p ξ ) exhibit the phase curve on which the particle freely moves, it is a branch of a hyperbola. The hyperbola branch gets traversed at a constant speed in one of the two possible ways, either from ξ = −∞ to ξ = +∞ or the other way round. Notice that in the process of getting (8) — or (10) — we have regularized the problem, the singularity at q = 0 has disappeared from the Hamiltonian. It is not possible to transform the problem to a
Fig. 1. Phase portrait (p vs q) of the system with Hamiltonian (2) in arbitrary units. Notice how the phase trajectories avoid crossing from negative to positive values of q and vice versa. Classically, the particle remain confined in just one side of the singularity for ever.
free particle moving on a compact region, the motion is necessarily unbounded. The two possible ways of moving on the hyperbola correspond to the bouncing motions that can occur in the original problem, the particle can come towards the origin starting from the right at an initial position q0 = + (k/ ) > 0 with zero speed then move towards the origin and, instead of getting to it, change its velocity from −∞ to +∞, bouncing back towards its starting position in a finite time. This can be explicitly proved just noting that the following Poisson bracket
H , sgn(q) = 0,
(11)
always vanishes. Hence, S = sgn(q) is a classical constant of the motion. The two motions with the same energy are both parityrelated and independent from one another. This feature can be appreciated from the phase portrait, in the original coordinates [used in (2)] shown in Fig. 1, which clearly exhibits how the system avoids crossing through the singularity. For proving the constancy of S we used the original Hamiltonian (2), therefore the constant is valid both for the E 0 case and for the E < 0 case and does not depend on the canonical transformation or the regularization performed. The dynamical invariance group of both the original classical problem (1) and of the free particle on a hyperbola is the same, O (1, 1), with both translations ξ → ξ + ξ0 , p ξ → p ξ and reflections ξ, p ξ → −ξ, − p ξ [30] as can be seen explicitly in Eq. (10). The constancy of the S implies that the classical motion is wholly to the right or wholly to the left of the singularity at q = 0. Parity is already broken at the classical level. As we show below, in the quantum case the operator associated to Sˆ generates a superselection rule. This operator breaks parity. We now pinpoint some quantum properties of the system that are related to the previous classical features. As one may infer from equation (10), the quantum system has eigenfunctions ψ(ξ ) = N exp(±i βξ ), where N a normalization constant and β is any real number. No discrete negative energy levels are found for there is no way to quantize the energy in an unbounded region with no singularities. The energy levels are continuous, given a state with negative energy 0 there are infinitely more with lower energies — as can be also argued from dimensional considerations. This feature implies that there is no ground state in the system
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[40]. If the Hamiltonian (2) is interpreted as a quantum operator, it can be factored as
ˆ = H
h¯ 2
2m
d dx
+
b+ x
1 2
b + 12 d − + , dx
(12)
x
where h¯ 2 (b + 1/2)(b − 1/2)/2m = k is the constant in the potential. ˆ cannot be negative if κ 1/4, The mean value of the energy H where κ ≡ 2mk/¯h2 ; hence κ > 1/4 to allow for negative energy eigenstates in Hamiltonian (2) [18]. Thus κ = 1/4 plays the role of a critical coupling and as long as κ is larger than the critical value the theory may admit bound states. However, it still requires of renormalization for producing the bound state that the regularized problem supports. Renormalization breaks the O (1, 1) scale symmetry of the problem and has been shown to produce a simple realization of an anomaly, namely, the binding of an electron to a polar molecule [17,19]. In the classical 1D case these properties are related to the splitting at the origin that breaks the symmetry of the problem. On the other hand, the symmetry under reflections of (1) and the above considerations, mean that the coordinate space eigenfunctions corresponding to the two possible classical bouncing motions, namely |φ+ and |φ− , should vanish, respectively, for q < 0 and for q > 0. Therefore the system has no eigenstates with well defined parity and thus parity is also spontaneously broken. We can prove this property as follows. Assume we were allowed to form parity states,
1 |φ+ + |φ− and |φo = √ |φ+ − |φ− ;
1
|φe = √
2
2
(13)
we should then conclude that such states describe exactly the same physics. To begin with, the probability densities would have the same value everywhere for both the even and the odd states |φe |2 = |φo |2 , and their Wronskian determinant would always vanish, meaning that |φe and |φo could not be regarded as describing two different states. But, if we cannot distinguish between |φe and |φo , it should be clear that the relative phase of the components along the left and right states, |φ− and |φ+ , is irrelevant. Therefore, any quantum state of the form
|Ψ = a− |φ− + a+ |φ+ ,
(14)
where a− and a+ are numbers, is necessarily devoid of meaning as a coherent superposition [31,32]. For (14) to represent a bona fide quantum state, the coefficients should vanish or the state (14) should have to be interpreted as an incoherent superposition [32, 33]. This feature is a manifestation of a superselection rule operating in the system [33–35]. Such rule forbids coherent superpositions of states |φ± on one side of the singularity with states |φ∓ on the other side. We are also forced to acknowledge that parity is broken. The superselection rule may be regarded as the quantum version of the classical property we have shown occurs in the classical problem, the particle is confined completely to the right or completely to the left of q = 0, as proven by the conservation of S = sgn(q). The superselection rule shows that the side of the singularity in which the motion occurs remains a classical property even in the quantum case. Note that we use the term ‘classical property’ meaning a property that can be always sharply and simultaneously measured with any other observable, as the mass in nonrelativistic quantum mechanics [36]. The quantum operator associated with the classically conserved quantity S and generating the superselection rule, is Sˆ =
f f f f φ φ − φ φ , +
+
−
−
(15)
f
where f is a label distinguishing the different states of the system and the sum (or integration if f is continuous) should include
all states of the system both the negative and the positive energy ones. This operator Sˆ commutes with every other observable in the system. However, it does not commute with the operator of inversion through the origin, Pˆ [30,37],
[ Sˆ , Pˆ ] = 0.
(16)
Hence, the associated property does not correspond to a good quantum number and the symmetry under inversion through the origin is broken. The eigenstates of the problem may be regarded as doubly degenerate. This apparent violation of the nondegeneracy theorem for bound states in one dimension, as in the example of the 1D hydrogen atom [31,39,41], is due to fact that the singularity at q = 0 separates the two regions q > 0 and q < 0. Even the quantum flux, j = i (¯h/2m)(ψ∂ψ ∗ /∂ q − ψ ∗ ∂ψ/∂ q), vanishes for all state at q = 0 [31]. There cannot be any relation between the region q > 0 and the region q < 0 as has been also established for the hydrogen atom in one-dimension [27,31,34]. As a quantum analog of the classical space splitting, the Hilbert space of the system H is splitted in two coherent sectors H = H+ ⊕ H− , where H+ (H− ) is the space where the left (right) vanishing eigenfunctions belong [35]. The feature has been termed a “space splitting” by some authors [38]. So, any physically realizable state of the ˆ and its eigenvalues +1, and system must be an eigenstate of S, −1 label the superselection sectors, H+ and H− , in which the Hilbert space splits. The superselection rule we have shown operates, though proved for the one-dimensional case, might nevertheless be useful in some applications [29]. The first step in the classical regularization we carried out, the π /2 rotation in phase space, corresponds to a Fourier transform in the quantum case. This suggest using the momentum representation for analysing the quantum problem. We therefore write, with the help of the correspondence
ˆ −1
q
→
i h¯
p
· · · dp ,
(17)
−∞
the Schrödinger equation of the problem in the p-representation as
p2 2m
+ φ( p ) +
k h¯ 2
p p
φ( p ) dp dp = 0,
(18)
−∞ −∞
where φ( p ) is the momentum eigenfunction. Such expression implies, according to (17) and (18), that the corresponding eigenfunction in the coordinate representation, ψ(q), has to vanish the origin ψ(0) = 0. Selecting (17) corresponds to a sort of quantum version of the regularization performed because it ameliorates the effect of the singularity in q-space. The solutions of (18) can be found by converting it to a differential equation. Every solution to such differential equation is expressible as a linear combination of 2 F1
u − , u + , 1/2, − p 2 /2m
and
2 F1
v − , v + , 3/2, − p 2 /2m p , (19)
where 2 F 1 [a, b, c ; −x ] is a hypergeometric function, and we have introduced u ± ≡ (3 ± χ )/4 and v ± = (5 ± χ )/4, with χ ≡ √ 1/4 − κ . As it should be clear from our discussion, we cannot impose any further limitation on the solutions. Therefore, there also no limitation on the possible energy values. Energy is not quantized as we have already guessed from the classical analysis, it can take therefore any negative value. As the classical analysis has suggested, the system does not have a ground state. Notice that the nature of the solutions must change when κ attains values greater than 1/4 as χ becomes purely imaginary. Thus, κc ≡ 14 is a critical value in the coupling constant. 2
M. Ávila-Aoki et al. / Physics Letters A 373 (2009) 418–421
As we have anticipated the solutions change about κc , which is a threshold separating two regimes, one with no bound states (κ < κc ) and other where bound states may exist (κ κc ). The functions (19) are the negative energy component functions in the p-representation. Such components √ limit the possible √ values the momentum variable can take as − 2m p + 2m in any negative energy state. To finalize, we should check that our component solutions (19) are capable of expressing the single bound state obtained by renormalization [17,19]. Using the expression given for such state given in [19] we get, after Fourier transforming it, that the only bound state of the system in the p-representation is
φbs ( p ) = A 1 (3/4)2 2 F 1 3/4, 3/4, 1/2, − p 2 /2m (20) + A 2 p (5/4)2 2 F 1 5/4, 5/4, 3/2, − p 2 /2m , √ 1/ 2 1/ 4 1/ 2 1/ 4 where √ A 1 ≡ (2/π ) (m ) (1 + −i ), A 2 ≡ i /(2π (m ) ) × (1 + −i ) are constants. Hence, the bound state wave function of the problem can be expressed — as it should be — in terms of the momentum components (19) at precisely the critical case. The existence of the single bound state (20) breaks the scale symmetry of the problem. Hence an anomalous breaking ensues, as has been explicitly proved in [17,19]. We have shown that the classical motion under an inverse square potential is equivalent to free motion on a branch of a hyperbola. We then uncovered the O (1, 1) hidden symmetry of the problem. The impenetrability of the origin also follows since S = sgn(q) is a classically conserved quantity, hence the system exhibits the so-called “space splitting” phenomenon [38]. We have also proved that the quantum operator associated to the classical quantity S induces a superselection rule in the corresponding quantum problem. We solved the quantum problem in the momentum representation exhibiting that its general solution is able to reproduce the previously reported regularized solution in the coordinate representation. We point out that the breaking of the scale symmetry of the problem has been interpreted as a manifestation of a physicochemical anomaly [17,19,24]. Acknowledgements This work has been partially supported by a PAPIIT-UNAM research grant (number IN 115406-3). We dedicate this work to the loving memories of Holda Yolanda Brito-Ancona, Darío Moreno, Oralia Vela-Carolán, M. X’Sac, and P.A. Koshka.
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