Superintegrability of the Caged Anisotropic Oscillator
aa r X i v : . [ n li n . S I] A ug Superintegrability of the Caged Anisotropic Oscillator
N. W. Evans ∗ and P. E. Verrier † Institute of Astronomy, Madingley Rd,University of Cambridge, CB3 0HA, UK (Dated: October 24, 2018)
Abstract
We study the Caged Anisotropic Harmonic Oscillator , which is a new example of a superinte-grable, or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillatorwith rational frequency ratio ( l : m : n ), but additionally with barrier terms describing repulsiveforces from the principal planes. This confines the classical motion to a sector bounded by theprincipal planes, or cage. In 3 degrees, there are five isolating integrals of motion, ensuring thatall bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in themomenta, the remaining two are polynomials of order 2( l + m −
1) and 2( l + n −
1) . In the quan-tum problem, the eigenstates are multiply degenerate, exhibiting l m n copies of the fundamentalpattern of the symmetry group SU (3). PACS numbers:Keywords: Classical mechanics, quantum theory, integration ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The subject of accidental degeneracy has fascinated physicists since the early days ofthe quantum theory. It has long been known that the trajectory in an r -squared force (theCoulomb or Kepler problem) is a conic section, so that every bound orbit is an ellipse andtherefore closed and strictly periodic. The advent of the quantum theory saw the deductionof the eigenfunctions for the Hydrogen atom and thence the realization that the boundstates of the Coulomb problem are degenerate [1, 2, 3]. This accidental degeneracy is aconsequence of a hidden symmetry group SO(4) that is not manifest to the eye. It causesall bound trajectories to be closed in the classical problem.There is a fundamental connection between accidental degeneracy and the separabilityof the Schr¨odinger or the Hamilton-Jacobi equations in more than one coordinate systemand therefore the existence of additional conserved quantities or integrals of motion. This ismentioned in a number of the famous texts of the old quantum theory – such as Born’s Me-chanics of the Atom and Sommerfeld’s
Atomic Structure and Spectral Lines . For example,the Coulomb problem is separable in both spherical polar and rotational parabolic coordi-nates. The former leads to the conservation of the angular momentum vector, the latterto the conservation of the Laplace-Runge-Lenz vector. The quantum mechanical operatorsclose to form the algebra of SO(4) (see, for example, [4] for a review). The accidental de-generacy is a consequence of additional integrals of motion [5] and so such systems are oftencalled superintegrable [6].Systematic investigations of all the possible combinations of coordinate systems for whichthe Schr¨odinger and Hamilton-Jacobi equation can separate have now been carried out [7, 8,9]. In three degrees of freedom, this yields 13 distinct superintegrable systems, all of whichhave classical integrals of motion or quantum operators that are quadratic in the canonicalmomenta and all of which exhibit accidental degeneracy. This includes familiar systems suchas the Coulomb problem and the isotropic harmonic oscillator.Nonetheless, this does not provide a comprehensive explanation of the phenomenon of su-perintegrability. For example, in three degrees of freedom, the Hamiltonian of the anisotropicharmonic oscillator with rational frequency ratio is H = 12 ( p x + p y + p z ) + k ( l x + m y + n z ) , (1)where l, m and n are integers and k is a constant. The Hamilton-Jacobi or Schr¨odinger2quations clearly separate in rectangular Cartesians. If l : m : n = 2 : 1 : 1, then theHamiltonian also separates in the rotational parabolic and elliptic cylindrical coordinatesystems [9], giving rise to additional conserved quantities and accidental degeneracy. How-ever, if l + m + n >
4, then the Hamiltonian is still superintegrable [10, 11, 12], even though itnow only separates in rectangular Cartesians. Further examples of systems which are super-integrable but not separable in more than one coordinate system include the Calogero-Moserproblem [13, 14] and the generalized Coulomb problem [15].The purpose of this paper is to introduce another superintegrable Hamiltonian, closelyrelated to (1), namely H = 12 ( p x + p y + p z ) + k ( l x + m y + n z ) + k x + k y + k z (2)We shall refer to this as “the Caged Anisotropic Oscillator”, as the presence of the barrierterms confines the motion to an octant defined by the principal planes. When l = m = n = 1,this becomes the Smorodinsky-Winternitz system, on which there is an extensive literature[7, 16, 17].Like the Smorodinsky-Winternitz system, the Caged Anisotropic Oscillator has five in-tegrals of motion and it exhibits accidental degeneracy in quantum mechanics. However,unlike the Smorodinsky-Winternitz system, the Hamilton-Jacobi and Schr¨odinger equationsonly separate in rectangular Cartesians. In this paper, we demonstrate that the Hamilto-nian (2) is superintegrable using the method of projection in §
2. Then we discuss both theclassical and quantum problems in some detail in § § II. PROOF OF SUPERINTEGRABILITY
Let us start with the observation that the commensurate anisotropic oscillator in N dimensions possesses 2 N − N energies of each individual oscillator and the N − H = X i =0 (cid:18) p i + kn i s i (cid:19) (3)3here the n i are positive integers and the s i are Cartesian coordinates. If we now introducecoordinates ( x, y, z, θ x , θ y , θ z ) according to s = x cos θ x , s = x sin θ x s = y cos θ y , s = y sin θ y s = z cos θ z , s = z sin θ z and let the frequencies n = n = l , n = n = m , n = n = n , we have the Hamiltonian H = 12 p x + p y + p z + p θ x x + p θ y y + p θ z z ! + k ( l x + m y + n z ) (4)Since the coordinates ( θ x , θ y , θ z ) are ignorable we can set the conjugate momenta to con-stants. Making the substitutions p θ x = k , p θ y = k , p θ z = k gives us the Caged HarmonicOscillator Hamiltonian (2).For the Hamiltonian in 6 dimensions given by (3), every bound trajectory is closed.Similarly, in the reduced 3 degrees of freedom Hamiltonian (2), every bound trajectory isalso closed and the system is still superintegrable. FIG. 1: A typical orbit for a frequency ratio of l : m : n = 1 : 1 : 3. IG. 2: A typical orbit for a frequency ratio of l : m : n = 1 : 2 : 3. III. CLASSICAL MECHANICSA. The Integrals of Motion
The Hamiltonian (2) clearly separates in rectangular Cartesian coordinates to give thefirst three integrals of motion as the three energies of oscillation I = 12 p x + kl x + k x (5) I = 12 p y + km y + k y (6) I = 12 p z + kn z + k z (7)As the system is separable in these coordinates, it is easy to see that the trajectories of theorbits are (c.f. [9]) x = I l k + (cid:18) I l k − k l k (cid:19) / cos( √ kl ( t − t )) (8) y = I m k + (cid:18) I m k − k m k (cid:19) / cos( √ km ( t − t ) + c ) (9) z = I n k + (cid:18) I n k − k n k (cid:19) / cos( √ kn ( t − t ) + c ) (10)where t and the c i are constants. The remaining two integrals are the phase differencesbetween the orbits, say c and c . If we say too that | m − l | < | n − l | < | n − m | , then the5ntegrals are also of lowest order possible in the momenta. First, let us define ξ . = x − αA = cos( √ kl ( t − t )) (11) η . = y − βB = cos( √ km ( t − t ) + c ) (12) ζ . = z − γC = cos( √ kn ( t − t ) + c ) (13)where α = I / (2 l k ), β = I / (2 m k ) and γ = I / (2 n k ) and A = (cid:18) I l k − k l k (cid:19) / , B = (cid:18) I m k − k m k (cid:19) / ,C = (cid:18) I n k − k n k (cid:19) / (14)The derivation of both integrals is similar and will be demonstrated with the case of c . Thefirst phase difference is given by (c.f., the discussion of the anisotropic oscillator in [12]) c = arccos η − ml arccos ξ (15)Taking the cosine givescos( lc ) = cos( l arccos η ) cos( m arccos ξ ) + sin( l arccos η ) sin( m arccos ξ )= T l ( η ) T m ( ξ ) + ˙ ξ ˙ η km l T ′ l ( η ) T ′ m ( ξ ) (16)where T l and T m are the Chebyshev polynomials of the first kind [18] and T ′ l and T ′ m aretheir derivatives with respect to the arguments η and ξ . The time derivatives ˙ ξ and ˙ η areequal to 2 xp x /A and 2 yp y /B respectively. It is more convenient to express the integral as I = (2 k ) l + m A m B l cos( lc ) (17)which is of order 2( l + m ) in the momenta, but can be reduced to order 2( l + m −
1) sincethe two highest powers of the momenta can be removed though a combination of the energyintegrals.The corresponding integral for the second phase difference c is I = (2 k ) l + n A n C l cos( lc ) (18)where cos( lc ) = T l ( ζ ) T n ( ξ ) + ˙ ξ ˙ ζ kn l T ′ l ( ζ ) T ′ n ( ξ ) (19)6hich can be reduced to order 2( l + n −
1) in the momenta. It is easy to verify that both I and I are integrals of motion by showing that the Poisson bracket with the Hamiltonianvanishes. They are also functionally independent, as may be verified by computing the rankof the appropriate Jacobian. FIG. 3: A typical orbit for a frequency ratio of 2 : 3 : 4.FIG. 4: A typical orbit for a frequency ratio of 2 : 3 : 4 with no potential barriers ( k = k = k =0). The dotted lines show the field of view of the corresponding orbit with the potential barriersas shown in Fig 3. B. The Group Theoretic Approach
Rodriguez et al. [19] have also recently examined this system, and derived the classicalintegrals. Ingeniously, they look for invariants under SO (2) × SO (2) × SO (2) corresponding to7he transformation (4). They then search for combinations of these invariants that commutewith the Hamiltonian (3) under the Poisson bracket. Such quantities will also necessarily beintegrals of the reduced Hamiltonian of the Caged Anisotropic Oscillator. Rodriguez et alfind integrals of motion that are rational functions in the momenta, but here we show howto adapt their method to give integrals that are polynomial in the momenta.Let us start by introducing the complex variables z = p − iℓ √ ks , z = p − iℓ √ ks ,z = p − im √ ks , z = p − im √ ks ,z = p − in √ ks , z = p − in √ ks , (20)so that the Hamiltonian (3) is just H = 12 X i =1 | z i | . (21)Now, following Rodriguez et al, we look for invariants under the generators of rotations inthe ( s , s ), ( s , s ) and ( s , s ) planes. For the ( s , s ) plane, they include z + z , ¯ z + ¯ z , | z | + | z | . (22)with similar results holding for the ( s , s ) and ( s , s ) planes. Expressions like | z | + | z | clearly commute with the Hamiltonian (3) and are just the separable energies in theoscillation in the coordinate directions. The remaining quantities do not commute with (3),but it is possible look for an invariant that is a function of the two expressions ¯ z + ¯ z and z + z that does. Therefore, we require that { H , f (¯ z + ¯ z , z + z } = 0 . (23)Inserting H from (3), this gives the complex invariant R = (¯ z + ¯ z ) m ( z + z ) ℓ (24)whose real part R + ¯ R = (¯ z + ¯ z ) m ( z + z ) ℓ + ( z + z ) m (¯ z + ¯ z ) ℓ (25)8s a polynomial of order 2( ℓ + m ). Modulo an unimportant overall numerical factor, it isthe same as the polynomial invariant found earlier in eq (17). Similarly, the invariant (18)is just (¯ z + ¯ z ) n ( z + z ) ℓ + ( z + z ) n (¯ z + ¯ z ) ℓ (26)up to a numerical factor. FIG. 5: A typical orbit for a frequency ratio of 3 : 4 : 5.FIG. 6: A typical orbit for a frequency ratio of 3 : 4 : 5 with no potential barriers ( k = k = k =0). The dotted lines show the field of view of the corresponding orbit with the potential barriersas shown in Fig 5. C. The Orbits
It is interesting to plot out the orbit of a particle in the potential. As the Hamiltonianis superintegrable, all bound orbits must be closed curves. Using a standard Bulirsch-Stoer9ntegrator [20] to solve the equations of motion, some example orbits are plotted. These areshown for various frequency ratios in Figs 1, 2, 3 and 5. The orbits are confined to a box,defined by the limits α ± A , and similar. Figs 4 and 6 show the corresponding cases to Figs3 and 5, but with no potential barriers. Here, the trajectories are the well-known Lissajousfigures [21], and it can be seen how the orbit in the general case is a reflection and slightdistortion in the x , y and z axes. IV. QUANTUM MECHANICS
The Schr¨odinger equation is separable in rectangular Cartesians, and reads: −∇ + X i =1 h ω i kx i + 2 k i x i i! Ψ = 2 E Ψ (27)where ω i are the integer multipliers of the frequencies, corresponding to l, m, n in the previoussection, and ~ = 1. The separable solution has wavefunction Ψ = Q i =1 ψ n i where theindividual wavefuntions are (c.f., [7, 16, 22]) ψ n i ( x i ) = N n i e − ( w i √ k/ x i x / ± ν i i L ± ν i n i ( w i √ kx i ) (28)where N n i is the normalisation constant given by N n i = w / i (2 k ) / q (2 w i k ) ± ν i / Γ( n i + 1) / Γ( n i + 1 ± ν i ) (29)and L νn i are associated Laguerre polynomials, the Γ are Gamma functions and ν i = (1 +8 k i ) / . The quantised energy is given by E = 2 √ k X i =1 w i (cid:18) n i + 12 ± ν i (cid:19) (30)The degeneracy of each energy level with quantum number N = w n + w n + w n istherefore the same as that of the three dimensional anisotropic harmonic oscillator withrational frequency ratio w : w : w (listed for example in [27]) In the simplest case, if thefrequency ratio is 1 : 1 : n then the degeneracy is given by g ( N ) = (cid:20) Nn (cid:21) + 1 ! N + 1 − n (cid:20) Nn (cid:21) ! (31)where [ N/n ] denotes the integer part of
N/n . The allowed states for three degrees of freedomand the frequency ratio 1 : 1 : 2 are shown in Fig. 7.10o look at the group structure, the annihilation and creation operators can be constructedas b i = − w i √ k (cid:18) w i √ kx i ∂∂x i + 2 w i kx i − k i x i + w i √ k + ∂ ∂x i (cid:19) (32) b † i = − w i √ k (cid:18) − w i √ kx i ∂∂x i + 2 w i kx i − k i x i − w i √ k + ∂ ∂x i (cid:19) (33)which annihilate and create quanta of energy in the i direction, that is[ H, b i ] = − w i √ kb i (34)[ H, b † i ] = 2 w i √ kb † i (35)and, representing ψ n i as | n i i , act in the following way b i | n i i = p n i ( n i ± ν i ) | n i − i (36) b † i | n i i = p ( n i + 1)( n i ± ν i + 1) | n i + 1 i (37)which are identical to those for the Smorodinsky-Winternitz system given by [16]. As suchthe number operator given by ˆ n i = ([ b i , b † i ] ∓ ν i −
1) can be used to again to construct theoperators T ij = 12 { b † i ( ˆ n i ± ν i + 1) − / , b j ( ˆ n j ± ν j ) − / } (38)which close under commutation [ T ij , T rs ] = δ jr T is − δ is T rj (39)and give the Lie algebra u(3).In the case of the isotropic harmonic oscillator, it is well known that the degeneracy of the N th energy level is ( N +1)( N +2) /
2, which corresponds to the dimensions of the irreduciblerepresentations of SU (3) . Even though the symmetry group of the anisotropic harmonicoscillator is also SU (3), it is no now longer the case that the degeneracy levels follow thepattern 1 , , , , ... . This was already noted as a complication by Jauch & Hill [10], andthere have been a number of possible resolutions proposed in the literature [23, 24, 25, 26, 27]Following the lines of argument put forward in [24], we define˜ n = n mod ( w w ) , ˜ n = n mod ( w w ) , ˜ n = n mod ( w w ) (40)11rom which it follows that n = ˜ n w w + r , n = ˜ n w w + r , n = ˜ n w w + r (41)where 0 ≤ r < w w , 0 ≤ r < w w and 0 ≤ r < w w . This divides the energy levelsinto w w w subsets according to the values of r , r and r . From eq. (30), the energy levelsbecome E = 2 √ k X i =1 (cid:20) ( w w w )˜ n i + w i ( r i + 12 ± ν i (cid:21) (42)so that the energy levels within each subset ( r , r , r ) have the characteristic degeneracy ofSU(3). This is illustrated by the color coding in Fig. 7. n x + m n y + n n z n z n y + n n z FIG. 7: Energy levels for the frequency ratio 1 : 1 : 2. Objects belong to different sets of ( r , r , r )values are shown in different color. For each color, the degeneracies are the dimensions of theirreducible representations of SU(3). V. SUMMARY AND CONCLUSIONS
The
Caged Anisotropic Harmonic Oscillator is a new superintegrable Hamiltonian,namely H = 12 ( p x + p y + p z ) + k ( l x + m y + n z ) + k x + k y + k z . (43)12f the frequency multipliers are integers, then the Hamiltonian is superintegrable. We havefound the five isolating integrals for the classical motion in three degrees of freedom. Three ofthe integrals of motion – the energies in each oscillation – are quadratic in the canonical mo-menta and arise from separation of the Hamilton-Jacobi equation in rectangular Cartesians.The other two integrals are still polynomial in the momenta, but now of order 2( l + m − l + n −
1) respectively. If l = m = n = 1, the Hamiltonian becomes the well-studiedSmorodinsky-Winternitz system [7, 8, 17, 28], and all the integrals are then quadratic andarise from separability of the Hamilton-Jacobi equation.The system is interesting for at least three reasons. First, from the perspective of integra-bility, there are still very few systems known with integrals of motion that are polynomials inthe momenta of higher order than 2 [29]. Systematic searches for Hamiltonian systems withhigher order polynomial invariants have been performed, confirming the impression thatthey are rare [30, 31]. Given this sketchy and disparate information, we have no unifyingtheory of the conditions for the existence of such integrals of motionSecond, from the perspective of superintegrability, if the integrals of motion are allquadratic in the momenta, then a classification theorem exists and all systems in flat spacehave been found [7, 8, 9]. Such systems always arise from separability of the Hamilton-Jacobiequation in more than one coordinate system. However, the Caged Anisotropic Oscillatorjoins the Toda Lattice and the Generalized Kepler Problem as an example of a system forwhich some of the integrals are cubic polynomials or higher, and then the superintegra-bility does not arise from separability in more than one coordinate system. It would beinteresting to classify such systems and find all examples in flat space. In particular, theCaged Anisotropic Oscillator is the second superintegrable Hamiltonian to be deduced bythe method of projection introduced in [15]. Essentially, the idea is to view superintegrablemotion in three degrees of freedom as a projection of a higher dimensional superintegrablesystem, such as the Coulomb or Kepler problem, or the harmonic oscillator. Are there anymore such systems to be found?Third, from the perspective of group theory in quantum mechanics, the proper interpre-tation of the symmetry or degeneracy group remains unclear. Already in 1940, Jauch &Hill [10] noted that the quantum mechanical problem of the anisotropic oscillator presentsproblems which leaves its symmetry group in doubt. Since that day, there have been a num-ber of different suggestions in the literature as to the proper interpretation of the symmetry13roup [24, 25, 26, 27]. Although these procedure seem reasonable, they are more along thelines of a posteriori justification than compelling argument. [1] Pauli W., 1926, Z. Phys., 36, 336[2] Fock V., 1935, Z. Phys., 98, 145[3] Bargmann V., 1936, Z. Phys., 99, 576[4] Abarbanel, H., 1976, in “Studies in Mathematical Physics: Essays in Honour of ValentineBargmann”, eds E.H. Lieb, B. Simon, A.S. Wightman, Princeton, University Press, Princeton,p. 3[5] Weigert, S. Thomas H., 1993, Am J. Phys., 61, 272[6] Tempesta P., Winternitz P., Harnad J., Miller Jr, W., Pogosyan G., Rodriguez M., 2005,Superintegrability in Classical and Quantum Systems, American Mathematical Society[7] Fris J., Mandrosov V., Smorodinsky Y. A., Uhl´ı M., & Winternitz P. 1965, Physics Letters, 16,35[8] Makarov A. A., Smorodinsky Y. A., Valiev K., & Winternitz P., 1967 Nuovo Cimento 52, 1061.[9] Evans N. W. 1990, Phys. Rev. A, 41, 5666[10] Jauch J. M., & Hill E. L. 1940, Phys Rev, 57, 641[11] Amiet J.-P., & Weigert S. 2002, Journal of Math. Phys, 43, 4110[12] Boccaletti D., & Pucacco G. 1996, Theory of Orbits. Volume 1: Integrable Systems andNon-perturbative Methods, Springer Verlag, New York[13] Adler M., 1977, Comm. Math. Phys., 55, 195[14] Wojciechowski, S. 1983, Physics Letters A, 95, 279[15] Verrier P. E., & Evans N. W. 2008, Journal of Math. Phys, 49, 2902[16] Evans N. W. 1991, Journal of Math Phys, 32, 3369[17] Ballasteros A., Herranz F.J., 2007, Journal of Phys A: Math Gen, 40, 51[18] Erd´elyi A., Magnus W., Oberhettinger F., Tricomi F.G., 1953, Higher Transcendtal Functionsvol 2, McGraw-Hill, New York[19] Rodriguez M., Tempesta P., Winternitz P. 2008, ArXiv e-prints, 807, arXiv:0807.1047[20] Press W. H., Teukolsky S. A., Vetterling W. T., & Flannery B. P. 2002, Numerical recipes inC++ : the art of scientific computing, Cambridge University Press
21] Symon K.R. 1960, Mechanics, Addison-Wesley, Reading, Massachusetts, Section 3.10[22] Winternitz P., Smorodinsky Ya., Uhlir M., Fris I., 1967, Sov J Nucl Phys 4, 444[23] Demkov Yu. N., 1963, Soviet Phys.JETP 17, 1349[24] Louck J. D., Moshinsky M., Wolf K. B. 1973, Journal of Math Phys, 14, 692[25] King G. M. 1973, Journal of Phys A: Math Gen, 6, 901[26] Rosensteel, G., & Draayer, J. P. 1989, Journal of Physics A : Math Gen, 22, 1323[27] Bonatsos, D., Kolokotronis, P., Lenis, D., & Daskaloyannis, C. 1997, Int J of Modern PhysicsA, 12, 3335[28] Evans N.W., 1990, Phys. Lett. A., 147, 483[29] Hietarinta J.,1987, Phys Reports, 147, 87[30] Thompson G, 1984, Journal of Math Phys, 25, 3474[31] Evans N.W, 1990, Journal of Math Phys, 31, 60021] Symon K.R. 1960, Mechanics, Addison-Wesley, Reading, Massachusetts, Section 3.10[22] Winternitz P., Smorodinsky Ya., Uhlir M., Fris I., 1967, Sov J Nucl Phys 4, 444[23] Demkov Yu. N., 1963, Soviet Phys.JETP 17, 1349[24] Louck J. D., Moshinsky M., Wolf K. B. 1973, Journal of Math Phys, 14, 692[25] King G. M. 1973, Journal of Phys A: Math Gen, 6, 901[26] Rosensteel, G., & Draayer, J. P. 1989, Journal of Physics A : Math Gen, 22, 1323[27] Bonatsos, D., Kolokotronis, P., Lenis, D., & Daskaloyannis, C. 1997, Int J of Modern PhysicsA, 12, 3335[28] Evans N.W., 1990, Phys. Lett. A., 147, 483[29] Hietarinta J.,1987, Phys Reports, 147, 87[30] Thompson G, 1984, Journal of Math Phys, 25, 3474[31] Evans N.W, 1990, Journal of Math Phys, 31, 600