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- Different quantum spectra for the same classical system Vladimir Cuesta † Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de o e M´ xico, M´ xico e e Abstract. Using the bra, ket and ascent and descent formalism, I show that the usual one dimensional isotropic harmonic oscillator admits two quantum spectra. Additionally, I ﬁnd that for the case of the two dimensional isotropic harmonic oscillator there are four different quantum spectra. The importance of the result lies in the fact that at classical level the one and two dimensional isotropic harmonic oscillator correspond to a pair of systems, only. 1. Introduction Classical mechanics is one of the more important branches in modern physics, we can study that subject using different approaches, using Newton’s equations, using Lagrange or Hamiltonian formulations (see [1] for instance). In all these cases one key point is to obtain the equations of motion for a given system, in the case of Newton’s equations or Lagrange point of view we work with coordinates and time derivatives of coordinates (or velocities) and the equations of motion are of second order for the time variable. In the case of Hamiltonian formulations, the phase space is labeled by coordinates and momentum variables and the set of differential equation are of ﬁrst order in the time variable. However, if we are interested in the equations of motion for a chosen system, these can be obtained using variational principles, too. Particularly Hamilton’s equations need a Hamiltonian and the canonical symplectic structure (see [1] for systems without constraints and [2], [3] for constrained systems). A lot of modern physical theories are formulated in terms of geometrical and advanced mathematical theories, one of these is symplectic geometry and in fact there exist different formulations of Hamilton’s equations with non canonical symplectic structures (see [4], [5] and [6]). The present work will be focus in the study of the one and two dimensional isotropic harmonic oscillator at classical and quantum level, at classical level a hamiltonian formulation for the ﬁrst case † vladimir.cuesta@nucleares.unam.mx
- Different quantum spectra for the same classical system 2 is the following: the phase space coordinates are (xµ ) = (x, px ) with Hamiltonian 1 2 mω 2 2 H= p + x, (1) 2m x 2 and equations of motion px x = , px = −mω 2 x, ˙ ˙ (2) m as second case we will study the two dimensional isotropic harmonic oscillator, at classical level the Hamiltonian is 1 2 2 mω 2 2 H= px + py + x + y2 , (3) 2m 2 where the coordinates (xµ ) = (x, y, px , py ) label the phase space, the set of equations of motion are px py x = , y = , px = −mω 2 x, py = −mω 2 y, ˙ ˙ ˙ ˙ (4) m m at classical level we will study this pair of systems and like the reader can be note there exist two sets of equations of motion, only. Quantum mechanics is another branch of modern physics with great importance for the understanding of nature, it was formulated in essentially two ways, the ﬁrst one is in terms of a Hilbert space of square-integrable functions, partial differential equations, wave functions and so on (see [7] for instance). The second point of view is the formulation in terms of an abstract Hilbert space of vectors (kets) and its duals (bras), abstract operators and so on (see [8] and [9] for a profound review of the theory), with this in my mind I present my results following this second approach. The purpose of this paper is to prove that the one and two dimensional isotropic harmonic oscillators have not an unique spectrum, the importance of these systems is undeniable, it appears like textbook exercises in basic classical mechanics, in quantum mechanics, in quantum ﬁeld theory appears when electromagnetism is quantized and in innumerable physical theories. To mention a pair of different studies, in [10] is obtained a lot of different couples of Hamiltonians and symplectic structures and using algebraic methods the spectrum of the system is obtained and is only one. In [11], using different couples of Hamiltonians and symplectic structures Heisenberg’s uncertainty principle is studied, this paper makes a different study and like I said, for the same classical system I ﬁnd more than one quantum spectra. 2. Quantum one dimensional isotropic harmonic oscillator 2.1. Case I: Usual case The hamiltonian for the one dimensional isotropic harmonic oscillator is given by ˆ 1 2 mω 2 2 H= p + ˆ x, ˆ (5) 2m x 2
- Different quantum spectra for the same classical system 3 now, following the procedure given in [9] we can deﬁne the following operators mω ı mω ı ax = ˆ x+ ˆ p x , a† = ˆ ˆx x− ˆ px , ˆ (6) 2¯ h mω 2¯ h mω ˆ and the commutation relations between the number operator Nx = a† ax , ax and a† are ˆx ˆ ˆ ˆx ˆ ˆ [Nx , ax ] = −ˆx , a ˆ ˆ [ N x , a† ] = a† , ˆx [ˆx , a† ] = 1, a ˆx (7) x ˆ if I suppose that the number operator has as eigenkets |nx then we must have Nx |nx = nx |nx , using the previous equation we can obtain ˆ ˆ Nx ax |nx = (nx − 1)ˆx |nx , a ˆ ˆ Nx a† |nx = (nx + 1)ˆ† |nx , ax (8) x and we can deduce ax |nx = α|nx −1 and a† |nx = β|nx +1 , where α and β are complex constants, ˆ ˆx choosing the constants suitably we ﬁnd √ √ ax |nx = nx |nx − 1 , a† |nx = nx + 1|nx + 1 , ˆ ˆx (9) the ax operator is the destruction operator and a† is the creation operator and nx = 0, 1, 2, . . ., now ˆ ˆx inverting the set of equations (6) we obtain h ¯ mω¯ † h x= ˆ ax + a† , p x = ı ˆ ˆx ˆ ax − ax , ˆ ˆ (10) 2mω 2 if I substitute (10) into (5) I obtain ˆ ˆ 1 ˆ 1 H = Nx + hω, H|nx = nx + ¯ hω|nx , ¯ (11) 2 2 and the spectrum of the one dimensional isotropic harmonic oscillator is 1 Enx = nx + hω, ¯ (12) 2 to ﬁnish, the vacuum ket is deﬁned like ax |0 = 0, ˆ (13) using the creation operator we can obtain √ √ a† |0 = |1 , a† |1 = 2|2 , ˆ ˆ a† |2 = ˆ 3|3 , a† |3 = 2|4 , . . . , ˆ (14) and in general a state |mx can be obtained with the help of the vacuum ket as 1 mx |mx = √ a† ˆ |0 . (15) mx !
- Different quantum spectra for the same classical system 4 2.2. Case II I will be interested in the hamiltonian 2 Hˆ = 1 p2 + mω x2 , ˆ ˆ (16) 2m x 2 now, I deﬁne the operators mω ı mω ı ax = − ˆ x− ˆ px , ˆ x+ ˆ a† = − ˆx px , ˆ (17) 2¯ h mω 2¯ h mω ˆ and the commutation relations between the new number operator Nx = a† ax , ax and a† are ˆx ˆ ˆ ˆx ˆ ˆ [ N x , ax ] = ax , ˆ ˆ ˆx [Nx , a† ] = −ˆ† , ax [ˆx , a† ] = −1, a ˆx (18) ˆ if I suppose that the new number operator has as eigenkets |nx then we must have Nx |nx = nx |nx , after a straightforward calculation ax |nx = ˆ |nx ||nx + 1 , a† |nx = ˆx |nx − 1||nx − 1 , (19) where nx = . . . , −2, −1, 0., ax is the ascent operator and a† is the descent operator, if I invert the set ˆ ˆx of equations (17) I ﬁnd h ¯ mω¯ † h x=− ˆ ax + a† , p x = ı ˆ ˆx ˆ ax − ax , ˆ ˆ (20) 2mω 2 and the hamiltonian will be ˆ ˆ 1 ˆ 1 H = Nx − hω, H|nx = nx − ¯ hω|nx , ¯ (21) 2 2 and the spectrum of the one dimensional isotropic harmonic oscillator is 1 Enx = nx − hω, ¯ (22) 2 in this case the vacuum ket is |0 and is deﬁned as a|0 = 0, using the ascent creation operator I ﬁnd ˆ √ √ a† |0 = |−1 , a† |−1 = 2|−2 , a† |−2 = 3|−3 , a† |−3 = 2|−4 , . . .(23) ˆ ˆ ˆ ˆ and in general 1 |mx | | − |mx | = a† ˆ |0 , (24) |mx |! and with this I prove that the classical one dimensional isotropic oscillator has two different quantum spectra.
- Different quantum spectra for the same classical system 5 3. Quantum two dimensional isotropic harmonic oscillator 3.1. Case I: Usual case In this case the hamiltonian for the two dimensional isotropic harmonic oscillator is ˆ 1 mω 2 2 H= p2 + p2 + ˆx ˆy x + y2 , ˆ ˆ (25) 2m 2 the previous hamiltonian can be divided in a part that correspond with the x direction and another in the y direction, both hamiltonians are independent, for the x direction I can choose mω ı mω ı ax = ˆ x+ ˆ p x , a† = ˆ ˆx x− ˆ px , ˆ (26) 2¯ h mω 2¯ h mω and for the y direction mω ı mω ı ay = ˆ y+ ˆ p y , a† = ˆ ˆy y− ˆ py , ˆ (27) 2¯ h mω 2¯h mω ˆ with commutation relations between the number operator Nx = a† ax , ax and a† ˆ ˆ ˆ ˆ x x ˆ ˆ [Nx , ax ] = −ˆx , a ˆ ˆ [ N x , a† ] = a† , ˆx [ˆx , a† ] = 1, a ˆx (28) x ˆ and for Ny = a† ay , ay and a† ˆy ˆ ˆ ˆy ˆ ˆ [Ny , ay ] = −ˆy , a ˆ ˆ [ N y , a† ] = a† , ˆy [ˆy , a† ] = 1, a ˆy (29) y using the previous equations I ﬁnd √ √ ax |nx , ny = nx |nx − 1, ny , ˆ a† |nx , ny = ˆx nx + 1|nx + 1, ny , √ ay |nx , ny = ny |nx , ny − 1 , ˆ a† |nx , ny = ˆy ny + 1|nx , ny + 1 , (30) ˆ ˆ where nx = 0, 1, 2, . . ., ny = 0, 1, 2, . . . and Nx |nx , ny = nx |nx , ny , Ny |nx , ny = ny |nx , ny , with inverse equations h ¯ mω¯ † h x= ˆ ax + a† , ˆ ˆx px = ı ˆ ax − ax , ˆ ˆ (31) 2mω 2 h ¯ mω¯ † h y= ˆ ay + a† , p y = ı ˆ ˆy ˆ ay − ay , ˆ ˆ (32) 2mω 2 in such a way that the hamiltonian can be expressed as ˆ ˆ ˆ H = Nx + Ny + 1 hω, ¯ (33) and the eigenvalue equation can be solved ˆ H|nx , ny = (nx + ny + 1) hω|nx , ny , ¯ (34) and the spectrum for this case is Enx ,ny = (nx + ny + 1) hω, ¯ (35)
- Different quantum spectra for the same classical system 6 where nx = 0, 1, 2, . . ., ny = 0, 1, 2, . . ., the vacuum state is obtained as ax ay |0, 0 = 0 and a general ˆ ˆ state is given by 1 mx my |mx , my = a† ˆ a† ˆ |0, 0 . (36) (mx !)(my !) 3.2. Case II The hamiltonian is ˆ 1 mω 2 2 H= p2 + p2 + ˆx ˆy x + y2 , ˆ ˆ (37) 2m 2 the previous hamiltonian can be divided in a part corresponding with to the x direction and another in the y direction, both hamiltonians are independent, for the x direction I can choose mω ı mω ı ax = ˆ x+ ˆ p x , a† = ˆ ˆx x− ˆ px , ˆ (38) 2¯ h mω 2¯ h mω and for the y direction mω ı mω ı ay = − ˆ y− ˆ p y , a† = − ˆ ˆy y+ ˆ py , ˆ (39) 2¯h mω 2¯ h mω ˆ with commutation relations between the number operator Nx = a† ax , ax and a† ˆx ˆ ˆ ˆx ˆ ˆ [Nx , ax ] = −ˆx , a ˆ ˆ [ N x , a† ] = a† , ˆx [ˆx , a† ] = 1, a ˆx (40) x ˆ and for Ny = a† ay , ay and a† ˆy ˆ ˆ ˆy ˆ ˆ [Ny , ay ] = ay , ˆ ˆ ˆy [Ny , a† ] = −ˆ† , ay [ˆy , a† ] = −1, a ˆy (41) using the previous equations I ﬁnd √ √ ax |nx , ny = nx |nx − 1, ny , ˆ a† |nx , ny = ˆx nx + 1|nx + 1, ny , ay |nx , ny = ˆ |ny ||nx , ny + 1 , a† |nx , ny = ˆy |ny − 1||nx , ny − 1 , (42) ˆ ˆ where nx = 0, 1, 2, . . ., ny = . . . , −2, −1, 0 and Nx |nx , ny = nx |nx , ny , Ny |nx , ny = ny |nx , ny with the inverse equations h ¯ mω¯ † h x= ˆ ax + a† , ˆ ˆx px = ı ˆ ax − ax , ˆ ˆ (43) 2mω 2 and h ¯ mω¯ † h y=− ˆ ay + a† , ˆ ˆy py = ı ˆ ay − ay , ˆ ˆ (44) 2mω 2 in such a way that the hamiltonian is ˆ ˆ ˆ ¯ H = Nx + Ny hω, (45)
- Different quantum spectra for the same classical system 7 and the eigenvalue equation can be solved ˆ H|nx , ny = (nx + ny ) hω|nx , ny , ¯ (46) and the spectrum for this case is Enx ,ny = (nx + ny ) hω, ¯ (47) where nx = 0, 1, 2, . . ., ny = . . . , −2, −1, 0, the vacuum state is obtained as ax ay |0, 0 = 0 and a ˆ ˆ general state is 1 mx |my | |mx , −|my | = a† ˆ a† ˆ |0, 0 . (48) (mx !)(|my |!) 3.3. Case III In this case the hamiltonian is 2 Hˆ = 1 p2 + p2 + mω x2 + y 2 , ˆ ˆy ˆ ˆ (49) 2m x 2 the previous hamiltonian can be divided into a part that correspond with a x direction and another in the y direction, both hamiltonians are independent, for the x direction I can choose mω ı mω ı ax = − ˆ x− ˆ p x , a† = − ˆ ˆx x+ ˆ px , ˆ (50) 2¯ h mω 2¯h mω and for the y direction mω ı mω ı ay = ˆ y+ ˆ p y , a† = ˆ ˆy y− ˆ py , ˆ (51) 2¯ h mω 2¯ h mω ˆ with commutation relations between Nx = a† ax , ax and a† ˆx ˆ ˆ ˆx ˆ ˆ [ N x , ax ] = ax , ˆ ˆ ˆx [Nx , a† ] = −ˆ† , ax [ˆx , a† ] = −1, a ˆx (52) ˆ and for Ny = a† ay , ay and a† ˆy ˆ ˆ ˆy ˆ ˆ [Ny , ay ] = −ˆy , a ˆ ˆ [ N y , a† ] = a† , ˆy [ˆy , a† ] = 1, a ˆy (53) y using the previous equations I ﬁnd ax |nx , ny = ˆ |nx ||nx + 1, ny , a† |nx , ny = ˆx |nx − 1||nx − 1, ny , √ ay |nx , ny = ˆ ny |nx , ny − 1 , a† |nx , ny = ˆy ny + 1|nx , ny + 1 , (54) ˆ ˆ where nx = . . . , −2, −1, 0, ny = 0, 1, 2, . . . and Nx |nx , ny = nx |nx , ny , Ny |nx , ny = ny |nx , ny with the inverse equations h ¯ mω¯ † h x=− ˆ ax + a† , ˆ ˆx px = ı ˆ ax − ax , ˆ ˆ (55) 2mω 2
- Different quantum spectra for the same classical system 8 and h ¯ mω¯ † h y= ˆ ay + a† , p y = ı ˆ ˆy ˆ ay − ay , ˆ ˆ (56) 2mω 2 in such a way that the hamiltonian can be expressed as ˆ ˆ ˆ ¯ H = Nx + Ny hω, (57) and the eigenvalue equation is ˆ H|nx , ny = (nx + ny ) hω|nx , ny , ¯ (58) and the spectrum for this case is Enx ,ny = (nx + ny ) hω, ¯ (59) where nx = . . . , −2, −1, 0, ny = 0, 1, 2, . . ., the vacuum state is obtained as ax ay |0, 0 = 0 and a ˆ ˆ general state is given by 1 |mx | my | − |mx |, my = a† ˆ a† ˆ |0, 0 . (60) (|mx |!)(my !) 3.4. Case IV The usual hamiltonian for the two dimensional isotropic harmonic oscillator is ˆ 1 mω 2 2 H= p2 + p2 + ˆx ˆy x + y2 , ˆ ˆ (61) 2m 2 it can be divided in a part corresponding with the x direction and another in the y direction, both hamiltonians are independent, for the x direction I can choose mω ı mω ı ax = − ˆ x− ˆ p x , a† = − ˆ ˆx x+ ˆ px , ˆ (62) 2¯h mω 2¯ h mω and for the y direction mω ı mω ı ay = − ˆ y− ˆ p y , a† = − ˆ ˆy y+ ˆ py , ˆ (63) 2¯h mω 2¯ h mω ˆ with commutation relations between the operators Nx = a† ax , ax and a† ˆ ˆ ˆ ˆ x x ˆ ˆ [ N x , ax ] = ax , ˆ ˆ ˆx [Nx , a† ] = −ˆ† , ax [ˆx , a† ] = −1, a ˆx (64) ˆ and for number operator Ny = a† ay , ay and a† ˆy ˆ ˆ ˆy ˆ ˆ [Ny , ay ] = ay , ˆ ˆ ˆ [Ny , a† ] = −ˆ† , ay [ˆy , a† ] = −1, a ˆy (65) y using the previous equations I ﬁnd ax |nx , ny = ˆ |nx ||nx + 1, ny , a† |nx , ny = ˆx |nx − 1||nx − 1, ny , ay |nx , ny = ˆ |ny ||nx , ny + 1 , a† |nx , ny = ˆy |ny − 1||nx , ny − 1 , (66)
- Different quantum spectra for the same classical system 9 ˆ where nx = . . . , −2, −1, 0, ny = . . . , −2, −1, 0 and Nx |nx , ny ˆ = nx |nx , ny , Ny |nx , ny = ny |nx , ny with the inverse equations h ¯ mω¯ † h x=− ˆ ax + a† , ˆ ˆx px = ı ˆ ax − ax , ˆ ˆ (67) 2mω 2 and h ¯ mω¯ † h y=− ˆ ay + a† , p y = ı ˆ ˆy ˆ ay − ay , ˆ ˆ (68) 2mω 2 in such a way that the hamiltonian can be written as ˆ ˆ ˆ H = Nx + Ny − 1 hω, ¯ (69) and the eigenvalue equation can be solved ˆ H|nx , ny = (nx + ny − 1) hω|nx , ny , ¯ (70) and the spectrum for this case is Enx ,ny = (nx + ny − 1) hω, ¯ (71) where nx = . . . , −2, −1, 0, ny = . . . , −2, −1, 0, the vacuum state is obtained as ax ay |0, 0 = 0, with ˆ ˆ the ﬁnal expression for a general state 1 |mx | |my | | − |mx |, −|my | = a† ˆ a† ˆ |0, 0 . (72) (|mx |!)(|my |!) 4. Conclusions and perspectives In the present paper I have deduced that two different classical systems have more than one quantum spectra. This differs of my proper knowledge of quantum mechanics that I have learnt along my physical science studies, it was a great surprise to ﬁnd this result, I veriﬁed all these mathematical results and all are correct. My personal point of view is that there are more systems with this characteristic and in fact, using similar arguments for the case of n-dimensional isotropic oscillators the number of different spectra is undeniable not unique. To ﬁnish, the possibility to make further studies is open, for example, the study of the Kepler problem, a charged particle in a constant gravitational ﬁeld, problems in quantum ﬁeld theory and so on. However, about the uniqueness of the spectrum and the physical interpretation of my results I can not say more. References [1] V. I. Arnold, Mathematical methods of classical mechanics, Moscow, Springer-Verlag, New York, (1980). [2] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University, Princeton, NJ (1992). [3] K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 69, Springer-Verlag, Berlin, (1982).
- Different quantum spectra for the same classical system 10 [4] Ricardo Amorim, J. Math. Phys. 50, 052103 (2009). [5] Vladimir Cuesta, Merced Montesinos and Jos´ David Vergara, Phys. Rev. D 76, 025025 (2007), e [6] David Hestenes and Jeremy W. Holt, J. Math. Phys. 48, 023514 (2007). [7] W. Heisenberg, The Physical Principles of the Quantum Theory, University of Chicago Press, Chicago, (1930). [8] P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, (1981). [9] J. J. Sakurai, Modern Quantum Mechanics, Late, University of California, Los Angeles, Addison-Wesley Publishing Company, Inc. (1994). [10] G. F. Torres del Castillo and M. P. Vel´ zquez Quesada, Rev. Mex. F´s. 50 (6) 608-613. a ı [11] Merced Montesinos and G. F. Torres del Castillo, Phys. Rev. A 70, 032104 (2004).

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