- Email: [email protected]
A q-deformed quantum mechanics
12 November 1998
Physics Letters B 440 Ž1998. 66–68
A q-deformed quantum mechanics Jian-zu Zhang
a,b,c
a
c
Max-Planck-Institut fur Ring 6, D-80805 Munchen, Germany ¨ Physik (Werner-Heisenberg Institut), Fohringer ¨ ¨ b Department of Physics, UniÕersity of Kaiserslautern, P.O. Box 3049, D-67653 Kaiserslautern, Germany School of Science, East China UniÕersity of Science and Technology, 130 Mei Long Road, Shanghai 200237, China Received 24 April 1998; revised 5 August 1998 Editor: P.V. Landshoff
Abstract Within a q-deformed quantum mechanical framework, features of the uncertainty relation and a novel formulation of the Schrodinger equation are considered. q 1998 Elsevier Science B.V. All rights reserved. ¨ PACS: 03.65.Bz
In searching for possible new physics at short distances Žor high energy scale. consideration of the space structure is a useful guide. Quantum groups are a generalization of symmetry groups which have been used successfully in physics. A general feature of spaces carrying a quantum group structure is that they are noncommutative and inherit a well-defined mathematical structure from quantum group symmetries. In applications in physics, questions arise whether the structure can be used for physics at short distances and what phenomena could be linked to it. Recently, starting from such a noncummutative space as configuration space a generalization to a phase space is obtained w1x. This noncummutative phase space is derived from the noncommutative differential structure on configuration space w2x. Such noncommutative phase space is a q-deformation of the quantum mechanical phase space and thus all the machinery used in quantum mechanics can be applied in q-deformed quantum mechanics w1,3–5x. In this letter we discuss the essential new features of a q-deformed quantum mechanics: Ži. A q-de-
formed uncertainty relation: Here we find that the lowest limit of the Heisenberg uncertainty is undercut. Žii. A q-deformed dynamical equation which is found to be non-linear. The perturbative expansion of the latter shows complex structure. In the lowest order approximation this equation is just the Schrodi¨ nger equation. The characteristics of the new equation are essentially non-perturbative. The qualitative behavior of its non-perturbative solutions is different from that of the Schrodinger equation. For example, ¨ the spectrum of the q-deformed harmonic oscillator is exponentially spaced w5x.
1. A q-deformed uncertainty relation The starting point of our investigation is the following q-deformed Heisenberg algebra w1,6x: q 1r2 XP y qy1 r2 PX s iU
Ž 1.
UX s qy1 XU,UP s qPU
Ž 2.
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 7 9 - X
J.-z. Zhangr Physics Letters B 440 (1998) 66–68
Ž 4.
conclude that in the state < n, s :< s :, D X D P s 0. This is a surprising qualitative deviation from Heisenberg’s uncertainty relation. It therefore raises the question whether or not the conventional uncertainty relation is recovered at large scales. Unfortunately because of the complicated relations among X, P and U, an explicit form of the right-hand side of the uncertainty relation as a function of D X and D P is not obtained at this stage. Thus when q is some fixed value not equal to one it is still an important open question whether this q-deformed quantum mechanics does at all reproduce ordinary quantum mechanics at large scales. Ž7. may prove to be an important result within this formulation of a q-deformed quantum mechanics. Perhaps insights of a possible new physics just come from here. The issue of uncertainty relations in the context of quantum group symmetric Heisenberg algebras was first considered by Kempf w8x.
Ž 5.
2. The q-deformed dynamical equation
with the conjugation properties P†sP,
X †sX,
U † s Uy1 .
Ž 3.
This q-deformed algebra is derived from the noncommutative differential structure on configuration space w2x. However, if X is assumed to be a hermitean operator in a Hilbert space, the usual quantization rule p ™ yi E X does not yield a hermitean momentum operator w1x. In order to define conjugation of E X , and then a hermitean momentum operator P, it is necessary to introduce a unitary scaling operator U satisfying Ž2. 1. In Eqs. Ž1. and Ž2. the parameter q is real and q ) 1. For the case of q s 1, the scaling operator U is reduced to a unit operator, and Eq. Ž1. reduces to the Heisenberg algebra. The non-trivial properties of the operator U lead to a richer structure of algebra Ž1. and Ž2. than the Heisenberg algebra. From Ž1. and Ž3. we obtain XP y PX s iC
67
where C s Ž U q Uy1 . r Ž q 1r2 q qy1 r2 . .
In order to show the special characteristics of the q-deformed uncertainty relation, we prove the following lemma. Lemma. In any state the expectation value of the operator C in Ž5. satisfies <² C :< F 1 .
Ž 6.
We note that <²U q Uy1 :< F 2. Because q 1r2 q qy1 r2 G 2 for any q ) 0, we obtain lemma Ž6.. Eq. Ž4. gives 1 2
D X P D P G <² C :< .
Ž 7.
(
2 where D A s ² Ž A y ² A: . : . Lemma Ž6. shows that the lowest limit of the Heisenberg uncertainty relation D X P D P s 12 is undercut. It is interesting to show w7x that for the irreducible eigenstate < n, s :< s : of P w1x, P < n, s :< s : s P0 < n, s :< s : with P0 s s sq n , Ž s s "1; 0 - s - 1; n s 0,1,2, . . . ., we have D P s 0, but D X is still finite. In fact, using Ž1. – Ž3., we obtain D X s Ž q y qy1 .y1 Ž q q qy1 .1r2 Py1 0 . Thus we
1 The definitions of the scaling operator U, the conjugate E X and E X and the hamitean momentum operator P are w1x: Uy1 s q 1r 2 w1qŽ q y1. XE X x, E X sy qy1r2 UE X , P sy 2i Ž E X y E X ..
The variables of the q-deformed algebra Ž1. and Ž2. can also be expressed in terms of the variables of an undeformed algebra. There are three pairs of canonically conjugate variables w1x: Ži. The variables x, ˆ pˆ of the undeformed quantum mechanics; they satisfy w x, ˆ pˆ x s i. Žii. The variables x, ˜ p; ˜ which are obtained by canonical transformation of xˆ and p: ˆ p˜ s f Ž zˆ. p, ˆ x˜ s xf ˆ y1 Ž zˆ. where fy1 Ž zˆ . s
w zˆ y 1r2x i , zˆ s y Ž xp ˆˆ q px ˆˆ . , zˆ y 1r2 2
Ž 8.
and w A x s Ž q A y qyA .rŽ q y qy1 . for any A. The variables x˜ and p˜ also satisfy w x, ˜ p˜ x s i. Žiii. The q-deformed variables X and P where X, P and the scaling operator U are related to x˜ and p˜ in the following way P s fy1 Ž z˜ . p, ˜ y1 Ž
X s x˜ ,
U s q z˜ .
Ž 9.
in Ž9. z˜ and f z˜. are defined, by the same Eqs. Ž8. for zˆ and fy1 Ž zˆ.. It is easy to check that X, P and U in Ž9. satisfy Ž1. – Ž3.. Our starting point is to use the q-deformed variables X and P to write down the Hamiltonian, then using Ž9. to represent X and P by x˜ and p. ˜ Because of w x, ˜ p˜ x s i, Žthus in the x˜ representation p˜ s yi E˜,
J.-z. Zhangr Physics Letters B 440 (1998) 66–68
68
where E˜s ErE x˜ . all the machinery of quantum mechanics can be used for the Ž x, ˜ p˜ . system. The q-deformed Hamiltonian of the system with potential V Ž X . is H Ž X, P . s P 2r2 m q V Ž X .. Using Ž1. – Ž3. and Ž9. the stationary dynamical equation of q-deformed quantum mechanics reads as
½
1 y 2m
Ž q y qy1 .
y2
x˜y2
˜
˜
= q Ž qy2 x˜ E y 1 . q qy1 Ž q 2 x˜ E y 1 .
5
qV Ž x˜ . c Ž x˜ . s Eq c Ž x˜ . .
Ž 10 . Acknowledgements
Eq. Ž10. is a non-linear equation which is a q-generalisation of the Schrodinger equation. ¨ For the case of q is close to 1, we let q s e f , 0 - f < 1. The perturbative expansion of the Hamiltonian is then Hsy
1
y2
Ž 2 f q 13 f 3 q . . . . x˜y2 2m
= 4 f 2 x˜ 2E˜2 q 13 f 4 Ž 4 x˜ 4E˜4 q 16 x˜ 3E˜3 q10 x˜ 2E˜2 . q . . . q V Ž x˜ . .
Ž 11 .
Thus to the lowest order in f, Eq. Ž10. reduces 1 y 2m
If q-deformed quantum mechanics is a correct theory, its corrections to the undeformed theories must be very small at the energy range which can be reached by present-day experiments. In view of the present accuracy of tests of quantum electrodynamics at least down to 10y1 7 cm, the effects of q-deformed quantum mechanics would show up at distances much smaller than 10y1 7 cm. We hope that the q-deformed uncertainty relation might show some evidence in present-day experiments.
E˜2 q V Ž x˜ . c Ž x˜ . s Ec Ž x˜ . .
Ž 12 .
This is just the Schrodinger equation of the Ž x, ¨ ˜ p˜ . Ž . system. In 11 the next order correction of H shows a complex structure which amounts to some additional momentum dependent interaction. In the above we constructed the q-deformed Hamiltonian of the variables X and P in analogy with the undeformed system. Another possible way to construct a q-deformed Hamiltonian is that the system should act in accordance with a special algebra. An example is a q-deformed harmonic oscillator w5x.
The author would like to thank Prof. J. Wess very much for giving the author the opportunity to join his project on quantum groups and for many stimulating helpful discussions and Prof. H.J.W. Muller¨ Kirsten for many helpful comments. He would also like to thank the Max-Planck-Institut fur ¨ Physik ŽWerner-Heisenberg-Institut. for financial support and the Sektion Physik, Universitat for ¨ Munchen, ¨ warm hospitality. His work has also been supported by the Deutsche Forschungsgemeinschaft ŽGermany., the National Natural Science Foundation of China under Grant No. 19674014, and the Shanghai Education Development Foundation.
References w1x M. Fichtmuller, A. Lorek, J. Wess. Z. Phys. C 71 Ž1996. 533. ¨ w2x J. Wess, B. Zumino, Nucl. Phys. ŽProc. Suppl.. 18B Ž1990. 302. w3x J. Schwenk, J. Wess, Phys. Lett. B 291 Ž1992. 273. w4x A. Lorek, J. Wess, Z. Phys. C 67 Ž1995. 671. w5x A. Lorek, A. Ruffing, J. Wess, Z. Phys. C 74 Ž1997. 369. w6x A. Hebecker, S. Schreckenberg, J. Schwenk, W. Weich, J. Wess, Z. Phys. C 64 Ž1994. 355. w7x J. Wess, private conversation. w8x A. Kempf, J. Math. Phys. 35 Ž1994. 4483.
Physics Letters B 440 Ž1998. 66–68
A q-deformed quantum mechanics Jian-zu Zhang
a,b,c
a
c
Max-Planck-Institut fur Ring 6, D-80805 Munchen, Germany ¨ Physik (Werner-Heisenberg Institut), Fohringer ¨ ¨ b Department of Physics, UniÕersity of Kaiserslautern, P.O. Box 3049, D-67653 Kaiserslautern, Germany School of Science, East China UniÕersity of Science and Technology, 130 Mei Long Road, Shanghai 200237, China Received 24 April 1998; revised 5 August 1998 Editor: P.V. Landshoff
Abstract Within a q-deformed quantum mechanical framework, features of the uncertainty relation and a novel formulation of the Schrodinger equation are considered. q 1998 Elsevier Science B.V. All rights reserved. ¨ PACS: 03.65.Bz
In searching for possible new physics at short distances Žor high energy scale. consideration of the space structure is a useful guide. Quantum groups are a generalization of symmetry groups which have been used successfully in physics. A general feature of spaces carrying a quantum group structure is that they are noncommutative and inherit a well-defined mathematical structure from quantum group symmetries. In applications in physics, questions arise whether the structure can be used for physics at short distances and what phenomena could be linked to it. Recently, starting from such a noncummutative space as configuration space a generalization to a phase space is obtained w1x. This noncummutative phase space is derived from the noncommutative differential structure on configuration space w2x. Such noncommutative phase space is a q-deformation of the quantum mechanical phase space and thus all the machinery used in quantum mechanics can be applied in q-deformed quantum mechanics w1,3–5x. In this letter we discuss the essential new features of a q-deformed quantum mechanics: Ži. A q-de-
formed uncertainty relation: Here we find that the lowest limit of the Heisenberg uncertainty is undercut. Žii. A q-deformed dynamical equation which is found to be non-linear. The perturbative expansion of the latter shows complex structure. In the lowest order approximation this equation is just the Schrodi¨ nger equation. The characteristics of the new equation are essentially non-perturbative. The qualitative behavior of its non-perturbative solutions is different from that of the Schrodinger equation. For example, ¨ the spectrum of the q-deformed harmonic oscillator is exponentially spaced w5x.
1. A q-deformed uncertainty relation The starting point of our investigation is the following q-deformed Heisenberg algebra w1,6x: q 1r2 XP y qy1 r2 PX s iU
Ž 1.
UX s qy1 XU,UP s qPU
Ž 2.
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 7 9 - X
J.-z. Zhangr Physics Letters B 440 (1998) 66–68
Ž 4.
conclude that in the state < n, s :< s :, D X D P s 0. This is a surprising qualitative deviation from Heisenberg’s uncertainty relation. It therefore raises the question whether or not the conventional uncertainty relation is recovered at large scales. Unfortunately because of the complicated relations among X, P and U, an explicit form of the right-hand side of the uncertainty relation as a function of D X and D P is not obtained at this stage. Thus when q is some fixed value not equal to one it is still an important open question whether this q-deformed quantum mechanics does at all reproduce ordinary quantum mechanics at large scales. Ž7. may prove to be an important result within this formulation of a q-deformed quantum mechanics. Perhaps insights of a possible new physics just come from here. The issue of uncertainty relations in the context of quantum group symmetric Heisenberg algebras was first considered by Kempf w8x.
Ž 5.
2. The q-deformed dynamical equation
with the conjugation properties P†sP,
X †sX,
U † s Uy1 .
Ž 3.
This q-deformed algebra is derived from the noncommutative differential structure on configuration space w2x. However, if X is assumed to be a hermitean operator in a Hilbert space, the usual quantization rule p ™ yi E X does not yield a hermitean momentum operator w1x. In order to define conjugation of E X , and then a hermitean momentum operator P, it is necessary to introduce a unitary scaling operator U satisfying Ž2. 1. In Eqs. Ž1. and Ž2. the parameter q is real and q ) 1. For the case of q s 1, the scaling operator U is reduced to a unit operator, and Eq. Ž1. reduces to the Heisenberg algebra. The non-trivial properties of the operator U lead to a richer structure of algebra Ž1. and Ž2. than the Heisenberg algebra. From Ž1. and Ž3. we obtain XP y PX s iC
67
where C s Ž U q Uy1 . r Ž q 1r2 q qy1 r2 . .
In order to show the special characteristics of the q-deformed uncertainty relation, we prove the following lemma. Lemma. In any state the expectation value of the operator C in Ž5. satisfies <² C :< F 1 .
Ž 6.
We note that <²U q Uy1 :< F 2. Because q 1r2 q qy1 r2 G 2 for any q ) 0, we obtain lemma Ž6.. Eq. Ž4. gives 1 2
D X P D P G <² C :< .
Ž 7.
(
2 where D A s ² Ž A y ² A: . : . Lemma Ž6. shows that the lowest limit of the Heisenberg uncertainty relation D X P D P s 12 is undercut. It is interesting to show w7x that for the irreducible eigenstate < n, s :< s : of P w1x, P < n, s :< s : s P0 < n, s :< s : with P0 s s sq n , Ž s s "1; 0 - s - 1; n s 0,1,2, . . . ., we have D P s 0, but D X is still finite. In fact, using Ž1. – Ž3., we obtain D X s Ž q y qy1 .y1 Ž q q qy1 .1r2 Py1 0 . Thus we
1 The definitions of the scaling operator U, the conjugate E X and E X and the hamitean momentum operator P are w1x: Uy1 s q 1r 2 w1qŽ q y1. XE X x, E X sy qy1r2 UE X , P sy 2i Ž E X y E X ..
The variables of the q-deformed algebra Ž1. and Ž2. can also be expressed in terms of the variables of an undeformed algebra. There are three pairs of canonically conjugate variables w1x: Ži. The variables x, ˆ pˆ of the undeformed quantum mechanics; they satisfy w x, ˆ pˆ x s i. Žii. The variables x, ˜ p; ˜ which are obtained by canonical transformation of xˆ and p: ˆ p˜ s f Ž zˆ. p, ˆ x˜ s xf ˆ y1 Ž zˆ. where fy1 Ž zˆ . s
w zˆ y 1r2x i , zˆ s y Ž xp ˆˆ q px ˆˆ . , zˆ y 1r2 2
Ž 8.
and w A x s Ž q A y qyA .rŽ q y qy1 . for any A. The variables x˜ and p˜ also satisfy w x, ˜ p˜ x s i. Žiii. The q-deformed variables X and P where X, P and the scaling operator U are related to x˜ and p˜ in the following way P s fy1 Ž z˜ . p, ˜ y1 Ž
X s x˜ ,
U s q z˜ .
Ž 9.
in Ž9. z˜ and f z˜. are defined, by the same Eqs. Ž8. for zˆ and fy1 Ž zˆ.. It is easy to check that X, P and U in Ž9. satisfy Ž1. – Ž3.. Our starting point is to use the q-deformed variables X and P to write down the Hamiltonian, then using Ž9. to represent X and P by x˜ and p. ˜ Because of w x, ˜ p˜ x s i, Žthus in the x˜ representation p˜ s yi E˜,
J.-z. Zhangr Physics Letters B 440 (1998) 66–68
68
where E˜s ErE x˜ . all the machinery of quantum mechanics can be used for the Ž x, ˜ p˜ . system. The q-deformed Hamiltonian of the system with potential V Ž X . is H Ž X, P . s P 2r2 m q V Ž X .. Using Ž1. – Ž3. and Ž9. the stationary dynamical equation of q-deformed quantum mechanics reads as
½
1 y 2m
Ž q y qy1 .
y2
x˜y2
˜
˜
= q Ž qy2 x˜ E y 1 . q qy1 Ž q 2 x˜ E y 1 .
5
qV Ž x˜ . c Ž x˜ . s Eq c Ž x˜ . .
Ž 10 . Acknowledgements
Eq. Ž10. is a non-linear equation which is a q-generalisation of the Schrodinger equation. ¨ For the case of q is close to 1, we let q s e f , 0 - f < 1. The perturbative expansion of the Hamiltonian is then Hsy
1
y2
Ž 2 f q 13 f 3 q . . . . x˜y2 2m
= 4 f 2 x˜ 2E˜2 q 13 f 4 Ž 4 x˜ 4E˜4 q 16 x˜ 3E˜3 q10 x˜ 2E˜2 . q . . . q V Ž x˜ . .
Ž 11 .
Thus to the lowest order in f, Eq. Ž10. reduces 1 y 2m
If q-deformed quantum mechanics is a correct theory, its corrections to the undeformed theories must be very small at the energy range which can be reached by present-day experiments. In view of the present accuracy of tests of quantum electrodynamics at least down to 10y1 7 cm, the effects of q-deformed quantum mechanics would show up at distances much smaller than 10y1 7 cm. We hope that the q-deformed uncertainty relation might show some evidence in present-day experiments.
E˜2 q V Ž x˜ . c Ž x˜ . s Ec Ž x˜ . .
Ž 12 .
This is just the Schrodinger equation of the Ž x, ¨ ˜ p˜ . Ž . system. In 11 the next order correction of H shows a complex structure which amounts to some additional momentum dependent interaction. In the above we constructed the q-deformed Hamiltonian of the variables X and P in analogy with the undeformed system. Another possible way to construct a q-deformed Hamiltonian is that the system should act in accordance with a special algebra. An example is a q-deformed harmonic oscillator w5x.
The author would like to thank Prof. J. Wess very much for giving the author the opportunity to join his project on quantum groups and for many stimulating helpful discussions and Prof. H.J.W. Muller¨ Kirsten for many helpful comments. He would also like to thank the Max-Planck-Institut fur ¨ Physik ŽWerner-Heisenberg-Institut. for financial support and the Sektion Physik, Universitat for ¨ Munchen, ¨ warm hospitality. His work has also been supported by the Deutsche Forschungsgemeinschaft ŽGermany., the National Natural Science Foundation of China under Grant No. 19674014, and the Shanghai Education Development Foundation.
References w1x M. Fichtmuller, A. Lorek, J. Wess. Z. Phys. C 71 Ž1996. 533. ¨ w2x J. Wess, B. Zumino, Nucl. Phys. ŽProc. Suppl.. 18B Ž1990. 302. w3x J. Schwenk, J. Wess, Phys. Lett. B 291 Ž1992. 273. w4x A. Lorek, J. Wess, Z. Phys. C 67 Ž1995. 671. w5x A. Lorek, A. Ruffing, J. Wess, Z. Phys. C 74 Ž1997. 369. w6x A. Hebecker, S. Schreckenberg, J. Schwenk, W. Weich, J. Wess, Z. Phys. C 64 Ž1994. 355. w7x J. Wess, private conversation. w8x A. Kempf, J. Math. Phys. 35 Ž1994. 4483.