Quantum mechanics for q-deformed rotators

Physics Letters B 317 (1993) 359-362 North-Holland PHYSICS LETTERS B

Quantum mechanics for q-deformed rotators

T s u n e h i r o K o b a y a s h i a n d T a k a s h i S u z u k i 1

Institute of Physws, Umverstty of Tsukuba, Tsukuba-sht, Ibarakz- 305, Japan

Received 25 March 1993 Editor M Dine

We study a system for particles rotating on nngs, where only discrete changes of an angle variable ~0 are allowed. The momentum operator for such discrete changes p ~ - ) obeys a q-deformed commutator [ e w, p ~- ) ] q = lh A hamfltonmn Hq which coincides with that for a rotator m the hmat q--. 1 is introduced Finiteness of the energy spectrum and an mfimte-fold degeneracy for every state with a fixed energy appear

Recent deve lopments in technology made it possi- ble to trap some ions in a small regmn [1 ]. In the near future possibly ions well be arranged on a ring. In such a case, the ions will be arranged on the ring with equal dis tance because of repulsive forces between them. When an electron is added to the ring, the motmns o f the electron will approx imate ly be descr ibed by discrete changes f rom 1on to ion i f oscilla- t ions a round the ions are neghglble. One the other hand, a character is t ic feature of quan tum mechanics based on quan tum groups or q-deformed commuta - tors ~s represented by the fact that m o m e n t u m operators are not differential operators o f coordinates but difference opera tors of them, that is, the m o m e n t u m operators generate finite shifts of the coordinates [2 ]. In this sense, the above ment ioned system for the electron on the ring may be approx tmated to a quantum mechanics with an appropr ia te q-deformed commuta tor . In this let ter we shall investigate a quan tum mechanical t rea tment of such motions. We show that it is described by a quantum mechanics with a q-deformed commuta to r and an infini te-fold degeneracy appears m every state w~th a fixed energy

Let S~ be a ring which is pa ramete r lzed by an angle var iable q~ (0~<~<2~) and has a number o f k stable points arranged with equal distance. Taking into ac-

Address after 1 April 1993 Department ofMathemat~cs, Um- verslty of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK

count that elgenfunctions of a ro ta tor on the usual ring S l are wri t ten in terms of l inear combina t ions of ~m (~o) = e ' ' U x / ~ ( m e Z ) , we may guess that eigenfunctions o f parttcles on S}, are descr ibed by func- t ions of e '~, i.e., ~ ( e ' * ) . For such functions we can introduce an opera tor K which shifts ~0 in ~b(e'~) by A~o = 2n /k such that

K ~ ( e '~) = ~ ( e '(~+ 2~/k) ) . ( 1 )

We easily see that the opera tor

K = q-l°~

satisfies eq. (1) , where 0~=0/0~o and q=exp(x .2zr / k) is taken. Now we can int roduce a m o m e n t u m operator as follows:

P~(-) = - l h e - ' ~ Vq ( - ) , (3)

where

I - K Vq(-) = 1 - q ' (4)

and h may be replaced by a function h(q) which re- duces to h in the l imit q--, 1. Note that Vq ( - ) is related to the usual derlvatxve in the l imi t q ~ I as

1 0 hm Vq ( - ) q ~ - 10~0"

We can see that p(~-) obeys to the following q-deformed commuta tor :

0370-2693/93/$ 06 00 © 1993 Elsevier Soence Publishers B.V All rights reserved. 3 59

Volume 317, number 3 PHYSICS LETTERS B 11 November 1993

[e~, p~ - ) ]q=ih , (5)

where [A, B ] q - q A B - B A . Taking account o f the equation K f ( e ~) = f ( q e 1~) and introducing a com- plex variable z = e '~, we easily see that

p ~ - ) f ( e ~ ) = - i h D q f ( Z ) ,

where Dq is the so-called q-derivative defined by D q f ( Z ) = [ f ( z ) - f ( q z ) ] / z ( 1 - q ) , and then p~-~ is reduced to the usual momentum operator Pz = -lhO/ 0z in the limit q ~ 1. Even though p~-) ~s not a hermitian operator, we can write the hamiltonian in a hermitian form as

H q - ~ r n ( + ) " ( - ) (6)

where x is a real constant and

p~+) = ihVq(+) e ~ , (7)

with

1 - K - 1 v~ +~ (8)

- l _ q _ l •

The hamlltonian is written as

2 _ K _ K -1 Hq=tch 2 (9)

( l - q ) ( l - q - ' ) '

and then we immediately see that it is a hermite operator. It is noticeable that the hamiltonian ~s reduced to the well-known hamiltonIan for a rotator with the moment of inertm I = (2~:) -~ in the limit q ~ 1 as

h 2 02 HI-= lira Hq . . . . (10)

q~ 1 21 O~o 2 "

We may say that Hq represents the hamiltonian for q- deformed rotators.

Taking account o f the periodic boundary condi- tion, we can easily find the eigenvalues and the eigenfunctions o f H u as follows:

n q (I) rn ( ~O ) .=. E m (1) m ( ~O ) , (11)

where

Em = x h 2 s in2(zcm/k) ( 12 ) sin 2 ( n / k ) '

for

1 ~m(~0) = v / - ~ e ' ' ~ , (13)

with m = 0, + 1, + 2, .... The eigenfunctions for Hq are the same as those of HI, whereas the eigenvalues having q-dependence are quite different from those for H/. Of course, m the limit q--) l, they coincide with those of the usual rotator, limq~lEm = ( h E / 2 I ) m 2 for I = (2x)-1 . From eq. (12), we see that the relation

Em>~0 (14)

is satisfied. The important consequence of the q-de- formation with qk= 1 is that every eigenstate with a fixed energy degenerates infinitely. For instance, Et appears for an infinite sequence of eigenfunctaons q),~ with

m = l + n k , (15)

where 1/1 =0 , 1, ..., k - 1 and n~Z. Thus, in this system, the number of energy levels is finite. These sit- uations, the degeneracy and finiteness in the energy spectrum, never appear m the system represented by hi. The degeneracy arises from the symmetry o f infinite dimensions in the system we are discussing. In fact the hamlltonian (9) ~s commutable with operators j r = elkn~0~+ 1, that is

[Hq, J~n]=O, (16)

where the J r satisfy the following relations:

Jrn tPm( {O ) =mr+ ltI)m+nk( ~O ) ,

and

r s [J.,J,.]

max(r's' r ( r + 1 ) ( s + l ~ , , u + , ] t r + s _ u = y. ,, / . . + m .

u=o k \ / l + 1 \ /~+ 1] (17)

Eq. 17) shows that J~ for r = - 1, 0, 1, 2 .... form a W~ + oo algebra [ 3 ] without central extensions. Since the investigation of the algebraic structures in the present model is not the theme of this letter, it will be discussed elsewhere [ 4 ].

We find an interesting fact which follows from the degeneracy. Because the lowest energy states with Em= 0 are given by the infinite sequence written by m = nk, the q-deformed rotator can rotate even in the ground states in the sense that

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p ~ ( ~ ) ¢ 0 ,

where ~u is a hnear combinat ion of ~ , , with m = nk such that T = Z,~zc, dP,k ((o), and p~%f-1~ O/O~0 is the angular momen tum operator.

Note here that one should not confuse the motions discussed above with atomic vibrations described by a chain of identical atoms which have the maximum frequency known as the cut-off frequency and then do not have infinite degeneracy. We have discussed the motions o f electrons on the ring and never the vibrations of ions constructing the ring. The latter vibrations are quite similar to the atomic ones with cut- off. We will actually show that the infinite degeneracy can be dissolved by introducing gauge interac- tions in the following discussions

Now we shall study the introduction of gauge inter- actions in Hq. A simple way to introduce a gauge in- teraction is given by the replacement

p(_)~p(_)_ e cA~,

p(+)~p(+)_eA*, (18)

where A~ is in general a function o f q~ and q. Let us discuss a simple example corresponding to the choice A~=lAoe - '~ with Ao a real constant. In this case the hamiltonian is written as

e A o_,~'~ H~q=#e(P(~+)+leAoe")(P(~-)-i c o• ]

+ eAo ('--_" +

+ff q. (19)

Note that in the limit q--, 1 H~ is reduced to the well- known form for a hamiltonlan of a rotator with the charge - e in a constant magnetic field perpendicular to the plane where the rotator moves as follows:

lim a: _ : i eAo a ( e ) 2 ] ~ , L a~o ~ ~ + 7 A° '

(20)

where p , - - i h 0 / 0 ~ 0 . We see that eigenfunctlons for H~ are the same functions ~m(fP) for Hq and eigenvalues of H~ for q)m(q~) are obtained as

e m- 1 2nn E~=E,,+ 2-Aon4i ~ cos c ,=0 k +x(eA°/c)2

for m > 0 ,

Iml 2nn E~=Em-2 Aon'h ~ cos ~ +tc(eAo/c) 2

n= l

f o r m < 0 ,

E~m=tc(eAo/c) 2 for m = 0 . (21)

In the hmxt q--, 1, we have the same results as those for the Zeeman effect. The degeneracy between m and - m is dissolved, but those for m=mo+eNk ( N = 0 , 1 .... ), where I m01 =0 , 1, ..., k - 1 and e= + 1 (respectively - 1 ) for mo> 0 (respectively m 0 < 0 ) , still re- main. All degeneracies are dissolved for the general choice of the gauge function A~. For instance, if A~= lace - 'a~ with o/# integer is taken, the q)m given by ( 13 ) are no more eigenfunctions of H i. It is not easy to solve them exactly, but we can easily see that in the perturbative treatment for the gauge interac- tion matrix elements between the eigenstates described by ~ , , depend on the values m of q),,.

Note here that we can introduce the gauge interac- tion by replacing 10~0 in q,0, with 1 ~ ------ 10q~ - - ( e / c ~ ) A ~ . This choice also gives the same q-~ 1 limit given in (20), but it does not preserve the commutat ion relation (5) in general. The relation is, however, pre- served for the choice A~=Ao=const . In such a case

e _ e(+)p~(-) , wherep~ (+) are the hamiltonian H q = t~p ~

defined by the replacement o f iO~ with iO~ in p(~-+), has the same eigenfunctions q),. and the following eigenvalues for them:

Ee=/c.h2 (1--q(m+a))(1--q -('+a)) ( 1 - - q ) ( 1 - - q -L)

(22)

where a = (e/ch)Ao. Only the degeneracy between m and - m is dissolved, just as in (21). Note that the energy spectrum (22) is different from that given by (21), while both of them coincide with the known results E ~ , = ~ [ m h + (e/c)Ao] 2 for /-/i in the limit q- , 1.

Now we may understand that motions of a particle

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rotat ing on a discret ized ring can be represented by q-deformed quantum mechanics. We can expect to see the character is t ic features in the energy spec t rum of the q-deformed mot ions such as the finiteness of the energy levels and the infinite degeneracy of every state with a fixed energy, while such s i tuat ions never appear for the usual ro ta tor descr ibed by hl. The spec- t rum will be observed in absorpt ion and emission processes of photons. In part icular , the fact that the spectrum has the m a x i m u m value l¢'hE/sinE(R/k), der ived from finiteness, will easily be seen in the observat ion of the ioniza t ion energy of the part icle trap- ped on the ring. We can also expect very interest ing phenomena arising from the infini te degeneracy of every fixed-energy state. As was noted previously, the electron t rapped on the ring can rotate even in the lowest energy states descr ibed by ~,~ with m = nk ex- cept in the case with n = 0. General ly speaking, it is true in the ground state, expressed in terms o f a l inear combina t ion o f ~m. This means that currents possibly exist in the ground state. Taking account of the fact that the energy levels o f ~m for m > 0, m = 0 and m < 0 split In a constant magnet ic field as was given in (21 ) and that then the states with ei ther m > 0 or m < 0 can only be the lowest-energy states for suffi- ciently large magnetic fields, we can expect a non-zero expectation value for p , = -lhO/Otp in the ground state for the system observed in such a magnet ic field. In the lowest-energy states, of course, the electrons can- not emit any photons and then they can move on the ring without any loss of energies. That means that a pe rmanent current appears there. Fur thermore , the infini te degeneracy allows the macroscopic conden- sat ion o f particles, not only bosons but fermions as well, even in the ground state. It indicates that the pe rmanen t current may possibly be of macroscopic order as is usual for superconduct ing currents. In the present case, however, the charge unit in the currents can be - e for electrons, while it is - 2 e in usual superconduct ing currents. Observat ions of such phenomena will be interesting and it is impor tant to know whether physical phenomena descr ibed by q-deformed quan tum mechanics really exist or not. How- ever, it may be a little more compl ica ted in realistic processes for systems with many electrons where in- teract ions among the electrons will be not negligible. We also have to take account of the correct ions for the approx imat ions used in the present discussions such as the devia t ion from the ideal ized dlscretiza-

t ion of the ring, e.g. the dlscret izat ion is not a good approx imat ion for small numbers of k even i f qk= 1 is satisfied. That is to say, the q-deformed mot ions will be realized not exactly but only approximate ly in real physical processes. Anyhow it is very interesting to search for such q-deformed quan tum phenomena in physical processes even i f they are realized only approximately . We hope that technological develop- ments will allow to observe such phenomena, the finiteness o f the energy spectrum and the permanent current, in the near future. Finally, we also remark that we can extend the present formal ism to more general cases having many variables and also apply it to different phenomena, e.g. mot ions for an azimu- thal angle in two and three d imensional problems, mot ions descr ibed by variables with per iodic boundary condit ions, etc. We shall discuss them in another paper [4 ].

Note added. After the complet ion of this work we came across a talk presented by E.G. Flora tos at the Argonne Workshop on Quantum groups [ 5 ]. He discussed the relat ion between q-harmonic oscil lator systems and quan tum mechanical systems of a part i - cle moving on a discret ized circle a round a shielded magnet ic flux. The lat ter case for the charges on a discret ized circle is s imilar to that for the rotators we s tudied above. However , the theme discussed there is quite different from ours. In fact the hami l ton ian used there is different from ours and then it does not have any coincidence with the haml l ton lan for rotators even in the l imit q-~ 1.

References

[ 1 ] W Schlelch, plenary talk, m Proc II Intern Wlgner Symp (Goslar, 1991 ), to appear

[2] J Wess and B Zummo, Nucl Phys B (Proc Suppl ) 18B (1990) 302, B Zummo, Mod. Phys Lett A 16 (1991)1225; I Ya Aref'eva and I.V Volovlch, Phys Lett. B 268 ( 1991 ) 179, U. Carow-Watamura, M Schheker and S. Watamura, Z Phys. C49 (1991) 439, J. Schwenk and J Wess, Phys. Len. B 291 (1992) 273, Y.-Q. L1 and Z.-M Sheng, J. Phys. A 25 (1992) 6779.

[3] CN Pope, L.J Romans and X. Shen, Phys Lett B 242 (1990) 401

[4] T Kobayashl and T Suzuki, Umverslty ofTsukuba prepnnt UTHEP-254 (1993)

[5] E.G Floratos, m. Proc Argonne Workshop on Quantum groups, eds T Curtnght, D Falrhe and C. Zachos (1990) p. 158

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Quantum mechanics for q-deformed rotators