An instanton approach to quantum tunnelling for a particle on a rotating circle

Physics Letters A 168 (1992) 179-186 North-Holland PHYSICS LETTERS A

An instanton approach to quantum tunnelling for a particle on a rotating circle

Sayan Kar Physics Department, Indian Institute of Technology, Kanpur, Kanpur 208016, U.P., India

Received 30 December 1991; accepted for publication 24 June 1992 Communicated by J.P. Vigier

Quantum tunnelling of a particle on a vertical rotating circle in a constant gravitational field is investigated using instanton methods. Instamon solutions and Euclidean actions are constructed by exploiting a correspondence between the particle mechanics problem and certain nonlinear field theories. These Euclidean actions are used to evaluate the tunnelling amplitude for tran- sitions between degenerate vacua, in the Gaussian approximation. Finally a dilute gas Sum is performed to arrive at results for the low lying energy levels and the level-splitting.

1. Introduction

This paper investigates the quantum mechanical features of the problem of a particle on a vertical rotating circle in a uniform gravitational field. An exact solution of the Schr6dinger equation with the corresponding potential function is a fairly difficult problem. We have therefore concentrated on figuring out certain qualitative features using the instanton methods available to us. Thus, the results quoted in this paper are valid only in the semiclassical limit.

The problem of a particle on a vertical rotating circle (henceforth referred to as PORC) is, in a sense, a generalization of the simple pendulum (or particle on a circle, POC). The quantum mechanics of the latter has been discussed by various authors [ 1,2 ]. On the other hand, PORC has not been studied to that extent and the advantage of getting qualitative results using instanton methods has been overlooked. At the level of classical mechanics, PORC has fairly different characteristics depending on whether co > COo or to < 090 (w is the angular frequency of rotation of the circle and 090 = ~ is a certain constant). In the former case, there exist two nontrivial stable points and the particle can be found to be static at any one of these, when observed from the rotating frame. For 09 < 090 there is only one stable point. The two stable points for 09 > 090 are separated by two different barriers. Quantum mechanically, tunnelling takes place between these two degenerate minima and therefore the degeneracy is lifted. The situation is very close to that in the familiar double-well problem. We know that in the Gaussian approximation, the path integral representation for the tunnelling amplitude is dominated by the contribution from instanton solutions. Evaluating the amplitude for the tunnelling across each barrier separately and adding them gives us the total amplitude. We then perform a dilute-gas sum to arrive at a formula for the two low lying levels and the level-splitting.

We shall also see that it is not necessary to work out the instanton solutions and Euclidean actions. This is because there exists a correspondence between the 1 + 1 dimensional double-sine-Gordon field theory (DSG) and the PORC. In fact, the full picture of the correspondence between PORC, POC on the one hand and the SG and DSG field theories on the other will be established. Utilizing this correspondence we can directly write down the instanton solutions and Euclidean actions from a knowledge of the soliton solutions and energies for the 1 + 1 dimensional field theories.

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Volume 168, number 3 PHYSICS LETTERS A 24 August 1992

The organization of the paper is as follows. In section 2 we discuss the classical mechanical problem briefly. Section 3 establishes the correspondence mentioned above. Section 4 is devoted to the tunnelling amplitude, the dilute-gas approximation and the level-splittlng formula. Section 5 is a conclusion with suggestions for future work along these directions.

2. The classical mechanical problem 13,4]

We consider a particle of mass m constrained to move on a circle of radius r. The circle (which can be a loop made out of wire with the particle being a bead free to slide on it) is placed vertically on the surface of the earth, with gravity acting downwards and is made to rotate about the vertical axis at a constant angular frequency co. The only generalized coordinate required to describe the system is 0 which is an angular coordinate (see fig. 1 for details). The Lagrangian of the system is given as

L = ½mr202- (mgr cos 0 - ½mr2co 2 sin20) ( 1 )

The term in ( 1 ) in brackets is the effective potential function. It has certain interesting features which we state below:

(a) For co>co0, O=cos-l(-g/co 2r) and 0 = 2 ~ - c o s - ~ ( - g / c o 2 r ) are minima and 0=0, ~ are maxima. (b) For co<coo, 0 = n is the only minimum and 0 = 0 is the maximum. This means that for co>coo the particle can be stable at any of the two points O=cos-l(-g/coEr) and

0= 2 ~ - c o s - l ( - g / c o 2 r ) . These two stable points are separated by two kinds of barriers. For co< coo there is only one stable point which is at the bottom of the loop. The V(O) versus 0 plots for the potential function for the two cases are shown in figs. 2a and 2b. The fact that the nature of the potential changes completely as co crosses coo has been used to show that this problem has features similar to the well known Higgs potential. Therefore PORC can serve as a classical illustration of spontaneous symmetry breaking [ 5 ]. From the nature of the potential function we can conclude that quantum tunnelling will take place between the degenerate minima in case (a). In case (b) also, one can argue that there is a tunnelling phenomenon associated with the multiple windings of the particle around the circle.

From the Lagrangian given in ( 1 ) we can get the following two Lagrangians for the two different cases mentioned above by adding certain constant terms which do not affect the equations of motion. For co> coo we have

La = ½mr202- ½mr2coE(cos O-k g/coEr) 2 • (2)

For co<coo

Lb = ½rarE02- ½mrEco2 (cos 0-- 1 ) (cos 0+ 1 --2g/co2r) . (3)

)

/ / / / / / / / / / / / /

Fig. 1. The particle on a rotating circle in a constant gravitational field g acting downwards, to is the angular frequency vector, along the vertical pointing upwards.

180


8

to

- - - 0

2Tr

(T) Z>

t i•• ~ c°s'= (- a) 21T- c ° ~ l ( - ° ) Y

-----"-- e

211"

Fig. 2. V(O)-Oplotforthe V(O) in expression (1). (a) to<COo. (b) co>COo.

We notice that as to--, 0 Lb goes over to the Lagrangian of a particle on a circle (POC) or the simple pendulum. This is given as

Lc = ½mr202- mgr( 1 - c o s 0) . (4)

We could have written (3) in the form (2) but then it is not possible to take the o2--*0 limit. Thus for o2<o20 we have to use either (3) or (1) primarily to demonstrate that (3) goes to (4) as co-+0. It also goes without saying that we could have chosen to work with ( 1 ) throughout. The results would be similar. We stress that the choice of ( 2 ) - ( 4 ) is only a matter of convenience.

3. The correspondence

The periodic potential generalizations of ( 2 ) - ( 4 ) are made by replacing 0 (0 ~< 0< 2n) with q, where q now can take all values between - 0o and 0o ( - 0o < q < ~ ). We basically go over from a problem on SI to a problem on ~ which is the universal covering space for S ~. The reason for defining such periodic potential generalizations is related to the fact that the corresponding field theoretic problems are defined such that the field can take all values of~. However, the instanton solutions and Euclidean actions for the periodic potential problem are the same as the ones for the particle mechanics problem on $1. The only difference is that in the periodic potential cases we have a band into which the apparent ground state splits, whereas for the problem on S I we have the periodic boundary condition for the wave function in the form ¢~(0= 2 ~ ) = ~ (0= 0) which chooses a single energy level from the band.

The Lagrangians corresponding to the periodic potential problems related to ( 2 ) - ( 4 ) are given as

l L'~ =½q2- ~aa (c°s q+a)2 ' 0 < a < l , (5)

l L{,= ½q2- ~aa (cos q - 1 ) (cos q+ 1 - 2 a ) , a > l , (6)

L c = ½ q 2 - ( l _ c o s q) . (7)

In the above expressions a-g/co2r. Each of the three Lagrangians shown above are essentially one-dimensional analogs of 1 + 1 dimensional non-

linear field theories. The Lagrangian densities for these field theories are given as follows:

181


, l 1 ~"=~(OaO)(OUO)- ~a (cos O + a ) 2, 0 < a < 1 ,

1 i f { , = ½ ( 0 , 0 ) (OuO) - ~a ( c o s 0 - 1 ) ( c o s 0 + l - Z a ) ,

~'c = ½ (a ~o) ( o " o ) - ( l - c o s 0) •

a > 1,

(8)

(9)

(lO)

As is clear, ( 8 ) - ( 10 ) correspond to ( 5 ) - ( 7 ) respectively. ( 8 ) and ( 9 ) are the double-sine Gordon Lagrangian densities whereas (10) is the familar s ine-Gordon one. The full relationship between PORC, POC and the DSG, SG field theories is shown diagrammatical ly in fig. 3. The crucial point here is the parameter a in the 1 + 1 field theories and the angular frequency to ( a = t o 2 / t o 2) in PORC. For finite values of a we get two ver- sions of the DSG theory. As a ~ 0o we obtain the SG Lagrangian density f rom .Y ;. In a similar way to > too and to< too correspond to the two cases of PORC. As to- ,0 we get POC. The simple fact that the rotating circle becomes a static one as to- ,0 allows us to arrive at the fact that the 1 + 1 field theory corresponding to ff~, is in a sense a generalization of SG theory. On the other hand, L~" leads to a field theory which is quite different from SG theory in almost all respects.

Till now, we have shown a correspondence between Lagrangians only. We know, however, that the static solitons and their energies of 1 + 1 nonlinear field theories are the same as the instanton solutions and Euclidean actions for the corresponding particle mechanics problem (when suitable replacements like f ie lds~coordinates and coordinates--, Euclidean t ime are made) . Thus f rom a knowledge of the former we can get the latter. We utilize this for P O R C and POC to get the instantion solutions and Euclidean actions f rom the solitary waves and energies for the 1 + 1 nonlinear D S G and SG theories.

We first quote f rom the literature [ 6-8 ] the solitary wave solutions and energies for the three 1 + 1 field theoretic Lagrangian densities given in ( 8 ) - (10) .

(a ) For the Lagrangian density ~¢ ~ there are two kinds of solitary wave solutions - one representing a solitary wave across the 0 = 0 hill and the other across the 0 = ~ hill. These are

_ , r LN/ \ ~ " / x / 1 - a 2 xTa'~] ,]J (i) O , ( x )=+2 tan_ / / l _~ tanh / . ~ x . (11)

Restriction on AF

I+1 Field theory I ÷ 1 F T

r

o el

I + l F T

Fig. 3. The full correspondence between PORC, POC and the 1 + 1 dimensional DSG and SG field theories. Here PPP denotes periodic potential problem. FT denotes field theory.

182


Its energy is

E l =4N//a [ - ~ 2 + 2 t a n - t ( ~ + ~ ) 1 . (12)

l + a ~ x ( i i ) ¢ u ( X ) = + 2 t a n - [ ~+_~co th ( . ~ x/~) ] . (13,

It energy is

E 2 = 4 x / / a E ~ 2 2tan -1( l ~ a a ) ] , t a " (14)

(b) For £~b there is only one solitary wave solution given by

~= +2 tan-l(~ 1--~)]2exp(x/P)~, P= N/aa- 1" (15)

Its energy is

E= 4~/-a [tan- ' ( a ~ - ~ ) + x//~- 11 " a (16)

(c) For £P ~ which is the SG theory we have

¢~= +4 tan-~(e x) , (17)

E=8. (18)

The minus sign in the expressions for ¢ refers to antisolitary waves. From this set of expressions we can directly write down the instantons and Euclidean actions for the periodic potential problems mentioned in (5)-(7 ) and hence for PORC and POC. We do it for the latter using the following replacements,

X--*tooZ, 0--*0, Ei--.SE=mgl/2r3/2Ei. (19)

We get the following results: (a) PORC for to> too. (i) Instanton/anti-instanton representing tunnelling between 0= cos- 1 ( _ a) and

0 = 27r - cos- ~ ( - a ) across the 0 = 0 barrier,

1 +a ~ ogz]). O~(z)-- + 2 tan- ( ~l~atanh[x/~(1-a2) (20)

Its Euclidean action is

SE,=4mg'/2r3/2[~aa2 +2x//-atan-'(~l~a) ] . (21)

(ii) Instanton/anti-instanton representing tunnelling between 0 = cos- 1 ( - a) and 0= 2re- cos- 1 ( _ a) across the 0=n barrier,

,/ ~ . tozl) 0u(Z)= +2 tan- ~N/l---~a cotla[ ~ x / ~ - a 2) . (22)

Its Euclidean action is

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SE2 -- ~ 1 -- a = 4 m g ' / 2 r 3 / 2 [ ~ a a 2 - - 2 x / a t a n - ( ~ l ~ a ) l .

(b) PORC for tO < tOo. The instanton/anti-instanton solution is

03(z)= _ + 2 t a n - l ~ 2exp(tOoZ/p) 1 - e x p ( 2 t O o z / p ) } "

and its Euclidean action is

S E 3 = 4 m g , / 2 r 3 / 2 [ x / / - ~ t a n _ l ( ~ + x / a - - l ] t \ x / a - l / -'--x'x'x'x'x'x'x'x~-J"

(c) POC. The instanton/anti-instanton is

04(z) = + 4 tan -1 [exp(tOoZ) ]

and its Euclidean action is

SE4 = 8mg t/2r3/2 .

We notice once again that as a--,oo, i.e. tO-,0, 03(z)-~O4(z) and SEa-,SE4.

(23)

(24)

(25)

(26)

(27)

4. Tunnelling amplitude, dilute-gas sum and level-splitting [9,10]

Once we have the instanton solutions and Euclidean actions it is easy to get the tunnelling amplitude for the transition between degenerate vacua. We shall use the Gaussian or semiclassical approximation to the path integral representation of the amplitude. In this approximation the amplitude for transition between the two vacua labelled as Ii> and I J) is given as

/ \ 1 / 2

T= ,~lim < j l e x p ( - H r / h ) l i > , l , o ) = l ~ h ) exp(-½0tr)x/~E K z e x p ( - S E / h ) , (28)

where ot 2= d2V/dx21 ij and K is a constant independent of z. ( 1, 0) denotes one instanton and no anti-instanton. For PORC (09 > tOo) we have two vacua separated by two different kinds of barriers. The amplitude would

therefore be

( o t ~ ./2 T= lina <Jl exp( - H z / h ) l i > (l,O)= \~--~] exp( - ½0tz)r[K, X/~E~ exp( -SE, /h ) + K= X/~E2 exp(-SE2/h) ] ,

a = x / ~ tOr( l - a 2 ) = x / ~ x / ( l - aZ ) / a , (29)

We can write (29) in a slightly different form by using the fact that

SEI--Se2- 4nmgr =4~mgl/2r3/Ex//-~=__A . (30) tO

Thus we have

T= lim <2~-cos-1 ( - a ) l e x p ( - H z / h ) Icos-I ( - a ) >(1.o), T~oo

= \ - ~ ] e x p ( - - ½ a z ) K 2 x / - ~ E 2 e x p ( - - S E E / h ) [ l + ( K , / K 2 ) ( l + A / S E E ) w E e x p ( - - A / f i ) ] . (31)

184


We now look at an interesting limit of (31). If we consider a-}0, i.e. o~-}oo, which means the rotation is taking place at a very high angular frequency, the minima are close to 0= ½n and/7= ~ n. The barrier heights at 0= 0 and O=n vary as [ ( 1 + a)/x//-a] 2 and [ ( 1 - a ) / x / ~ ] 2. This leads us to the fact that as a-}0 the barrier heights not only become very large but equal too. It is therefore obvious that there would be no communication between the two degenerate minima. The tunnelling amplitude should go to zero. From the expressions for zt, ot and T we get the following,

a-}0 =~ A-}0, a-}oo, SE2-}oo. (32)

Ot and SE2 go tO infinity as

~ - - 1 i/2 3"2 I ot~~/mgr----~SE2~4mg r / ~SEI (33) X/a ~ '

T f ~ r ~ 1/2 [ ~/m-g-rz~(4mgi/2r3/2~l/2exp(_4mgl/2r3/2 ~ ~~,~r~-~) e x p ~ - - - - ~ ) ~ , x/~ ] x/ra h ] , (34)

which shows clearly that tunnelling is suppressed as expected. For finite a we need to perform a dilute-gas approximation to arrive at a formula for the energy-levels and

the splitting. We proceed along the lines of the double-well problem. From the single instanton contributions we can get the expressions for the amplitudes,

TI =(cos-I(-a)lexp(-H'r/h)lcos-l(-a)), T2=(2n-cos-l(-a)lexp(-H'c/h)lcos-i(-a)) ,

(35)

resulting from the multi-instanton configurations. These turn out to be

l f a ~ '/2 Tl = i\~--h,/ exp ( -½ar ) (exp{[K1 exp(-SEl/h)x/r-~Et+K2 exp(--SE2/h).v/'-~E2]r}

+exp{- [Ki exp( -SEl/h)x/~m +Kz exp( --SE2/h)v/'-~E2 ]r}) , (36)

/'2 = g \~-~,] exp( - ½otr)(exp{ [Kl exp( -SEilh)x/~m +K2 exp( --SE21h)v/~E2 lr}

-exp{ - [Kl exp( --SE1/h)x/~E~ +K2 exp( -- SE2/h)x/~E2 )~}). (37)

Comparing with

lim ( - c o s - l ( - a ) +2hi e x p ( - H ~ / h ) Icos-I ( - a ) ) T ~ O O

= lim ~ ( 2 n - c o s - l ( - a ) I n ) ( n l c o s - l ( - a ) ) exp(-E,~:/h), (38) T ~ O O n

we get

E+ = ½hot+h[Klx/~E~ exp(--SEl/h) +K2x/~E2 exp(-SE2/h) ] , (39)

E_ = ½ho~-h [K1 ~ exp( - SEi/h) + K2X/~EZ exp(-SE2/h) ] . (40)

Thus the ground state is split into two separate levels, the splitting p being

P= 2K2 x/~E2 exp(-SE2/h) [ 1 + (Ki/KD ( 1 +J/SE2)1/2 e x p ( - J / h ) ] . (41)

In a similar way, we can do a dilute-gas sum for PORC (co<o~) and arrive at a formula for the quantum ground state. This turns out to be

185


Eo = ½hy- 2hK3x/~r3 e x p ( - & 3 / ~ ) •

We quote the result for the ground state for POC from ref. [ 9 ],

E'o = ½hy'- 2K4x/~E4 e x p ( - S E 4 / h ) ,

where

y=x/~gr[ (a-1) /a] 1/2, y'=v/-m-gr.

(42)

(43)

(44)

5. Conclusion

Let us now summarize the results obtained. We have seen that there exists a correspondence between PORC, POC and the 1 + 1 dimensional DSG and

SG field theories. In fact, PORC turns out to be a one-dimensional analog of the 1 + 1 dimensional DSG theory. Such analogs are not easily available in the context o f field theory and particle mechanics. Moreover, we have obtained a very clear picture o f interrelationships between the field theories and the particle mechanics problems in terms of the ranges of the parameter a given above.

Apart from the correspondence, the other central result is the analysis o f the quantum features o f PORC using instanton methods. In this, we have utilized the correspondence to write down the relevant instanton solutions and Euclidean actions. Thereafter, employing the standard techniques o f the Gaussian approximation and the dilute-gas sum we have shown how the apparent degeneracy of the ground state is lifted due to quantum tunnelling. This demonstrates once again that there can be no spontaneous symmetry breaking in quantum mechanics. In the a ~ 0 limit we have seen that the splitting fl as well as T both go to zero. Since a ~ 0 implies co--,oo we can safely say that for all finite o9 (however large) there is a finite but small fl which enables the symmetry to be restored.

It would be interesting to study the generalization o f this problem by making the particle charged (say, an electron) and introducing a magnetic field. More specifically, the question can be framed as - if we have a magnetic monopole sitting at the centre of the circle what happens to the classical and quantum mechanics of the system?

Work along these lines is in progress and will be communicated later.

Acknowledgement

The author wishes to thank B. Chakraborty, H.S. Mani, S. Ramaswamy, V. Ravishankar, D. Sahdev and R. Shankar for discussions and comments on various issues related to the paper.

References

[ 1 ] E.U. Condon, Phys. Rev. 31 (1928) 891. [2] T. Pradhan and A.V. Khare, Am. J. Phys. 41 (1973) 59. [ 3 ] H. Goldstein, Classical mechanics (Addison-Wesley, Reading, 1985 ) p. 68. [4 ] V.I. Arnold, Mathematical methods of classical mechanics (Springer, Berlin, 1978) p. 87. [5] J. Sivardiere, Am. J. Phys. 51 (1983) 1016. [ 6 ] R.K. Bullough and P.J. Caudrey, eds., Topics in current physics, Vol. 17. Solitons (Springer, Berlin, 1980) p. 107. [7] A. Bishop, D. Campbell and B. Nicolaenko, eds., Nonlinear problems: present and future (North-Holland, Amsterdam, 1982) p.

229 [8] M.J. Ablowitz et al., SIAM J. Appl. Math. 36 (1979) 428. [ 9 ] R. Rajaraman, Solitons and instantons (North-Holland, Amsterdam, 1982) ch. 10.

[ 10 ] S. Coleman, Aspects of symmetry (Cambridge Univ. Press, Cambridge, 1985 ) ch. 7.

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An instanton approach to quantum tunnelling for a particle on a rotating circle

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