Degeneracies of dyonic harmonic oscillator and effect of magnetic field of dyonium

Indian Journal of Pure & Applied Physics Vol. 40, August 2002, pp.569-573

Degeneracies of dyoni~ harmonic oscillator and effect of m _ agnetic fiel«! on dyonium _

...

-

. \

S~!Joshi*, V P Pandey** & B S Rajput***

\J

Depnrtment of Physics, Kumaun University, Nainital, Uttaranchal 263 002 I E-mail: *scjku@yahoo.com;**vindhyeshwari@yahoo.com;***vc@vsnl.com]

Received 21 June 2001; revised I March 2002; accepted 15 May 2002

Study of dyonic harmonic oscillator and dyonium has been undertaken by incorporating magnetic coupling parameter for dyons in the generalised expression of the Hamiltonian. It is shown that, degeneracy levels for such a system are modificd from the usual degeneracy levels of quantum electrodynamics due to the presence of magnetic charge on dyon. Study of dyonium under the influence of external magnetic field also shows that, energy levels are modified due to the presence of mngnetic charge on dyon and degeneracy of energy levels is lifted, causing splitting of energy levels in the usual way. The radiation from the transition between Ihese levels may indicate the existence of monopoles.

1 Introduction

Tn order to demonstrate the complete behaviour and physical picture of monopoles l-³ and dyons, several mathematical and physical models have been considered by different authors4-7. One such model is the anharmonic oscillator model, which plays important role in the evolution of a large number of areas^X-11 in physics. These models have been used for investigating several interesting physical problemslw and to obtain energy eigenvalues of the systeml~. Following the approach of Cabbibo & Ferrari of two four-potentials, the authors have made successful attempts to develop a self-consistent quantum field theory of generalized field produced by dyonsI5-1~. The bound state of two dyons have also been analyzed, which demonstrated that these bound state solutions are different from the usual bound state solutions of hydrogen atom²⁰ due to the presence of magnetic charge on dyon.

Extending this work to dyonic oscillator, the authors fi nd that, energy levels are modi fied from the usual energy eigenvalues of harmonic oscillator of quantulll electrodynamics, due to the presence of additional magnetic charge on dyon. The degree of degeneracy of the dyonic oscillator is also different from the degree of degeneracy of usual harmonic osci lIator, due to the presence of magnetic charge, and SUO), therefore, cannot be the invariance algebra to describe these particles. This work has been extended to study the behaviour of dyoniulll

(i.e. bound state of two dyons) also under the influence of some external magnetic field. It is seen that, energy levels corresponding to dyonium split into three lines under the influence of applied magnetic field.

2 Energy Spectrum and Degeneracies Associated with Dyonic Oscillator

In order to study the degeneracies, which arise in a system of dyonic charge oscillator, due to the presence of magnetic charge on dyon, one can consider the Hamiltonian as:

-2 2 2 ?

- 1C l1ij mr w- H = -+- - + - --

2m 2mr2 2 ... (2.1)

where, ii is the gauge invariant linear momentum operator defined asl~:

... (2.2) and ~ljj is the magnetic coupling parameter defined as 11·· = e.g. - c .g. and

V-

^{r IS} the spatial

IJ I J J I

component of four-potential Vp defined asl'^L

VII = AII-iBII ... (2.3)

where, All is the electric four-potential and BII is the magnetic four-potential.

The Schrodinger equation for this system:

... (2.4)

570 INDIAN J PURE & APPL PHYS, VOL 40, AUGUST 2002

where, B~ is given by Eq. (2.1), leads to infinite set of equidistant energy levels given by:

[

2

]1/

²

E

= ~(i)

^{(4n+2)+(2,+,F}+4(J1 .. )

11 2 IJ

.. . (2.5)

with n = 0, I ,2,---.

These reduce to the energy eigenvalues of ordinary isotropic harmonic oscillator in the absence of magnetic charge on dyon.

For special case of

... (2.6) so that f.1ij = 0 . Then Eq. (2.5) reduces to:

Thus, additional degeneracies will occur for the identical dyons and for dyons having the same magnitude of the product of electric and magnetic charge.

The degree of degeneracy of the level k is given by:

(k -J.1ij /2)

f (k )

=

L.

(21 +

I)

11=0

... (2.8)

when

I - Ip

^ij

I

^{is even.}

... (2.9)

when

1 - I Pijl

^{is odd.}

It clearly states thaL, in dyonic oscillator the deuree of deoeneracy is different from those of

b b

usual electronic oscillator due to the presence of magnetic charge on dyon, since, the lowest

admissible value of l is

lJ1ij I

instead of zero. In the presence of dyon lJ1ij

I (Ip

^ij

I-I ) /2

^{states have}

angular momentum lower than

!P ij !,

which are not allowed and one gets degree of degeneracies as given by Eqs (2.8) and (2.9). From this equation it is clear that, SU(3) cannot be the invariance algebra to take into account all degeneracies.

3 Energy Spectrum and Degeneracies Associated with Dyonium

In this section, the authors shall explore the energy spectrum and degeneracies associated with dyonium (i.e. bound states of a point-like dyon moving in the field of another point-like dyon). The Hamiltonian for this system is given by:

H ^A = -^{I (}P-Rc^- p.V

- T)2

+V(r)

2J1 IJ ... (3.1)

where IL is the reduced mass of the system. Potential VCr) is defined as²⁰.21

:

... (3.2)

where ct·. =e.e.+g.g. and p .. =e.g . -g.e. are

IJ I J I J 1/ 1.1 1 .1

electric and magnetic coupling parameters²⁰ respectively. Schrodinger equation for this system

IS:

... (3.3) with

lr

^{given by the Eq. (3.1) yields the following}

energy eigenvalues for the system:

Ell = -2ct&m[ (211 + 1)+ t21 + If + 4p i7]12

r

^{2 ... (1.4)}

where n = 0, I, 2.

The energy eigenvalues, given by Eq. (3.4), depend only on n and hence they are degenerate with respect to both l and m since for each value of

1/, l can take values from 0 to 1l-1 and 111 can take values from - I to + I. As such, the total degeneracy of the energy level En is:

... (3.5)

JOSHI ef al.:DYONIC HARMONIC OSCILLATOR 571

where, right hand side shows additional degeneracy, which can be accounted for, by taking a rotationally symmetric angular momentum operator:

[ ]

A _

- - T r r

L=rx P- RcJ1V +J1.-=rxli+J1 .-

IJ IJ r IJ

I rl

^{... (3.6)}

where, Ii IS gauge invariant linear momentum operator.

Runge-Lenz vector

A

of the system is:

... (3.7) Eqs (2.2) and (3.12) lead to following commutation relations 1<1:

[A A] . T

rr rr _{k' I} ^{= I}

J1

^{I) .. E} kllll ~ ^{'I III '}

lA2 A J lA2 A J tA2 A J L , Lk

=

L , L,

=

L , L'II

=

⁰

From Eq. (3.6), one can get^lY:

iix L = -(Lxii)+ i2rr - iJ1ijV T

... (3.8)

Hence, the Range-Ienz vector (3.13) takes the following form:

A I

r (- )] r

A =-~ii- Lxii ~+a .. -

m IJ ,. ... (3.9)

Using Eqs (3.1), (3.6) and (3.9), one may define the following operator:

L' = (L+ A')/2, LH = (L - A')/2 ... (3.10)

1/2 where, A'

= (-

2H

t

^A

it can also be written in the following form:

Diagonalizing them with fl, one can get the following values for L'2 and L,,2

L'2 =

± [~1 -

^{J1 ij}

f -

^{I ]}

C²

=± [~'+J1ij f+ d

The energy levels incorporate dimensions

~, 2

^-

^{J1i; )}

^{accommodating all possible degeneracies}

of the dyonium system.

4 Dyonium in an External Magnetic Field

In order to understand the behaviour of dyonium in an externally applied magnetic field, one can start with the following quantum mechanical Hamiltonian of the system, as

fl2 2 ill ( - -) iqfl . H

=

- - V + - \eA - g/3 - - -2 cp

2/11. mc mc

I [(? ?

y ?

^{2 \ - -]}

+--2 ~~e--g-AA--/3 r4egA./3 +qcp 2mc

... (4.1)

For total field along Z-direction, A and B are given by:

Ax

= - }';,

^{H y'} Ay

= }';,

^{H x'} Az

= OJ

Bx =-}';,Ey,By = }';,Ex,Bz =0

... (4.2)

where, E and H are electric and magnetic fields given as^lX:

- aA - - ]

E = -ecp+- - Vx /3

- at vB - -

H =-gcp+- - VxA

at

In this case:

... (4.3)

572 INDIAN] PURE & APPL PHYS, VOL 40, AUGUST 2002

... (4.4) and

- I - d

A-V=-f-I-

2

d ¢

^{... (4.5)}

... (4.6)

The Hamiltonian (4.1) then takes the following form:

+ I:'R (f-I}" ^- sin e)(Ersine^- ) ^] - ^Re(qq*) + ^{Im(qq*0 ) "} .(4.7)

r ^21111'-

Since

RC( fl r/' ) Im(

^{qq* )}

qcp = - + ?

}" 21111'-

... (4.8)

In this equation, the term

12

R c( ^{fl e/' )} lm( qr/ )

~

V

^{2 -} + forms the Hamil-

fi,n2111 ,. 2111}"2

tonian of the field-free system and other terms, occurring due to the effect of magnetic field, may be considered as perturbation. The second term is field- dependent term and effecti ve potential of the system is changed by the third term, which includes H~ and

E~ .For weak magnetic fields, this term produces second order correction. which can be neglected.

The unperturbed Hamiltonian can be written as:

... (4.9)

'vvhere VCr) is given by Eg. (4.8) and the first order perturbation can also be written as:

(I) -ill ( ) d

f-I

= - -

ef-l + ~E -

47rJIIC J¢

... (4.1 0)

The energy eigenvalue of the Hamiltonian (4.9) are gi ven by Eq. (3. 10) and the fi rst order perturbation theory gives the perturbation energy:

Ii ( \. ' EJ

= - -

^ef-l + gEJ/IL

4nmc ^{... (4.11)}

where /ii' =111 cose is the magnetic quantum number of the system of dyon moving in the field of another dyon2021. The total energy of the system then is:

11111' ( ) tl ( )

E = Eo + - - ef-l + f!.E = Eo + - ef-l + gE cose

47rJIlC 2c

... (4.12) Let the transition occur from an energy state E2 to another energy state E1, then, the frequencies of lines emitted during the transition will be given by:

E - E v = 2 I

Ii

E(O) E(o)

--=.2_-_~-,-1_+ ef-l + gE (~IIO

h 47rJlIC

... (4.13)

5 Conclusion

For identical dyon the authors have obtained the energy eigenvalues given by Eq. (2.5), which are modified from the energy eigenvalues of simple harmonic oscillator due to the presences of magnetic charge on dyon. In the absence of magnetic charge, these equations reduce to the energy levels of usual harmonic oscillator. Magnetic charge on dyon introduces certain anharmonicities in the system of a dyon, moving in the field of another dyon. Such anhannonicities change the degeneracy levels of dyonic oscillator. 11 implies that, for identical dyon, coupling is mainly due to magnetic charge (as magnetic coupling parameter for identical dyons is zero, but, electric coupling parameter takes the value e2+ g2, where e2 is much smaller than g2). This shows that, magnetic charge plays a significant role in the case of dyonic oscillator, even if the magnetic couplir.g parameter vanishes.

The study of degree of degeneracy of dyonium (i.e. bound state of two dyons) reveals that, degeneracy is modified from those of hydrogen atom. As such, the energy levels incorporate (Il- PiiNI+!liJ dimensions encompassll1g all possible degeneracies of the system The motion of a dyonium in externally applied magnetic field, as studied in the fourth section, leads to the emission of frequencies given by Eq. (4.13). These frequencies and energy levels are different fro111 the llsual

JOSHI el al.:DYONIC HARMONIC OSCILLATOR 573

frequencies and energy levels of quantum electrodynamics. This difference gives the signal for the existence of monopoles.

Acknowledgement

One of the author S C Joshi is thankful to DSN COS 1ST (SAP), New Delhi, for financial assistance.

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