T H E R I G I D R O T A T O R A C C O R D I N G T O R E L A T I V I S T I C Q U A N T U M M E C H A N I C S

BY K. N . SR1NIVASA RAO*

(Department of Physics, Central College, Bangalore)

AND

D. SAROJA

(C.S.LR. dunior Research Fellow, Department of Physics, Central College, Bangalore)

Received September 22, 1964

(Communieated by Dr. K. S. Viswanathan, r.A.Se,)

AFTER the classic application of the Relativistic Wave Equation of Dirac to the theory of hydrogen ¡ structure, ~-3 there were several other applications of the equation to problems solved by non-relativistic Quantum Mechanics. Sauter ~ treated the problem of a particle in a uniform ¡ V (x) ---- ax and Nikolsky 5 considered the problem of the harmonic oscillator with V (x) = ~ox z. The general case when the potential V (x) is a polynomial of arbitrary degree in x or 1/x was treated by Plesset. 6 However, the problem of the rigid rotator does not appear to have been considered and the purpose of this note is to determine the energy eigenvalues of a rigid rotator according to both the Klein-Gordon and Dirae equations.

For the sake of completeness, we first of all consider the rather trivial case of a single Klein-Gordon particle of test mass m at a fixed distance ' a ' from a centre of rotation with moment of inertia I = m a l The stationary states of this system must satisfy the Klein-Gordon equation,

_ = • _ _ h V * ~ b + ~ c r ( W 2 g2)~b 0; g = m c 2 ; --2*r (1)

which, in sphe¡ co-ordinates with r = a, becomes

sin �91 _ _ sin20 ~4 z + ~~-g (W -- g*) 6 = 0. (2)

Comparing (2) with the corresponding non-relativistic Schrodinger equation, we immediately obtain

* Present Address : Reader in the Department of Post-Graduate Studies and Researr in Physics, Manasa Gangotri, Mysore.

398

The R i g i d R o t a t o r according to Re la t i v i s t i c Q u a n t u m M e c h a n i c s 399

I (W 2 -- ~2) = I (1 + 1)" 1 integral /ja a

from which follows

W -- Wz ---- g J 1 + l (1 +igl) ~a (3)

One observes that, with W = E + �91 (3) goes over, in the non-relativistic approximation, to the well-known expression

El -- I (l + 1) h ~ 8~ai (4)

We next consider the more realistic case of a dumb-bell shaped rotator. It will be seen that the problem can be solved ir we assume that the two particles of the non-interacting

H = c

rotator are of equal mass. The Hamiltonian f o r a pair of particles o f rest masses mi and m~ is

~/m~ + ~:+ . V / ~ + ~:

---~ C %/mi2C 2 - ~2VIz + C v 'm22e a - - ~"V2 ~ (5)

where we make the usual substitution ~ = -- i~V and V~ and V~ refer to the differr particles. J u s t a s in the non-relativistic case we transform to the relative co-ordinates

x = x x - - x a ; y-----yx--Ya; z-----zt--za

and the co-ordinates of the centre of mass,

X = m l x l + max2 . y _ mxya + m2ya . t o t + m a ' m l + m~ '

Z - - mlZ1 + m2z2 mi + mi

wr then obtain

axl = M ~ - X + ~ ' etc.; M = r o l + m 2

~ m~ ~ ~ etc." ~x~ M ~X ~x ' '

~2 m12 ~z ~2 2m~ ~2 ~xt ~ - M ~~X 2 + ~ + M ~X~x' etc.;

~-~a ~ = ~ ~ + "~x 2 M ~X~x' etc. (6)

400 K . N . SRINIVASA RAO AND D. SAROJA

Since we are not interested in the translational motion of the system, we hold X, Y, Z constant and we obtain

(~ ~ ~~) V~ = - - V ~ = V - , ~y,

V,'= V,' = V'.

I f now we take

mx = m~ ---- m ; m c 2 =

(5) becomes

(6')

%/i H = �91 2~2V 2 (7) m g

From the relativistic wave equation

~ - w ) ~ = 0

we have on multiplying by H + W on the left

( H ' - - W 9 ~ = 0

and substituting from (7), we finally obtain

( w ' - a 9 m V ~ + ~ ~ - ~ - 4 = 0 (8)

whieh is the appropriate Klein-Gordon equation for our system.

With r = a ; I = p~ ' where the reduced mass q = m / 2 a n d a = 2 m e ~

the test energy o f the system, (8) goes over into (2) and we recover the formula (3) for W.

We now proceed to determine the eigenvalues of W according to the Dirar equation. As before, we first consider the case of a single particle at a distance ' a ' from a centre of rotation. The Dirac equation

c (a,pz + a~~ + a3pz + c~4mc ) # = W~; aia$+ aj~ --- 28i$ (9)

The Rigid Rotator according to Relativistic Quantum Mechanics 401

with the representation

( '~ _ -,~ 1 ( i �9

": ) t ) 1 ' a2 = ,

1 i

(" 1"~--1 ;' 1 ") ":t' ) ' ~ - '

fea& to the set of differential equations (see, for example, Persico, Funda-

mentals o f Quantum Mechanics, p. 430)

•162 ~ - i ~ ~ ,+~ ~ o

i (mc 2 - -W)~b~q- ( ~ i ~ ) ~- ~+ ~ r162

~-? ( - mc~ - W) es + ~ - i r + ~ r = 0

, (~ ~) ~-~ (-- mc 9 -- W) r + ~ + i ~b 1 -- ~-z ~b~ = 0. (10)

In analogy with the solution of the hydrogen problem, but with r = a, we attempt to solve the equations (10) by taking each of the four Cp ~ = 1, 2, 3, 4) in the forro

4 4,r (l + m) v (l -- m) v Y~m; ap const. ~p = apZL ~ = ap 2 ~ . .

where Y~m is a surface harmonic. Taking r = a, one can ve¡ the following relations for Z~m:

(~+i~)~, l (I - - ra) (1 - - m - - 1) (I + 1) Z~.m+ ,

- - (21 + 1) a Z•+lm+l - - (21 + 1) a

402 K . N . Sml, aVASA RAO AND D. SAROJA

(~-,;)z,. 1 Z . m-x ( l + m ) ( l + m - - 1 ) ( l + 1)

-- (2l + 1) a ~-1 + (21 + 1) a Z l - l m - I

~z Zz=

1 (21 + 1) a

I f now we take

Zz+l m + (l + m) (l - - m) (1 + 1) Zt_I m (2l + 1) a (11)

~1 = ia lZ l+lm ; ~z = dalZl+l m+l

$s = (l + m + 1) a~Zzm; $4 = - - ( l - - m) asZzm+l;

substitution in (10) eliminates the spherical harmonics entirely; two of equations (10) yield

W - m c 2 1 ~e a l - - a a~ = 0

al, a~ # 0

(12)

the first

(13)

and the second two equations yield

l + 2 a 1 _ W + mc ~ ~ ~ - a2 = 9. (13')

In order that the system of equations (13) should have a non-trivial solution for al and a2, the detcrminant of the system should be zero, i .e. ,

W ~ _ r ~~c 9 _ l ( l + 2 ) ~ ~ g a ~ - - I (14)

or, we have for the eigenvalues of W

W~ = g 4 1 + 1 (l + 2 ) / f i r ; l = 0 , 1 , 2 , . . . (15)

On the other hand, taking

~b x = i (1 + m) blZ/_l m ; ~b 2 = -- i (l -- m -- 1) blZl_l ra+l

~b3 ----- b~Z~ m ; ~b4 = b#Z~ m+l ; bx, b~ :# 0

givr another solution of the equations (10) which results in

(16)

The Rigid Rota tor according to Relat iv&tic Quantum Mechanics 403

W z = $ 4 1 + ( l " 1) - ( l+ l ) ¡ a i e ; l = 1, 2, 3 . . . . (17)

whieh is the same as (15) with l - 1 written for l.

Recalling the ideas of angular momentum in Dirae theory, we see that We can introduce the inner quantum number J defined by

J = l + �89 for the solutions (12) and

= l - �89 for the solutions (16) and in terms of J, (15) and (17) both can be written in the forro

Wj = � 9 1 + [j (j _1._ 1) _ t] ~12 1 3 I# ; J = 2 ' ] ' "'" (18)

As is well known, the non-relativistie description of this situation is that the spin of the rotating particle is either parallel of antiparaUel to the orbital angular momentum.

Before discussing the formula (15), we prove its validity as in the Klein- Gordon case for a system of two non-interacting particles of equal mass. One observes that taking the sum of two Dirac Hamiltonians as that of the system, leads in view of relations (6) to the uninteresting result that the total Hamiltonian refers merely to the centre of mass. We, therefore, start with the Klein-Gordon equation (8) in the forro

( W : W 4/izV 2 - 4 m Z c ~ ) r = 0

and seek the corresponding Dirac equation (19)

( W - - f l lpx - - f l~p, - - flspz - - fl4mc ) ~b -= O

which is equivalent to (8). The operators fli will obviously have to satisfy the commutation relations

fli~~ + fl~~~ = 88i~. (20)

Hence f l i= 2cti where ai are the Dirac operators of (9). Taking W' = W/2 (19) beeomes

(w, ) alpx __ a ~ p l ; - aap z - - a4mc ~b = 0

404 K . N . SRINIVASA RAO AND D. SAROJA

whieh gives, as in the case of a single particle [cf. Equation (14)],

W , ~ _ m~c4 = 1 ( l + 2) /l~c ~' a a (21)

With

W' W m a~ = --2 ; g = 2mc~; I = t~a 2 = -2

we again arrive at (15) for the eigenvalues of the energy.

We now proceed to discuss the formulae (3) and (15). We have already seen that the formula (3) leads to the well-known wavemechanical formula (4) in the first order approximation. Carrying out the approximation a stage further, we have

l" (l + 1)~ h~cg"B " W t = g + l ( l + l ) hcB - - - - (22) 2g

where

h B = 8~r~ic

is the so-called "rotational eonstant" ( see Herzberg, Spe r o f D i a t o m i c

Mo lecu l e s , p. 71). Introducing the wave number

Wt+l -- Wz vl = he

we obtain (in conformity with the selection rule Al = + 1), for the transition l + 1 - - ~ 1

2hcB z v l = 2 ( l + l ) B g ( l + 1 ) 3 ; 1 = 0 , 1 , 2 , . . . (23)

The seeond terna of (23) is very much smaller than 4D (l + 1)8 ( see Herzberg) which would arise if one assumed that the rotator is non-rigid, and may not be detectable. However, in the empirical formula

v = f m - - g m a,

we must therefore have

2hcB" f = 2 B ; g = 4 D + ~ ; m = l + l . (24)

The Rigid Rotator according to Relativistic Quantum Mechanics 405

The constant g is extremely small in comparison with f and neglecting the second term in (24) would lead to a spectrum consisting of nearly equidistant lines. Employing the more exact formula (23) for the "Rigid ro ta tor" (D ----- 0) we obtain, for the line separation,

vl+x-- v i = 2B 6hcB ~.

g - - [ ( 1 + 1 ) ( 1 + 2 ) + 3 ] ; l = 0 , 1 , 2 , . . .

(25)

We next consider the formula (15) or equivalently (18) in terms of the quantu/n number J = l 4- �89 which obviously takes into account the " s p i n " of the rotating particle. To the second order of approximation, (18) leads to

W j $ -}- [J (J -~- 1) -- �88 hcB - - [J (J q- 1) - - �88 ----- 2g

from which we have

v j = 2 ( J + 1)B

J ~--- �89 ~, . . . (26)

hcB ~ 2r (J + 1) (4J~ + 8J + 1); J = �89 t , . . .

(27)

Neglecting the second term, for the moment, we immediately find that the ¡ rotational line v lies at 3B and not at 2B which is the case according to the non-relativistic or even the Klein-Gordon equation. This circumstance would show whether the atoms of a homonuclear diatomic molecule should be considered as Dirac particles or Klein-Gordon particles. To this order of approximation the line separation is 2B in this case also. On the other hand the formula analogous to (25) is

vj+l __ vj _~_ 2B 6hcB' [ 1 ] a ( J + 1 ) ( J q - 2 ) + ~ ; J

We wish to express our gratitude to the Council Industrial Research for the award of a Fellowship and to Dr. Syed Ziauddin for kind interest.

= � 8 9 t , . . .

(2S)

of Scienti¡ and to o n e o f us (D.S.)

REFERENCES

1. Dirae, P. A. M.

2. Datwin, C. G. _ Proc. Roy. Soc., 1928, 117 A, 610. . , Ibid., 1928, 118, 654.

406 K . N . SRINIVASA RAO A~~) D. SAROJA

3. Gordon, W.

4. Sauter, F.

5. Nikolsky, K.

6. Plesset, M. S. 7. Enric, o Perisco

8. Herzberg, G.

. . Zei ts . F. Physik. , 1928, 48, 11.

.. /b/al., 1931, 69, 742.

. . Ibid., 1930, 62, 677.

. . Phys. Rey. , 1932, 41 (2), 278.

. . Fundamentals o f Quantum Meclumies.

. . Spectra o f Diatomic Molecules.

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