The Thomas-Reiche-Kuhn Sum Rule and the Rigid Rotator

Transcription

1 Fairfield Uiversity Egieerig Faculty Publicatios School of Egieerig The Thomas-Reiche-Kuh Sum Rule ad the Rigid Rotator Evagelos Hadjimichael Fairfield Uiversity, W. Currie S. Fallieros Peer Reviewed Repository Citatio Hadjimichael, Evagelos; Currie, W.; ad Fallieros, S., "The Thomas-Reiche-Kuh Sum Rule ad the Rigid Rotator" (1997). Egieerig Faculty Publicatios Published Citatio Hadjimichael, E.; Currie, W.; ad Fallieros, S The Thomas-Reiche-Kuh Sum Rule ad the Rigid Rotator. Am. J. Phys. 65, 4: This Article is brought to you for free ad ope access by the School of Egieerig at It has bee accepted for iclusio i Egieerig Faculty Publicatios by a authorized admiistrator of DigitalCommos@Fairfield. For more iformatio, please cotact digitalcommos@fairfield.edu.

2 The Thomas Reiche Kuh sum rule ad the rigid rotator E. Hadjimichael Departmet of Physics, Fairfield Uiversity, Fairfield, Coecticut William Currie a) ad S. Fallieros Departmet of Physics, Brow Uiversity, Providece, Rhode Islad Received 7 November 1995; accepted 25 September 1996 It is show that the Thomas Reiche Kuh sum rule, associated with the photoabsorptio cross sectio from quatum systems, appears to be violated i the case of the quatized rigid rotator. The origi of the apparet violatio is ivestigated, ad its resolutio is preseted o the basis of a related system, i.e., a particle i a spherical -fuctio potetial whose eergy spectrum approaches that of the rigid rotator whe the stregth of the potetial becomes large America Associatio of Physics Teachers. I. INTRODUCTION It is well kow that for relatively low eergies, the iteractio of electromagetic radiatio with a microscopic system is domiated by electric dipole (E1) trasitios. The matrix elemet for the E1 amplitude, E1 f D ˆ i, I1 is expressed i terms of the dipole operator for the charged system, i.e., Dˆ j e j r r r j, I2 where ˆ is the polarizatio of the exteral electric field i a arbitrary directio, ad e j is the electric charge of the jth particle. There is a well kow sum rule that is associated with the E1 amplitude, the Thomas Reiche Kuh sum rule, 1 which we proceed to derive. Startig from the fudametal commutatio relatio i ˆ * r,ˆ p 1, I3 which is itimately related to the ucertaity priciple, ad the idetity Ĥ,ˆ r i m ˆ p, I4 where Ĥ is the Hamiltoia of the system uder cosideratio, assumed to cotai o velocity-depedet forces, we obtai m 0 2 ˆ * rˆ,ĥ,ˆ rˆ 0 1, I5 where 0 represets the groud state of the system, ad ˆ * ˆ 1. We ote that for the case of liear polarizatio ˆ is a real uit vector, i.e., ˆ *, ad Eq. I5 ca be expressed i Cartesia coordiates; for circular polarizatio, ˆ *ˆ, ad Eq. I5 would cotai spherical compoets of the positio operator rˆ, i.e., those proportioal to ry l,m (,). By expadig the commutator i Eq. I5 ad usig the closure relatio for the complete set of eigestates of the Hermitia operator Ĥ, i.e., *r r r r, I6 we fially obtai the result f 0 2m 2 E E 0 ˆ r 0 2 1, I7 335 Am. J. Phys. 65 4, April America Associatio of Physics Teachers 335

3 which is the celebrated Thomas Reiche Kuh TRK sum rule. 1 The quatities f 0 are the well-kow oscillator stregths, so called because of their appearace i the amplitude for scatterig of light from a harmoic oscillator 2 i the log wavelegth electric dipole approximatio. I Eq. I7 we have restricted ourselves to the case of a sigle particle of mass m boud i a potetial. For a two-body system, r would be the separatio distace betwee the two particles ad m would be replaced by the reduced mass ; for a maybody system, r would be replaced by a sum over particles. As the derivatio of Eq. I7 idicates, the TRK sum rule is a direct cosequece of the fudametal laws of quatum mechaics. I additio, it implies some importat results cocerig directly measurable quatities, such as the scatterig amplitude ad the cross sectio for photo absorptio by a boud system. Before proceedig further we preset here examples of such results. It ca be readily show 2 that the itegrated cross sectio for E1 photo absorptio is E1 abs EdE2 2 e 2 0 mc f e 2 mc, I8 where the TRK sum rule was used i the last step. E1 abs (E) is the cross sectio for absorptio of photos of eergy E by a boud particle of charge e i the electric dipole (E1) approximatio. Equatio I8 shows that the itegrated cross sectio has a fiite upper boud idepedet of the ature of the bidig forces. 2 4 Furthermore, we fid that the amplitude for forward scatterig of photos of eergy E from a boud system, i the E1 approximatio, is e 2 f E1 EE 2 c 2 ˆ r 0 2 2E 0 EE 0 2 i, I9 with E 0 E E 0 ad 0. It has a high eergy limit f E1 E e2 E mc 2 f 0 e2 mc 2 I10 Agai, the TRK sum rule was applied i the last step. Equatio I10 idicates that at high eergies, this scatterig amplitude becomes idetical to the Thomso amplitude describig the scatterig of photos from a free particle with the same charge ad mass. 5 Equatios I8 ad I10 appear i may textbooks, yet a commet is i order at this poit with regard to the compatibility betwee the log wavelegth approximatio which implies a low eergy regime, ad the high eergy limit employed i these equatios. It is importat to remember that both equatios are orelativistic. I this cotext, high eergy implies that the eergy of the photo is large compared to typical excitatio or ioizatio eergies, but still small compared to the mass of the boud particle. A atomic system is typical of this situatio where the high-eergy limit is still compatible with the orelativistic approximatio ad the domiace of the E1 trasitio. I the followig we examie the saturatio of the TRK sum rule, Eq. I7, ad idetify the eergy regio which cotais most of the sigificat oscillator stregth. Thus, i Sec. II we preset three typical examples which illustrate the saturatio of the sum rule. I Sec. III, we cosider the case of the rigid rotator which exhibits a behavior that appears at first to be idiosycratic. I Sec. IV, we achieve a uderstadig of this behavior by cosiderig the rigid rotator as the limit of a regular three-dimesioal system composed of a particle boud i a spherical -shell potetial, thus avoidig the costrait of perfect rigidity. The stregth of the -shell potetial that leads to the appearace of a eergy spectrum approachig that of the rigid rotator is ivestigated. We discuss briefly some additioal sum rules i Sec. V, ad we preset cocludig remarks i Sec. VI. II. THREE EXAMPLES 1. The three-dimesioal harmoic oscillator The Schrödiger equatio for this system is 2 p 2 2m 1 2 m 0 2 r E, ad yields eergy eigevalues E ( 3 2) 0, so that (E E 0 ) 0. Furthermore, it is readily show 2 that the electric dipole matrix elemet is, i this case, ˆ r 0 2m 0,1 so that f 10 1, as see from Eq. I7, ad all the other oscillator stregths are equal to zero. The sum rule is saturated by exactly oe sigle trasitio to the first excited state of the harmoic oscillator. 2. The ifiite spherical well For a particle of mass m trapped i a spherical, ifiitely deep, potetial well of radius R, i.e., Vr 0, 0rR,, rr there is a ifiite umber of boud states. The calculatio of the oscillator stregths is straightforward ad yields: 6 f , f , f , f , f , with all the other oscillator stregths beig egligibly small. 5 The above umbers add up to 1 f 0 1. That is, the TRK sum rule is saturated by excitatios to the first five states ad is domiated by the cotributio of the first excited state. Note that the idices 1, 2, 3, 4, 5 i f 0 represet excitatios of oe agular mometum state, the l1 state, as is clear from the ature of the electric dipole operator i Eq. I7. 3. The hydroge atom. It is show i Ref. 6 that i the case of a electro proto system, the sum of the oscillator stregths from trasitios to boud exited states amouts to f , boud ad the equivalet sum from trasitios to uboud states f , uboud so that the TRK sum rule is ideed f 0 1, ad is thus rigorously satisfied. The cotributio from the uboud 336 Am. J. Phys., Vol. 65, No. 4, April 1997 Hadjimichael, Currie, ad Fallieros 336

4 states i the cotiuum is rather well cocetrated at eergies ot far from the ioizatio threshold. III. THE RIGID ROTATOR The rigid rotator is evisioed as two particles of mass m 1 ad m 2, rigidly separated at a fixed distace rr. Hece, this system is ofte used as a simplified model describig the rotatioal excitatios of, say, a diatomic molecule. The reduced mass of the system is m 1 m 2 /(m 1 m 2 ), ad its momet of iertia is IR 2. The simplified Hamiltoia describig the system is H R L 2 /2R 2 where L 2 is the square of the agular mometum operator which is a purely agle-depedet operator. The eigevalues of the system are the familiar oes E l ll12 2R 2, III1 where l0, 1, 2,..., while the eigefuctios are the spherical harmoics Y l,m (,). There is oly oe eergy state for each value of the quatum umber l, ad the oly degeeracy is associated with the azimuthal quatum umber ml, l1, l2...l. Takig the polarizatio ˆ alog the z-axis, we fid the dipole matrix elemet to be Y l,m ˆ r Y 0,0 Y l,m R cos Y 0,0 R 3 l,1 m,0. III2 Evaluatig the TRK sum rule, i.e., the left-had side of Eq. I7, by meas of Eqs. III1 ad III2 we fid, f 0 f R 2 R III3 i apparet violatio of the TRK sum rule, Eq. I7. Give the fudametal ature of this sum rule ad its direct relatio to physical processes as illustrated i Sec. I, the missig 1/3 i the result of Eq. III3 deserves further discussio. To begi to uderstad the result III3, we review the derivatio of Eq. I7 usig the geeral Hamiltoia for a particle of mass m, H P 2 r 2m L 2 2mr 2 Vr, III4 where P r (rˆ p p rˆ)/2 is the radial mometum of the particle i a spherically symmetric potetial V(r), ad L 2 is the square of the agular mometum; rˆr /r is the uit vector. Equatio III4 shows explicitly the two kietic eergy terms, oe associated with the radial motio ad the other with rotatioal motio. A simple calculatio shows that 1/3 of the TRK sum rule origiates i the P r 2 term, while the remaiig 2/3 is obtaied from the cetrifugal term L 2 eve though the value of r 2 i the deomiator of this term is ot ecessarily costat. I compariso to Eq. III4, the Hamiltoia for the rigid rotator does ot have a term proportioal to the radial mometum; hece the missig 1/3 i the TRK sum rule, Eq. III3. Uderstadig the mechaics which leads to the result III3, however, does ot explai the physical implicatios of the uaccouted 1/3 of the sum rule i the case of the rigid rotator. This is a importat issue i view of the itimate relatio of this sum rule to physical processes. It is ivestigated i the followig. A first glimpse ito the questio at had is made possible by recallig the differece betwee a classical rotator ad a quatal rotator. I the former case, a rigidly fixed value of R implies that there is o radial mometum ad the P r 2 term disappears. I the quatal case, a costat value for R meas that the ucertaity i R is much smaller tha R, ad hece the ucertaity i the radial mometum, or the expectatio value of P r 2, becomes very large; ideed, it teds to ifiity i the limit of perfect rigidity. I cosequece, radial excitatios occur at very high eergies i the quatal case, approachig ifiitely high eergies i the limit of perfect rigidity ad zero ucertaity i R. We deduce from this ad the earlier discussio that the apparetly missig 1/3 i the TRK sum rule for the rigid rotator is actually to be foud at ifiitely high eergies. At this poit we eed a specific example which illustrates these cosideratios. We preset such a example i the followig sectio. IV. THE SPHERICAL -SHELL POTENTIAL Cosider a particle of mass i a -shell potetial V(r) g(rr). This potetial has at most oe boud state for each agular mometum value l ad a cotiuous spectrum above the ioizatio threshold. We shall show umerically that as g, the stregth of the potetial, icreases, the boud-state eergy spectrum will resemble that of a rigid rotator, ad that 1/3 of the stregth i the TRK sum rule will be located at high eergies i the cotiuum, i.e., above the dissociatio threshold. For the perfectly rigid rotator, this threshold is obviously at ifiite eergy, ad hece the missig sum rule stregth will be located at ifiity. The Schrödiger equatio for the radial wave fuctio R El u l (r)/r i the case of the particle i a -shell potetial takes the form, d 2 u l r dr 2 ll1 r 2 u l r 2g 2 rru l r 2 2 E l u l r. IV1 We shall ext obtai solutios for this equatio for both boud states ad states i the cotiuum. A. Boud states We write the eergy eigevalue E l i terms of the wave umber l, E l E l l IV2 ad defie a quatity 2g/ 2 ; the radial equatio becomes u l r ll1 r 2 u l rrru l r 2 l u l r. IV3 For 0rR ad for rr, this equatio takes the form u l r ll1 r 2 u l r l 2 u l r. IV4 337 Am. J. Phys., Vol. 65, No. 4, April 1997 Hadjimichael, Currie, ad Fallieros 337

5 This is the Bessel equatio with solutios u l r r C lj l i l r, 0rR A l j l i l rb l l i l ra l h 1 l i l r, rr, IV5 Itegratig Eq. IV4 over the rage 0r, we obtai a discotiuity i u l (r), the first derivative of u l (r), at rr; that is u l R u l R u l R IV6 where j l, l, ad h (1) l j l i l are the spherical Bessel, Neuma ad Hakel first kid fuctios, respectively. For the case 0rR we take ito accout the fact that the solutio must be regular at r0. with R R, 0. Dividig the above equatio by u l (R), ad usig the cotiuity of this fuctio at rr, ad Eq. IV5, we fid the result A l rh l 1 i l rrj l i l ra l rh l 1 i l rrj l i l r A l rh l 1 i l r. rr. IV7 The umerator o the left-had side of this equatio is the Wroskia of the fuctios rh l (1) (i l r) ad rj l (i l r), ad ca be readily show to be a costat idepedet of the value of r. Evaluatig it at r0, ad substitutig ito Eq. IV7 we fid l R R lr 2 h 1 l i l Rj l i l R. IV8 We will solve this equatio for the dimesioless product l R which defies the boud state eergy E l, Eq. IV2, i terms of the dimesioless variable yr. 1. Normalizatio ad eergy eigevalues The costats C l ad A l i the wave fuctio, Eq. IV5 are determied by meas of the wave fuctio ormalizatio coditio ad the cotiuity coditio at rr. We fid the result C l c l 3/2 l, IV9 A l a l 3/2 l. The values of c l ad a l for the agular mometum states l0 ad l1, ad for a rage of values of yr, idicative of potetial stregth, are show i Table I. Before we solve Eq. IV8, it is useful to use it to derive the coditio that a agular mometum state of quatum umber l be boud, for a give value of R. Clearly, this coditio is satisfied for l R 0. Employig the aalytic expressios for spherical Bessel fuctios ear the origi, we fid that states l are boud provided 2l1R. IV10 Table I. Normalizatio costats, Eqs. IV5 ad IV9, for the boud states l0 ad l1, for a rage of values of R. There is oly oe boud state for every value of l. With Eq. IV10 i mid, we solve Eq. IV8 umerically for a rage of values of R. We show the results for l R,0l4, i Table II. 2. Compariso of the relative distributio of eergy states ad of root-mea-square radii of the -fuctio potetial ad the rigid rotator Writig the eergy E l of the rigid rotator i terms of a wave umber l, ad usig Eq. III1, wefid 1 l1 R 2 l R 2 1 for the rigid rotator. 2 l1 IV11 This gives a measure of the eergy gap betwee successive agular mometum states i the case of the rigid rotator. At the same time, usig the results for the -fuctio potetial i Table II, we evaluate the left-had side of Eq. IV11 for this system also, ad we fid that as R grows larger, its value approaches 1, i.e., the value i Eq. IV11, rather quickly. Ideed, the two eergy distributios, for the rigid rotator ad for the -fuctio potetial, become essetially idetical whe R25. We show the results for the -shell potetial i Table III. Additioal iformatio regardig how the -fuctio potetial system approaches the rigid rotator is give by the values of the rms radius for the groud state i the former case, compared to the radius R of the rigid rotator. Usig the groud state wave fuctio, Eqs. IV5 ad IV9 ad Table I, for l0, we evaluate both the rms radius r 2, ad the Table II. Solutios l R of Eq. IV5 for the allowed boud states of agular mometum l, for a rage of values of R. l R R c 0 a 0 c 1 a 1 R l0 l1 l2 l3 l Am. J. Phys., Vol. 65, No. 4, April 1997 Hadjimichael, Currie, ad Fallieros 338

6 Table III. Distributio of boud state eergies for the -fuctio potetial similar to Eq. IV11 for the rigid rotator. R 1 l1 R 2 l R 2 2 l1 l0 l1 l2 l quatity r 2 R 2. The latter reflects the squeezig of the wave fuctio ito a progressively arrower regio as the potetial becomes deeper. We display these results i Table IV. It is oted agai that for R25, the rms radius of the -fuctio potetial is essetially equal to the radius R of the rigid rotator. 3. Cotributio to the TRK sum rule from the boud states of the -fuctio potetial Usig the wave fuctios, Eq. IV5, with the ormalizatios give by Eq. IV9 ad i Table I, we ow evaluate the cotributio to the TRK sum rule from boud-state excitatios, Eq. I7, i the case of the -shell potetial. The results are give i Table V for a rage of values of the potetial stregth R ad are compared to the result for the rigid rotator which was foud earlier to be 2/3, Eq. III3. We reiterate the fact that the results i Tables III ad IV show that the behavior of the -shell potetial system takes exactly the character of the quatized rigid rotator, i the eighborhood of R25. This is ow stregtheed by the results i Table V which show that at that poit, i.e., for R25, the cotributio to the TRK sum rule from the boud states of the -shell potetial is precisely the same as that for the rigid rotator, i.e., boud states f 0 2/3. Clearly, the ature of the TRK sum rule implies that there must be further excitatios which will provide the additioal 1/3 required to saturate the sum rule. This must come from excitatios to the cotiuum. While the rigid rotator has cotiuum states oly at ifiite eergy, the -shell potetial has calculable cotiuum states at fiite eergy. We proceed, i the ext subsectio, to verify umerically that the missig stregth i the TRK sum rule comes ideed from the cotiuum states. Table IV. The rms radius for the groud state, ad the cofiemet of the particle trapped i the -fuctio potetial. R 0 R r 2 R r 2 R R Table V. Cotributio to the TRK sum rule from boud states. R boud states f 0, -fuctio potetial Rigid rotator /3 2/3 B. Cotiuum states The radial Eq. IV1, with positive eergy eigevalues E l 2 k 2 l /2, has solutios r 2 D lj l k l r, 0rR r 1 2 2, e2i lh 1 l k l rh 2 l k l r, rr IV12 where h (2) l j l (k l r)i l (k l r) is the Hakel fuctio of the secod kid. These wave fuctios satisfy the required completeess relatio ad ru smoothly ito the free particle wave fuctios i the limit 0. For the evaluatio of the sum rule, Eq. I7, we eed oly the l1 wave fuctios, ad so the rest of the discussio focuses o these states oly. The amplitude D 1 ad the phase shift 1 i Eq. IV12 are foud from the cotiuity of the radial wave fuctio at rr, ad from a equatio equivalet to IV6 for the radial fuctio (r). The results are as follows: D e 1 2i 1i 1k 1 j 1 k 1 R 1i 1k 1 R j 1 k 1 R, IV13 ad k 1 R 2 j 1 k 1 R 2 ta 1 1k 1 R 2 j 1 k 1 R 1 k 1 R. I the evaluatio of Eq. I7, the sum is replaced by a itegral over the mometum k 1 of the l1 state. As the stregth of the potetial, R, ad the argumet k 1 R of the Bessel fuctios appearig i this itegral become very large, the umerical accuracy of the outcome suffers. We show the fial results i Fig. 1, where we display the cotributio to the sum rule from the boud states show also i Table V, ad that from the states i the cotiuum, ad the total cotributio which, as expected, is complete; i.e., 1 boud states cotiuum IV14 whe the two cotributios are added. We stopped the umerical evaluatio of the cotributio from excitatios to the cotiuum at about R12, because umerical istabilities sap the reliability of results at higher values of R. But the poit is adequately made, that the TRK sum rule is satisfied i the case of the spherical -shell potetial provided the cotributio from the cotiuum states is take ito accout. 339 Am. J. Phys., Vol. 65, No. 4, April 1997 Hadjimichael, Currie, ad Fallieros 339

7 for 1. 2 Hece, for high values of oly half the stregth of the sum rule, Eq. V2, is cotaied i the spectrum of the rigid rotator. Equally iterestig is the case of the moopole sum rule where we show that the etire sum rule stregth is missig. The moopole operator is r 2, ad the correspodig sum rule takes the form E E 0 r m 0r 20. V3 Fig. 1. The fractio of the TRK sum rule cotributed by boud states lie 1 ad cotiuum states lie 2 i the case of the -shell potetial, plotted versus the dimesioless variable yr which is a measure of the stregth of the potetial. At ay poit, the two cotributios add up to 1. The straight lie shows the icomplete saturatio, i.e., 2/30.667, of the TRK sum rule i the case of the rigid rotator. For large values of the potetial stregth, i.e., y25, the cotributio from the boud states of the -shell potetial becomes equal to that of the rigid rotator. V. ADDITIONAL SUM RULES It is worth extedig the cosideratios of the previous sectios to sum rules ivolvig multipoles higher tha the dipole. 7 We defie a operator Q r Y,,, V1 where Y, (,) with 1 are the familiar spherical harmoics. Followig a procedure similar to the oe described earlier for the dipole sum rule, usig the fudametal commutatio relatios betwee positio ad mometum variables ad the completeess relatio, we fid the followig sum rule i the case of a spherically symmetric groud state, 0: E E 0 Q r m 4 V2 It is iterestig to examie this sum rule also for the rigid rotator of legth R. The eergy E is give by Eq. III1, ad the matrix elemet o the left-had side of Eq. V2 is l,mq 0 l, m, R l /4, where we have substituted l,m for. Hece the left-had side of Eq. V2 is 2 /2m)/((1)/4)R 22. Furthermore, with 0R 22 0R 22, the right-had side of V2 takes the form 2 /2m)((21)/4)R 22. Comparig these two results, we fid that the fractio of the sum rule limit, Eq. V2, exhausted by the boud states of the rigid rotator is 1/21. Obviously, for 1, this correspods to the previously foud result of 2/3 for the dipole case, Eq. III3. For quadrupole trasitios, 2, this fractio is 3/5, ad, i geeral, For the rigid rotator, the right-had side of Eq. V3 is 2 2 /mr 2, while the left-had side is zero for the groud state, 0, because of Eq. III1, ad remais zero for 0 due to agular mometum coservatio. Hece, the cotributio to the moopole sum rule from all the rotatioal states of the rigid rotator is zero, ad the etire stregth of the sum rule is to be foud at ifiite eergies. This is a strikig result, but hardly surprisig. The excitatios iduced by the moopole operator r 2 do ot ivolve chages of shape or orietatio, but chages of size oly. These, however, require ifiite eergy due to the rigidity of the rotator. VI. CONCLUSION We showed through a direct calculatio that the TRK sum rule for dipole excitatios by photo absorptio is apparetly ot saturated i the case of the rigid rotator. All the eergy states of the rigid rotator are boud ad their cotributio to the TRK sum rule, i.e., the left-had side of Eq. I7, is oly 2/3 of the expected value. At the same time, we oticed that i a spherically symmetric Hamiltoia, the cetrifugal term i the kietic eergy cotributes 2/3 of the sum rule value, while the radial mometum term P r 2 /2m cotributes the additioal 1/3. I the case of a perfectly rigid rotator, the radial mometum term is missig ad, cosequetly, 1/3 of the sum rule should also be missig. Hece, the result of the direct calculatio was cofirmed. I the quatum regime, however, the radial mometum term P r 2 /2m i the rotator Hamiltoia must be very large, istead of beig zero, due to the ucertaity priciple. We deduce that the apparetly missig 1/3 stregth of the TRK sum rule, istead of beig truly abset, will be foud i the very high eergy regio of the spectrum. Ideed, we illustrated this fact by examiig a threedimesioal system uder the actio of a -shell potetial. The rigid rotator is the limitig case of this system whe the potetial becomes very strog. As expected, we foud that as the rigidity of the potetial was icreased, e.g., at R25 see Tables III V, the behavior of the system approached that of the rigid rotator, ad the TRK sum rule was satisfied by cotributios from boud ad cotiuum states, providig 2/3 ad 1/3 of the sum rule, respectively, as illustrated i Fig. 1. We coclude that i the case of the rigid rotator as well, the balace of 1/3 is cotributed to the TRK sum by states i the cotiuum. The differece is that for the rigid rotator these states are at ifiite eergy. ACKNOWLEDGMENT EH was supported i part by NSF Grat No. PHY Am. J. Phys., Vol. 65, No. 4, April 1997 Hadjimichael, Currie, ad Fallieros 340

8 a Preset address: Echo Systems Ceter, Marie Biological Laboratory, 7 MBL Street, Woods Hole, MA W. Thomas, Über die Zahl der Dispersioelektroe die eiem statioäre Zustad Zugeordet sid, Naturwisseschafte 13, ; F. Reiche ad W. Thomas, Über die Zahl der Dispersioelektroe die eiem statioäre Zustad Zugeordet sid, Z. Phys. 34, ; W. Kuh, Über die Gesamtstärke der vo eiem Zustade ausgehede Absorptiosliie, ibid. 33, E. Merzbacher, Quatum Mechaics Wiley, New York, 1961, p H. A. Bethe ad R. W. Jackiw, Itermediate Quatum Mechaics Bejami, New York, 1968, 2d ed., p G. Baym, Lectures i Quatum Mechaics Bejami, Reddig, 1976, p L. D. Ladau ad E. M. Lifshitz, Relativistic Quatum Theory Pergamo, Oxford, 1961 Eq , p W. Currie, The Thomas-Reiche-Kuh Sum Rule ad Distributio of the Quatal Oscillator Stregths, seior thesis, 1983; some of the material discussed here was preseted i this thesis which was submitted i partial fulfillmet of the Sc.B. degree requiremet at Brow Uiversity. 7 T. J. Deal ad S. Fallieros, Models ad Sum Rules for Nuclear Trasitio Desities, Phys. Rev. C 7, ; S. Fallieros, i Nuclear Structure Studies Usig Electro Scatterig ad Photoreactios, edited by A. Ui Laboratory of Nuclear Sciece, Tohoku Uiversity, 1972, Vol. 5, pp Am. J. Phys., Vol. 65, No. 4, April 1997 Hadjimichael, Currie, ad Fallieros 341