Chapter. Quantum Mechancs Notes: Most of the matera presented n ths chapter s taken from Cohen-Tannoudj, Du, and Laoë, Chap. 3, and from Bunker and Jensen 5), Chap... The Postuates of Quantum Mechancs.. Frst Postuate At a gven tme t, the physca state of a system s descrbed by a ket t) usng Drac s notaton). From ths ket a wave functon dependent on poston and tme can be defned by the projecton onto a bass defned by the bra r. That s the wave functon s gven by r,t) r t)..) The symbo s usuay caed a bracket. Equaton.) s the resut of the foowng two defntons. Frst, the bracket s by defnton a scaar product x) Second, to the ket r s assocated a Drac dstrbuton such that d 3 x * x..) r x r),.3) r t = d 3 x * x r x,t = d 3 x x r) x,t) = r,t)..4) Note that the orthogonaty of the r kets s apparent from r r = d 3 x x r ) x r = r r d 3 x x r ) = r r)..5)
.. Second Postuate For every measurabe physca quantty A corresponds an operator Â, and ths operator s an observabe. It s often the case that a representaton of kets and operators s done through vectors and matrces, respectvey. The acton of the operator on the ket produces a new ket = Â,.6) and ths acton s the mathematca equvaent of the mutpcaton of a vector and a matrx...3 Thrd Postuate The outcome of the measurement of a physca quantty A must be an egenvaue of the correspondng observabe Â. Snce observabes are reated to physca quanttes, then the matrx assocated wth them must be Hermtan. Ths s because the egenvaues of Hermtan matrces are rea quanttes n a mathematca sense). Reca that a matrx s Hermtan when  j = Âj *,.7) where * stands for compex conjugaton. Aternatvey, a Hermtan operator s one that s sef-adjont. That s,  = Â..8) When the matrx s of fnte dmenson, then the egenvaues are quantzed a matrx of nfnte dmenson woud correspond to a contnuum; for exampe, a matrx actng on r woud have to be of nfnte dmenson as r encompasses the contnuum made of a possbe postons)...4 Fourth Postuate The ket, say t), specfyng the state of a system s assumed normazed to unty. That s, t) t) =..9) Aternatvey, the assocated wave functon s aso normazed, snce t) t) = d 3 x * x,t x,t) = d 3 x x,t) =..)
Ths ket can aso be expanded usng any sutabe compete) bass of kets. For exampe, usng the r bass we have where c r,t = d 3 r c r,t) r t,.) s the coeffcent t s actuay the wave functon tsef, see equatons.) on r. Equaton.) can therefore and.4)) resutng from the projecton of t be wrtten as = d 3 r r t) t Rearrangng ths ast equaton we have whch mpes that t r..) = d 3 r r r t),.3) d 3 r r r = ˆ,.4) where ˆ s the unt operator or matrx). Smary any other normazed ket that can be expanded wth a compete) dscrete and orthonorma bass u.e., u u j = j ) wth = u u = c u,.5) s aso normazed to unty, and we have the foowng reaton for the bass u u = ˆ..6) Equaton.6) as we as equaton.4)) s a competeness reaton that must be satsfed for the correspondng bass to be compete. In consderaton of these facts and defntons, we can state the fourth postuate of quantum mechancs as foows: In measurng the physca quantty A on a system n the state, the probabty of obtanng the possby degenerate) egenvaue a of the correspondng observabe  s 3
P a = u n,.7) for dscrete states, wth the degree of degeneracy of a, and = { u n } the set of degenerate egenvectors. For contnuum states the correspondng probabty s gven by dp a) = v a da,.8) where v a s the egenvector assocated wth the egenvaue a. We can appy ths postuate to the case of a dscrete degenerate state by startng wth a generazaton of equaton.5) = = c n u n..9) n Projectng ths state on the set of states u m j takng the square of the norm we get of degeneracy g m.e., j =,,, g m ), and g m = c n j = u m j g m j = g m j = n = = j u m u n = c n j mn n g m j = c m. j =.) We, therefore, see that the probabty of fndng the system n a state or group of states) j possessng a gven egenvaue s proportona to the square of the coeffcent c m that appears n the expanson defnng the state n term of the bass under consderaton as s evdent from equaton.9)). Aternatvey, we defne the projector ˆP n ˆP n u n u n.) = 4
as the operator that projects a gven state, say, on the subspace contanng the set of egenvectors on we fnd { u n } that share the same egenvaue. For exampe, usng equaton.) ˆP n = u n u n = c n u n,.) = = and t s cear from a comparson wth equaton.9) that the ony part of that s eft s that correspondng to the subspace contanng the egenvectors contnuum of states, the projector s on the subspace of egenvaues specfed by a a a s { u n }. For a { v a } correspondng to the doman a ˆP a = v a v a da..3) a..5 Ffth Postuate If the measurement of the physca quantty A on a system n the state gave the vaue a as a resut, then state of the system mmedatey foowng the measurement s gven by the new state such that for dscrete states) = ˆP n ˆP n..4) Ths postuate smpy ensures that the new ket or wave functon) descrbng the system after a measurement s sutaby normazed to unty. Indeed, we can verfy from equatons.9) and.) that = u n u n = = u n u n = c n u n c n =,.5) and therefore 5
= = c n j = u n j c n = c n = j c n ) * c n = =. u n = j c n ) * c n j j = = c n =.6)..6 Sxth Postuate Athough t s possbe to gve a dervaton of Schrödnger s equaton, t s suffcent for our purposes to ntroduce t as the sxth, and ast, postuate of quantum mechancs: The tme evouton of the state vector t) of a system s dctated by the Schrödnger equaton d dt t) = Ĥ t) t),.7) where Ĥ t) s the Hamtonan of the system,.e., the observabe assocated wth the energy of the system. In most cases, we w be deang wth a tme-ndependent Hamtonan that has so-caed statonary states whose norms do not change as a functon of tme. More precsey, consder the foowng ket where E s the energy assocated wth the state. It s cear that t) = e Et,.8) t) t) =,.9) and f we nsert equaton.8) nto Schrödnger equaton.e.,.7)), then fnd that Ĥ = E.3) Therefore, the energy E s the egenvaue of the Hamtonan Ĥ for the statonary state. Equaton.3) s often referred to as the tme-ndependent Schrödnger equaton. 6
It s mportant to note that snce the observabes ˆR and ˆP assocated wth the poston and the momentum, respectvey, do not commute.e., ˆR ˆP ˆP ˆR ) and that they usuay appear n expressons for the Hamtonan, rues of symmetry must be used when deang wth ther products. For exampe, the foowng transformaton woud be apped for the smpe product ˆR ˆP ˆR ˆP ˆR ˆP + ˆP ˆR )..3). Important Operators and Commutaton Reatons Two measurabe physca quanttes that are ncuded n the expresson for the cassca Hamtonan are the poston r and the momentum p vectors. Correspondngy, t s necessary to ntroduce the aforementoned operators ˆR and ˆP when settng up the quantum mechanca Hamtonan. These two operators can be broken down nto the usua three components ˆR = ˆXe x + ˆ Ye y + Ẑe z ˆP = ˆP x e x + ˆP y e y + ˆP z e z..3) The compete bass and { r } contans the egenvectors for ˆR. More precsey, we can wrte r = x y z,.33) ˆX x = x x.34) or ˆX r = x r..35) Smar reatons hod for ˆ Y and Ẑ. We can aso defne a compete bass egenvectors for the momentum operator such that { p } of p = p x p y p z ˆP x p x = p x p x ˆP x p = p x p,.36) 7
{ r } or and so on. The queston s: what s the representaton for ˆP when actng on the that of ˆR for { p } )? To answer ths queston we frst note that the momentum bass satsfes reatons that are smar to that satsfed by the poston bass. That s, p ) p p p p p = p d 3 p p p = ˆ p = p)..37) We now combne the second of these reatons.e., the one concernng the orthogonaty of the momentum kets) and equaton.4) for the competeness of the r { } bass p p = d 3 r p r r p = d 3 r v * r;p)v r; p = p p ),.38) where v r;p) r p and s to be determned. Aternatvey, we can combne equaton.5) and the thrd of equatons.37) and get r r = d 3 p r p p r = d 3 p v r;p)v * r ;p = r r )..39) We can consder equatons.38) and.39) as the reatons that defne v r;p). One form that satsfy these condtons s v r;p) = ) 3 epr,.4) where s some constant havng the dmenson of anguar momentum, and we dentfy t wth Panck s constant dvded by ). [Note: The fact that equaton.4) satsfes both equatons.38) and.39), can be verfed by consderng the Fourer transform par between a functon f r transform f p and ts 8
f r) = f p) = ) d 3 p f p)e pr 3 ) d 3 r f r)e pr. 3.4) Equatons.4) mpy the foowng duaty between the Fourer transform and ts nverse f r) f p) f p f r..4) For exampe, f we set f r f p) = = r r, then t s easy to show that ) d 3 r r r )e pr = 3 Therefore, from the second of equatons.4) we have p p ) = whch s the same as equaton.38) when v r;p we can frst set f p = p + p ) to get f r) = ) 3 ep r..43) ) d 3 r ) 3 ) 3 er p + ' * + epr,,.44) ) d 3 p p + p )e pr = 3 and from the frst of equatons.4) we have = r r ) = d 3 r ) 3 r + r whch s the same as equaton.39).] Havng estabshed that s gven by equaton.4). Smary, ) 3 e p r,.45) ) ) 3 e r p + ' * + epr,,.46) r p = ) 3 epr,.47) 9
consder equaton.) whe usng the competeness reaton for equatons.37)) { p }.e., the thrd of r,t) = r t) = d 3 p r p p t) = d 3 p e pr ) p,t). 3.48) It s therefore apparent from ths ast equaton that the two dfferent forms of the wave functon are reated through the Fourer transform r,t) p,t)..49) Fnay, consder the acton of the momentum operator on the ket t) wth r ˆP t = d 3 p r ˆP p p t) = d 3 p p r p p,t) = ' = ) ) = r t d 3 p e pr p,t) 3. ) d 3 p e pr p p,t 3 *, +,.5) We therefore fnd the fundamenta resut that the acton of the momentum operator n the poston bass r { } s represented by ˆP.5) In quantum mechancs the order wth whch measurements are made can be mportant. For exampe, the act of measurng the momentum of a system can affect ts poston, and vce-versa. It s therefore nterestng to cacuate the dfference between two sets of measurements. Consder the foowng,.5) = ˆRˆP ˆP ˆR the ket resutng from the dfference between measurng the momentum before the poston on a system and measurng the poston before the momentum. To proceed further, we project both sdes of equaton.5) on r, and use equaton.5) to get
r = r ˆRˆP ˆP ˆR ) = r r ˆP r ˆP ˆR = r r r ˆR ' = r r ˆ r r r = r r r r ' = ˆ r. '.53) We, therefore, fnd the foowng mportant resut or aternatvey where ˆR, ˆP ˆRˆP ˆP ˆR = ˆ,.54) ˆR j, ˆP k =.55) jk ˆR, ˆP s the commutator of ˆR and ˆP. It shoud aso be obvous that ˆR j, ˆR k = ˆP j, ˆP k =..56) Another fundamenta operator s that of the anguar momentum ˆL = ˆR ˆP,.57) whch has the foowng commutaton propertes Ensten s summaton conventon adopted) ˆR j, ˆL k = km ˆP j, ˆL k = km = km ˆR jm ˆR j, ˆR ˆPm = km { } ˆR ˆRj, ˆP m + ˆR j, ˆR ˆP m = jk ˆR ˆP j, ˆR ˆP m = jk ˆP, ˆP m = km ' j.58) and ˆL j, ˆL k = jmn ˆR m ˆPn, ˆL k = jmn = jmn nk ˆRm ˆP + mk ˆR ˆPn { } ˆR m ˆPn, ˆL k + ˆR m, ˆL k ˆP n { } { ˆR m ˆP + nk j ' n jk ) ˆR ˆPn } = jk m ' j mk = ˆR j ˆPk ' ˆR k ˆPj ),.59) where we used njm nk = jk m j mk. The ast equaton can aso be wrtten as
ˆL j, ˆL k = ˆL jk.6) If we consder the square of the tota anguar momentum ˆL = ˆL j ˆL j, then we have and ˆL, ˆL k = ˆL j ˆL j, ˆL k + ˆL j, ˆL k ˆL j = ˆ = jk ˆL j ˆL ˆL j ˆL = jk ˆRk ˆL + ˆRj ) ˆL, ˆR j = ˆL k ˆLk, ˆR j + ˆL k, ˆR j ˆL k = jk ˆLk ˆR + ˆR ˆLk = jk ˆR ˆLk km ˆRm ˆL, ˆP j = ˆP jk k ˆL + ˆPj )..6).6) That s to say, any component of the anguar momentum operator commutes wth the square of the tota anguar momentum. On the other hand, the components of the poston and momentum operators do not commute wth the components, or the square, of the anguar momentum. However, a tte more work w show that ˆL, ˆR ˆRj = ˆR j 3 ˆR ˆRj ˆL, ˆP ˆPj = ˆP j 3 ˆP ˆPj ), and most notaby usng aso equatons.58)).63) ˆL k, ˆR = ˆL k, ˆP = ˆL, ˆR = ˆL, ˆP =..64) It s mportant to note that other operators e.g., ˆF, Î, Ĵ, ˆN, and Ŝ ) to be ntroduced ater are aso used to represent other anguar momenta and spns..3 Hesenberg Inequaty Whenever two observabes ˆQ and ˆP satsfy the same commutaton reaton as the poston and momentum operators,.e., ˆQ, ˆP =,.65) then we say that they are conjugate operators. Let s assume that for both operators
Q = ˆQ = P = ˆP =..66) Ths s not a restrcton, snce we coud aways defne new operators by subtractng Q and P from ˆQ and ˆP, respectvey, n the event that they weren t nu. Now consder the foowng quantty I = ˆQ ˆP..67) If we defne new states such that ˆQ ˆP,.68) then usng the Schwarz Inequaty we can wrte I =,.69) snce s a scaar product. We can rewrte the ast part of ths nequaty as foows = + ) 4 = ˆQ ˆP + ˆP ˆQ 4 ' ˆQ ˆP ˆP ˆQ ) ˆQ ˆP + ˆP ˆQ ) ˆQ ˆP ˆP ˆQ ) = 4 ' = { ˆQ, ˆP } 4 ' ˆQ, ˆP..7) The quantty { ˆQ, ˆP } ˆQ ˆP + ˆP ˆQ s commony caed the ant-commutator, for obvous reasons. We shoud note that { ˆQ, ˆP } ) = { ˆQ, ˆP },.7) snce { ˆQ, ˆP } = { ˆQ, ˆP } * because ˆQ and ˆP are observabes.e., Hermtan operators). Takng ths resut nto account, we can now nsert equaton.65) nto equaton.7) and fnd 3
ˆQ ˆP 4.7) Ths ast equaton s a generazaton of the so-caed Hesenberequaty, whch s usuay wrtten as foows Q P,.73) wth Q ˆQ P ˆP..74).4 Dagonazng the Hamtonan Matrx Gven a bass { }, the eements of the Hamtonan are cacuated wth H j = Ĥ j..75) It s straghtforward to verfy ths equaton wth coumn and row vectors for the bass, and a matrx for the Hamtonan. Usng the poston bass and ts competeness reaton we can express equaton.75) wth the wave functons that correspond to the bass H j = Ĥ j = d 3 r r r Ĥ j = d 3 r * r) Ĥ j r),.76) where n the ast expresson t s mped that the Hamtonan s expressed usng ts representaton n the poston bass. For exampe, for a free partce of mass m whose cassca Hamtonan s H = p m,.77) the correspondng quantum mechanca Hamtonan operator n the ast expresson of equaton.76) s 4
Ĥ = m..78) Most tmes, we woud ke to fnd the bass of egenvectors correspondng egenvaues E j { } ) for the Hamtonan, such that { j } wth the Ĥ j = E j j,.79) where no summaton on the ndex j s mped. It s apparent from equaton.79) that the Hamtonan matrx n ths representaton s dagona, hence ts attractveness for our probem. However, t s often the case that we have a bass j compete, s not that of the egenvectors for Ĥ. However, we can wrte { } that, athough t s j = C jk k,.8) k.e., we can aways expand an egenvector of the Hamtonan as a functon of the { j } bass. We can further wrte agan no summaton on j ) Ĥ j = C jk Ĥ k = C jk Ĥ k k = E j j = E j C jk k = E j C j. k k.8) where Ĥ j = Ĥ j. Aternatvey, ths equaton can be wrtten as Ĥ k E j k ) C kj =,.8) k wth C s the transpose C.e., C kj = C jk ). A non-trva souton to the system of equatons specfed by.8) w be obtaned by settng the foowng determnant to zero Ĥ k E k =,.83) when the egenvaue E equas E j. In other words, ths s just a typca egenvaue probem where a egenvaues can be evauated wth equaton.83). Once the egenvaues E j have been found, the eements C jk of the transformaton matrx C can be 5
cacuated wth equaton.8). But the orthonormaty condton for the bass mpes that { j } j k = * m C jm C kn n = C * jm C kn m n = C * jm C kn mn m = C * jm C km = jk. m n m,n m,n.84) Ths ast equaton mpes that the transformaton matrx s untary, that s C = C..85) If we ntroduce the dagona matrx whose eements are the egenvaues E j.e., j = E j, no mped summaton), then from equatons.8) and.85) we can wrte wth H the Hamtonan matrx) or aternatvey Exampe H C = C,.86) C H C =..87) We consder the case of a two-eve system wth the correspondng two-dmensona Hamtonan matrx H = E * H H E,.88) where E and E are the egenvaues of the unperturbed kets and Straghtforward appcaton of equaton.83) for the determnaton of the egenvaues of the Hamtonan yeds E E) E E) H = E E + E wth the foowng roots =,.89) E H E E. 6
E, = E + E ± E + E ) + 4 H ± E E ) + 4 H = E + E E E )' '..9) If we set for convenence that E E, then we can wrte wth E E ) E = E S E = E + S,.9) where S = + 4 H ),.9) ' and = E E. To fnd the correspondng egenvectors, we nsert these energy eves, one at a tme, n equaton.8), and then use equaton.84). For exampe, n the case of E equaton.8) yeds E E ) C + H C = C =, * H C + E E.93) or from the second of these equatons whe equaton.84) adds the constrant C = C = H * S + C,.94) = C * C + C * C = C + H S + = C ' = C ' + 4 H ' + 4 H + '. + 4 H + + 4 H ' ' ' + 4 H + ' ' '.95) We are certany at berty to wrte H = H e, and we fnd that 7
C + C = + + 4 H ' C ) )C * = e * ) C = e* ) + 4 H ',.96) where the sgn and phase of C are dctated by that of H through equaton.94). A smar exercse for E can be shown to yed C = C = C + and C = C * = C. The egenvectors are thus gven by = C + = C + C * + C..97) It s straghtforward to verfy that equaton.86) s satsfed wth the transformaton matrx defned wth equatons.96) and.97). Let s now concentrate on cases where H s rea and sma,.e., H * = H. We can then cacuate the foowng approxmatons S = H + 4H + H ' * ) +, -. C + = + -. - + 4H. H C = -. - + 4H. + H ' * - ) +,. H ' * - ) +,..98) H. We see that the amount by whch the states and mx to form the new egenvectors s a functon of both H and. The smaer ther rato.e., H ) the more the states and energes of the true Hamtonan resembe that of the unperturbed two-eve system. It s aso apparent that the perturbaton has for effect to ncrease the energy dfference between the two eves. 8
.5 The Cassca Moecuar Hamtonan To set up the quantum mechanca Hamtonan for a moecue, we frst wrte down the cassca Hamtonan and then make the approprate coordnates changes before expressng the correspondng reaton wth the proper quantum mechanca operators. For a moecue made of a tota of partces, wth N nuce and N eectrons, the tota energy s gven by the sum of the knetc and potenta energes T and V, respectvey. These two quanttes are gven by usng Système Internatona unts) T = r = m r X r + Y r + Z r ) V = C r C s e, 4 r <s= R rs.99) where, for the r th partce, m r s the mass, the poston s specfed by X r,y r, and Z r and measured from some arbtrary space-fxed coordnate system, C r e s the charge, and R rs R rs = X r X s ) + Y r Y s ) + Z r Z s ).) s the dstance to partce s. Fnay, s the permttvty of vacuum. We know that the knetc energy can be decomposed nto two components: one T CM ) due to the moton of the centre of mass of the system, and another T rve ) arsng from the motons of the nuce and eectrons reatve to the centre of mass. To separate these two terms, we ntroduce X r,y r, and Z r the poston components of the r th partce reatve to the centre of mass ocated at X,Y, and Z such that X r = X r + X Y r = Y r + Y Z r = Z r + Z..) Obvousy, R rs s unchanged n quantty by the ntroducton of these new varabes, but ony n form wth R rs = X r X s ) + Y r Y s ) + Z r Z s )..) On the other hand, f we wrte the sum of the knetc energes from partces r =,, as a functon of X, Y, Z, X,Y,Z,, X,Y,Z and the assocated veoctes, we fnd that 9
m r X r + Y r + Z r ) = r = r = m r X r + Y r + Z r ) + m r X r X + Y r Y + Z r Z r = + r = m r ' X + Y + Z )..3) However, equaton.3) s not that for the tota knetc energy, snce t does not contan the contrbuton for the frst.e., r = ) partce. But because of equatons.) we can wrte that and m r X r = m r X r X =,.4) r = r = X = m r X r,.5) m r = wth smar equatons for Y and Z. Usng ths ast equaton and the ones for Y and Z ) we fnd m X + Y + Z ) = m r m s X r X s + Y r Y s + Z r Z s m r,s= m r X r X + Y r Y + Z r Z r =.6) + m X + Y + Z ). Addng equatons.3) and.6) together, we fnd that the tota knetc energy s T = T CM + T rve = M X + Y + Z ) + m r X r + Y + Z ) + m r r r m s X r X s + Y r m Y + Z Z ), s r s r = r,s=.7) where M = m r s the mass of the moecue, and r =
T CM = M X + Y + Z ) T rve = m r X r + Y + Z ) + m r r r m s X r X s + Y r m Y + Z Z ). s r s r = r,s=.8) The ndex rve stands for rovbronc rotaton-vbraton-eectronc), and s assocated wth the nterna knetc energy of the moecue, as opposed to the externa or transatona knetc energy. We see from equatons.8) that we have, as expected, a compete separaton of the transatona and nterna components of the energy. As far as we are concerned, we w not worry about the transatona knetc energy assocated wth the centre of mass of the moecue, and ony use the rovbronc part of the energy E rve when deang wth probems n moecuar spectroscopy. More specfcay, we have wth E tota = T CM + E rve,.9) E rve = T rve + V = m r X r + Y + Z ) + m r r r m s X r X s + Y r Y s + Z r Z s m r = + C r C s e. 4 r <s= R rs r,s=.) The rovbronc energy s the quantty that we need to consder to determne the spectroscopy of a moecue..6 The Quantum Mechanca Rovbronc Hamtonan To convert the cassca rovbronc energy E rve nto the needed quantum mechanca verson of the Hamtonan, we must frst express equaton.) as a functon of the coordnates X r,y r, and Z r and momenta P Xr, P Yr, and P Zr. We coud then repace these quanttes wth the correspondng quantum mechanca operators see equatons.3) and.5)) and obtan an expresson for the quantum mechanca Hamtonan. To accompsh our frst aforementoned task, we make use of the Lagrangan defnton for the momenta,.e., P Xr = L rve X r,,.) where L rve T rve V s the rovbronc) Lagrangan. In ths case, snce the potenta energy s not a functon of the veoctes, then equaton.) smpfes to
P Xr = T rve X r,.) and so on. Usng the second of equatons.8) we fnd P Xr = m r X r + m r m m s X s = m r X r + m r ' + m r m s X s,.3) s= m m s= sr wth smar equatons for P Yr and P Zr. Equaton.3) can be put n the foowng matrx form P X P X ' + m m m m * m X = ), ),,.4) ) m m + m m +, m X whch n prncpe can be nverted to get m r X r as a functon of P Xr. Athough a souton to ths probem.e., for an arbtrary number of partces) s not obvous at frst sght, t s easy to cacuate that n the smpe case, where r = or 3, that m X m 3 X 3 = m + m 3 m 'm m + M * ) 'm 3 m + m m -, = ' m M 'm M + * ) 'm 3 M ' m 3 M -, P X P X3 P X P X3,.5) wth M = m + m + m 3. We aso rewrte the ast of equatons.8), and use equaton.5) to fnd + m T rve = m X, m 3 X 3 ) m m ' m + X * ) m 3 X 3 +, + m m m 3 = P X,P X3 ) - m M - m 3 + M m m m - m - m M - m 3 + M - m M ' M m m m - m 3 3 M - m 3 ) M +, P X P X3 * +,.6) and mutpyng the matrces
T rve = P X,P X3 = P X,P X3 m M m 3 M m M m 3 M m M M ' m ' ' ' ' ') * ' m ' 3 M m 3 M ' ' ') * ' P X P X3 P X P X3 +, - +, +, - +.7) or 3 T rve = P Xr m + P + P Yr Zr r = r M 3 P Xr P Xs + P Yr P Ys + P Zr P Zs )..8) r,s= Athough we mted ourseves to ony three partces n the precedng exampe, we w take t on fath that ths resut can be extended to an arbtrary number of partces t can be), and wrte E rve = P Xr m + P + P Yr Zr r = r M + C r C s e, 4 r <s= R rs r,s= P Xr P Xs + P Yr P Ys + P Zr P Zs ).9) for the rovbronc energy of a moecue. We are now n a poston to wrte down the equaton for the quantum mechanca rovbronc Hamtonan. Usng equatons.5) and.9) we have wth Ĥ rve = r + r s + C r C s e.) m r M 4 r = r,s= as a representaton n the usua coordnate space. r <s= r = e X X r + e Y Y r + e Z Z r,.) R rs 3
.7 The Rovbronc Schrödnger Equaton To the rovbronc Hamtonan of equaton.) w correspond egenfunctons rve such that Ĥ rve rve X,Y, Z,, X,Y, Z ) = E rve rve X,Y, Z,, X,Y, Z ),.) where E rve s now an egenvaue of the Hamtonan. It s possbe to show usng equatons.64) and.5)) that the rovbronc Hamtonan commutes wth the tota orbta anguar momentum and ts square, and thus share the same set of egenfunctons wth these operators. More precsey, defnng Ĵ and ĴZ for the operators of the square and the Z-component of the tota orbta anguar momentum, respectvey, we have Ĥ rve rve = E rve rve Ĵ rve = J J + ) rve, J =,,, Ĵ Z rve = m rve, m =,,,, J.3) where J and m are the so-caed tota orbta anguar momentum and projecton quantum numbers, respectvey..7. The Fne Structure and Hyperfne Structure Hamtonans The rovbronc Hamtonan of equaton.) does not take nto account some nteractons due to the ntrnsc magnetc moment.e., spn) of the eectrons, and the ntrnsc magnetc and eectrc moments of the nuce. More precsey, f we ony consder the ndvdua eectron spns ŝ, and the so-caed eectron spn-spn and spn-orbt coupngs, then the eectron fne structure Hamtonan Ĥ es must be ntroduced Ĥ es 46 e ' 5 4 m e c ) 76 + e m e c 3 8, e ' + m e c ) C 8 e R 8 3 3 j > 3 3 j R j + ˆR ˆR - 8 ) *,- + ˆR ˆR j ) * ˆP ˆP '. - j ),- / ŝ ˆP m e ˆP 8 m 8 '. ) / ŝ + ŝ 3 ŝ j 3 + ŝ ˆR ˆR j R j R, j. + / ŝ j ˆR, ˆR ' - j ). /),-.4) 8 3 9 ˆR ˆR j ) ŝ ŝ j ). : / ;<, where abes the nuce, and and j abe the eectrons. In equaton.4), the frst term a spn-orbt nteracton) corresponds to the coupng of the spn of each eectron to the magnetc fed t fees n ts reference frame) because of the presence of the eectrc 4
Couomb feds due to the other eectrons n ther respectve reference frames). The second term s aso a spn-orbt nteracton but ths tme wth the Couomb fed of the nuce, and the ast term corresponds the spn-spn coupngs between the ntrnsc magnetc moments of each par of eectrons. The nterna moecuar Hamtonan then becomes Ĥ nt = Ĥ + Ĥ rve es, and the tota anguar momentum must ncude both the orbta and eectron spn momenta. It s usua to denote the orbta anguar momentum wth ˆN nstead of Ĵ ), and the tota anguar momentum wth Ĵ. We then wrte Ĵ = ˆN + Ŝ,.5) where Ŝ s the tota eectron spn operator. The assocated quantum numbers J and m refer to the sum of the orbta and spn anguar momenta, and are those wth whch the moecuar egenfunctons can be abeed. On the other hand, f the nteractons of the ntrnsc magnetc and eectrc moments of the nuce are taken nto account, then the tota anguar momentum s denoted by ˆF wth ˆF = Ĵ + Î = ˆN + Ŝ + Î,.6) where Î s the tota nucear spn anguar momentum. The correspondnucear hyperfne structure Hamtonan Ĥ hfs w ncude nuce spn nteractons smar n form to those of equaton.4), as we as terms due to the nuce eectrc quadrupoe feds. The nterna moecuar Hamtonan then becomes Ĥ nt = Ĥ rve + Ĥ es + Ĥ hfs,.7) and the good quantum numbers wth whch the assocated egenfunctons can be abeed are those correspondng to ˆF and ˆF Z,.e., F and m F m F =,,,,F ). 5