Harmonic Oscillator, a, a, Fock Space, Identicle Particles, Bose/Fermi

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1 mar_-8.nb: 8/3/04:::3:0 armonc Oscllator, a, a, Fock Space, Identcle Partcles, Bose/Ferm Ths set of lectures (Mar,3,5,8) ntroduces the algebrac treatment of the armonc Oscllator and apples the result to a strng, aprototypcal system wth a large number of degrees of freedom. That system s used to ntroduce Fock space, dscuss systems of dentcle partcles and ntroduce Bose/Ferm annhlaton and creaton operators. armonc Oscllator ü Classc SO The classcal amltonan for the smple harmonc oscllator s = ÅÅÅÅÅÅÅÅ m p + ÅÅÅÅ k x = ÅÅÅÅÅÅÅÅ m p + ÅÅÅÅÅÅÅÅÅÅÅ mw x Ths leads to smple harmonc moton wth frequency w = k I m. ü QM: wave mechancs Make the replacement p = - ÅÅÅÅÅ x, and solve for the wave functons. For example, Merzbacher chapter 5. ü QM: operator approach Introduce rasng and lowerng operators (a and a ) and solve smple algebrac egenvalue problem. Note: n some contexts (feld theory) a, a are also known as annhlaton and creaton operators. ü Set up of problem, ntroducton of a, a, and N ü For convenence smplfy Defne: p' = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p mw and x' = mw x, n whch case can be rewrtten as = w ÅÅÅÅÅ p' + x' ü Defne a and a. Further, defne the operator a = ÅÅÅÅÅÅÅÅ x' + p' and snce x and p are hermtan, the adjont s a = ÅÅÅÅÅÅÅÅ x' - p'.

2 mar_-8.nb: 8/3/04:::3:0 Also note that x' and p' can be rewrtten as x' = ÅÅÅÅÅÅÅÅ a + a and p' = ÅÅÅÅÅÅÅÅ a - a. The commutator of a and a s a, a = ÅÅÅÅ x' + p', x' - p' = p', x' = p, x = and a a = ÅÅÅÅ x' - p' x' + p' = ÅÅÅÅ x' + p' + x', p' = ÅÅÅÅ x' + p' - ÅÅÅÅ a a = ÅÅÅÅ x' + p' x' - p' = ÅÅÅÅ x' + p' - x', p' = ÅÅÅÅ x' + p' + ÅÅÅÅ ü Defne N and rewrte It s convenent to recast = ÅÅÅÅÅ w p' + x' = w a a + ÅÅÅÅ = w N + ÅÅÅÅ wherethe "number" operator s N = a a. It should be obvous that, N = 0, and so and N can be smultaneously dagonalzed. Determnng the spectrum of energy egenstates can be reclassfed as determnng the spectrum of N. ü The spectrum of states Defne Sn as the normalzed egenstates of N, and let t be understood that the states are labeled by the egenvalue,.e. N Sn = n Sn ü Show n s postve defnte Consder the quantty n N n = n n n = n. It s convenent to defne Sb = a Sn. Snce N = a a we also have n N n = n a a n = b b 0. It follows that n s real and postve-defnte. ü Show that a s a lowerng operator. a, N = a a a - a a a = a a a - a a a = a, a a = a

3 mar_-8.nb: 8/3/04:::3:0 3 Agan, let Sb = a Sn, so that N Sb = N a Sn = N, a + a N Sn = -a + a n Sn = n - a Sn = n - Sb,.e. Sb s an an egenstate of N, wth egenvalue n -, or a Sn = c n Sn - where c n s some as yet undetermned coeffcent. We can evaluate c n by consderng n = n N n = n a a n = Sc n W, whch gves c n = e f n. Conventonally f = 0, whch gves a Sn = n Sn -. ü and a s a rasng operator Smlarly N, a = a and a Sn = b n+ Sn +, where b n = n as well. a acts as a rasng operator - a Sn = n + Sn +. It s often more convenent n ths form. a Sn - = n Sn, where we can easly see N Sn = a a Sn = a n Sn - = n Sn. ü spectrum So far we have showed how to construct a set of states Sn wth n values separated by ntegers. There are, however, many such sets, but only one s a vable set of states for the SO. Recall that we have the constrant that n 0. Suppose n s n the nterval 0,. Then operatng wth a would gve a state wth n < 0, whch s not allowed. The only possblty s that operatng wth a gves 0, but that would volate the relaton for c n - unless n = 0. It seems the only possblty s for n to be ntegral. In ths case we can satsfy the boundary condton by a S0 = 0 S0 = 0. The spectrum of states s then gven by Sn, n = 0,,.... Ths seems very reasonable. As the amltonan s postve defnte, the expectaton value s requred to be postve. Even wth the extra contrbuton of ÅÅÅÅ t s not unreasonable that N s also postve. One can buld normalzed states by teratve use of the rasng operator. Sn = a ÅÅÅÅÅÅÅÅÅÅÅÅ n S0 n!

4 mar_-8.nb: 8/3/04:::3:0 4 ü Matrx form for operators, N, a, a. It s straghtforward to express the operators n matrx form (see Merzbacher). and N are dagonal. a and a are off dagonal. e.g ÅÅÅÅ 0 = w ÅÅÅÅ 0, a =, x =, ª 0 ª 0 0 ª 0 3 or a = S n n Sn - nw. ü Useful relaton A useful fact for dong some manpulatons (for example problem.8) s a, f a = ÅÅÅÅÅÅÅÅ a f a Ths s smlar to k, x = - ÅÅÅÅÅ x Fock Space - "nd quantzaton" The harmonc oscllator provdes a startng pont for dscussng a number of more advanced topcs, ncludng multpartcle states, dentcle partcles and feld theory. As an ntroducton, consder the problem of quantzng a classcal strng (e.g. a gutar strng). Ths s done by treatng the oscllatng modes of the strng as a set of harmonc oscllators. Each.O. can be quantzed, so that the quantum state of the strng s gven by specfyng the quantum state of each oscllator. Ths s sometme referred to as "second quantzaton". Presumably the term s meant to suggest that the frst quantzaton s determnng the egenmodes of the system, and the second quantzaton s determnng the exctaton level of each mode. An alternatve language for dscussng the quantzed strng s to label the exctatons of the ndvdual modes as partcles. Ths language naturally carres over to any number of classcal systems that exhbt oscllatory behavor, ncludng transverse phonons on a strng, longtudnal phonons through a medum, and photons as quantzed oscllatons of the electromagnetc radaton. These partcles are all examples of Bosons. Bose systems may exhbt a large degree of exctaton. In the partcle language ths s equvalent to dscussng a system wth a large number of partcles. the partcles are dentcle, although they may be found n dfferent states. Accordngly, t s natural to dscuss the quantum theory of dentcal partcles at ths tme. The concept of annhlaton and creaton operators for Bosons, can be extended to descrbe Ferm systems as well. ü Quantze smple strng The energy for a classcal strng s gven by the sum of the knetc and potental energes. E = T + U

5 mar_-8.nb: 8/3/04:::3:0 5 To be specfc, consder boundary condtons where the strng s stretched between two fxed end ponts at 0,, and let the strng have mass densty r, and tenson k. et poston of strng be y x, t T = r ÅÅÅÅ 0 y U = k ÅÅÅÅ 0 y' ü Expanson n bass states Next, expand y n terms of normalzed bass states for the strng. y x, t = S n y n t ÅÅÅÅ Sn ÅÅÅÅÅÅÅÅÅÅÅ n p x Then, the knetc and potental energy terms can be reexpressed as a sum over modes. 0 y = S y n ÅÅÅÅ 0 n Sn ÅÅÅÅÅÅÅÅÅÅÅ n p x S y m ÅÅÅÅ m = S y n y m ÅÅÅÅ n,m 0 Sn n p x m p x ÅÅÅÅÅÅÅÅÅÅÅ Sn ÅÅÅÅÅÅÅÅÅÅÅÅ = S n,m y n y m d nm = S n Sy n W 0 y' = S y n ÅÅÅÅ 0 n = S y n y m ÅÅÅÅ n,m ÅÅÅÅÅÅÅ n p ÅÅÅÅÅÅÅ n p = S y n y m n p m p ÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ n,m d nm = S n n p ÅÅÅÅÅÅÅ Sy n W = S n k n Sy n W Sn ÅÅÅÅÅÅÅÅÅÅÅÅ m p x n p x Cos ÅÅÅÅÅÅÅÅÅÅÅ S y m ÅÅÅÅ m ÅÅÅÅÅÅÅÅ m p m p ÅÅÅÅÅÅÅÅ 0 Cos n p x m p x ÅÅÅÅÅÅÅÅÅÅÅ Cos ÅÅÅÅÅÅÅÅÅÅÅÅ m p x Cos ÅÅÅÅÅÅÅÅÅÅÅÅ ü strng = sum over harmonc oscllators These results can be used to reexpress the amltonan as a sum over harmonc oscllators. = ÅÅÅÅ S n r Sy n W + k k n Sy n W where w = S ÅÅÅÅÅÅ n n p n + q n = S w n a n a n + ÅÅÅÅ n

6 mar_-8.nb: 8/3/04:::3:0 6 p n = y n J r w n q n = y n r w n a n = ÅÅÅÅÅÅÅÅ q n + p n a n = ÅÅÅÅÅÅÅÅ q n - p n w n = r I k k n = c k n = w 0 n w 0 = ÅÅÅÅÅÅ c p ü Fock space: bass states for the strng From the above dscusson, a contnuous strng can be descrbed by a large (nfnte) number of QM harmonc oscllators, one harmonc oscllator for each bass functon descrbng the strng's moton. Denote the oscllators by the subscrpt. The bass states for each oscllator can be chosen to be the egenstates of the number operator N, and the bass states for the strng may be taken to be a product of the ndvdual oscllator bass states. A strng bass state can therefore be descrbed by an nfnte dmensonal vector specfyng the state of each of the harmonc oscllators. Sn = Sn Sn Sn 3... = Sn, n, n 3,... and each of the n may be a non-negatve nteger. An arbtrary state of the strng Sa would be gven by a superposton Sa = S S S... cn n =0 n =0 n3 =0 Sn where the c n are complex numbers. The lowest energy state of the strng s one where all the n are 0. Note that the lowest energy state has nfnte energy, w from summng up all the zero pont energes for each of the ndvdual oscllators, E 0 = S ÅÅÅÅÅ =. It s conventonal to gnore ths nfnty, notng that absolute energy scales are not observable, only relatve energy scales. Wth ths subtracton, the energy of a strng state s E = n n = n S N w n = S n w The bass states of the strng can be bult up from the ground state by operatng wth approprate rasng operators. For example, S, 0, 0,... = a S0, 0, 0... The operator a rases the exctaton level of the lowest mode by. It s mportant to dstngush the effect of rasng the exctaton ampltude of a gven mode, from the effect of shftng exctatons to a hgher mode. In the context of the gutar strng, shftng to a hgher mode corresponds to a hgher frequency or a hgher "note". Increasng the exctaton of a gven mode, corresponds to makng that note "louder". For example, suppose the strng s n the n = state, S, 0, 0,.... The energy s w 0. We can ncrease the energy to w 0 ether by ncreasng the ampltude to the n = state, or by ncreasng the frequency by shftng the exctaton to the nd mode. S, 0, 0,... = a S, 0, 0,... or S0,, 0,... = a a S, 0, 0,...

7 mar_-8.nb: 8/3/04:::3:0 7 For a hghly excted strng there s a large amount of degeneracy all states where E = w 0 S n such that the n add up to the same total are degenerate. ü Partcles as exctatons At ths pont, each of the n specfes the degree of exctaton of the th mode of the strng. There s, however, a natural nterpretaton where the exctatons may be thought of as "partcles", and the n represent the number of partcles n a gven mode. In the case of exctatons on a strng, the oscllatons of the strng correspond to sound waves, and we call the partcles "phonons". The concept can be extended to any "feld". In the present case, the feld s the lateral or transverse dsplacement of the strng. The restorng force whch provdes for the potental energy s the strng tenson. In the case of sound waves n a lqud or gas, the feld s the longtudnal dsplacement of the atoms n the materal, and the restorng force s pressure n the flud. When consderng electromagnetc radaton, the feld s the vector potental A, and the potental energy s assocated wth gradents of A provdng tenson n the feld (also known as E and B). Such partcles are called photons. We wll develop the concept of photons more thoroughly when dscussng radatve transtons. The analogy s a bt more dffcult when dscussng electrons. What, after all, s supposed to play the role of the feld? A couple of comments are n order. Frst, phonons and photons are massless. As a result the spectrum of modes ncludes long wavelength, low frequency oscllatons. We can observe the effects of such modes wth macroscopc nstruments. Electrons, on the other hand, have mass. Even low momentum modes have energes comparable to the mass, and thus oscllate at a hgh frequency, not easly measured by humans. More mportantly, phonons and photons are bosons, and the spectrum of states ncludes occupaton numbers wth large values of ndvdual n. Such states are recognzed as classcal felds. Electrons, however, are fermons whch admt to n = 0 or (see below). arge occupaton numbers are not permtted, and there s no analog of the classcal feld. That does not mean electrons cannot be descrbed n terms of a feld theory, only that we do not have drect experence wth the feld n a classcal confguraton. Thus, at ths juncture, there s no reason not to consder electrons as exctatons of an underlyng feld theory. Instead of dscussng the state of an electron, we could dscuss the number of exctatons for a gven mode of the electron feld. When usng the partcle language, t s common to refer to the rasng and lowerng operators as creaton and annhlaton operators. In the example above, The operator a creates a phonon n lowest mode. Smlarly, a destroys or annhlates a phonon from the frst mode. If there are no phonons n the mode, the result of operatng wth a s 0. The total number of partcles n the state s gven by the number operator, N = S N, whch has the expectaton value N = n, n, S N n, n, ) = n, n, S n n, n, ) = S n n, n, n, n, = n Each of the N acts only on ts own mode. The energy of the state s the sum of the partcle energes n the ndvdual modes E = n, n, S N w n, n, ) = S n w

8 mar_-8.nb: 8/3/04:::3:0 8 For the strng, the frequences are evenly spaced. Ths result s a feature of a -dmensonal system wth a massless egenmode spectrum, E = c p. For a 3-dmensonal system, the lattce of momentum states leads to uneven spacng of energy levels. In addton, for a massve relatvstc (E = m c 4 + c p ) or non-relatvstc (E = p ÅÅÅÅÅÅÅÅ m ) spectrum, the splttng s not proportonal to p. Identcal partcles Consderng partcles as exctatons of an underlyng feld theory s one way to approach a dscusson of dentcal partcles. Consder the strng system above. Vewed as exctatons, there s no way to dstngush the phonons n the system, other than by whch mode they are n. The same s true when consderng two electrons. There s no way to dstngush whch s whch. A full descrpton of the system s gven by specfyng the occupaton numbers for the varous electron modes. The ndvdual electrons have no dentty. ü Tradtonal dscusson of exchange symmetry Before contnung wth the Fock space descrpton, t may be useful to revew the tradtonal dscusson, gven n terms of exchange symmetry. Consder a system consstng of two dentcal partcles. et one (the frst) be n state Sk ' and the other be n state Sk ''. To dstngush the two partcles, wrte the combned state n a specfc order Sk ', k ''. More generally, n ths language, an N partcle state s wrtten Sk, k,, k N, where the "slots" n the state vector dentfy the partcle, and the value of the slot dentfes the mode. (Note that ths s dfferent than the nomenclature for Fock states, where the slots dentfy the modes and the value of the slot ndcates how many partcles are n each mode.) The partcle nomenclature makes sense f the partcles are dstngushable, n whch case the two-partcle states are product states of the one-partcle states for the two partcles. When the partcles are dentcal, the stuaton s less clear. ow, for example, does one dstngush between the state Sk ', k '' and the state where the partcle denttes are exchanged Sk '', k '? It would seem that for any par of quantum numbers k ', k '', there s a degenerate two state system,.e. Sk ', k '', Sk '', k ', or any lnear combnaton of bass kets c d Sk ', k '' + c x Sk '', k ' where d or x ndcates "drect" or "exchange" should lead to the same observables. To ad the dscusson, ntroduce the exchange operator P whch acts to exchange the state of the two partcles P Sk ', k '' = Sk '', k ' Actng wth P twce gves one back the orgnal state. P =. Operatng on a general lnear combnaton P c d Sk ', k '' + c x Sk '', k ' = c d Sk '', k ' + c x Sk ', k '' The egenstates of P are the symmetrc and antsymmetrc lnear combnatons Ss = ÅÅÅÅÅÅÅÅ Sk '', k ' + Sk ', k '' Sa = ÅÅÅÅÅÅÅÅ Sk '', k ' - Sk ', k '' wth P Ss = +Ss and P Sa = -Sa.

9 mar_-8.nb: 8/3/04:::3: 9 Snce the partcles cannot be dstngushed, t s reasonable that P, = 0, n whch case the "exchange party" of the system s conserved. The space of states can be dvded nto symmetrc and antsymmetrc subspaces. Untl ths pont n the dscusson ether state, or a lnear combnaton, would be acceptable. In fact, however, Nature appears to choose ether Ss states or Sa states but not both for a partcular partcle speces. Partcles wth ntegral spn spn = 0,,,... form symmetrc states, and are known as bosons (after Bose). Smlarly, partcles wth half-ntegral spn spn = ÅÅÅÅ ÅÅÅÅ ÅÅÅÅ... form antsymmetrc states, and are called fermons (after Ferm)., 3, 5 From the perspectve of non-relatvstc quantum mechancs, I thnk these rules are not dervable, and must be treated as addtonal assumptons. ü Dscusson There s somethng a bt odd about ths dscusson. It starts wth the assumpton that the two-partcle vector space conssts of states Sk ', k '', Sk '', k ' whch are dfferent, but n the end t turns out these states are not vald. Rather the physcal vector space s restrcted to ether Ss or Sa. An alternatve approach s to assume from the begnnng that there s no exchange degeneracy, but rather that specfyng quantum numbers for a two partcle state yelds a unque descrpton of the state. In ths case, Sk ', k '', l fully specfes the state, where l = s the egenvalue of the exchange operator. Bosons have l = and fermons have l = -. P Sk ', k '', l = Sk '', k ', l = l Sk ', k '', l In ths pcture, Sk ', k '', l and Sk '', k ', l represent states that dffer only by a phase, whch s constraned to be by the condton P =. Ether state may be used as the bass ket for the two partcle system. For fermons, a two partcle state cannot be formed wth the same set of quantum numbers occuped twce,.e. Sk ', k ', - = -Sk ', k ', - = 0 ü Permutaton symmetry and totally symmetrc/antsymmetrc multpartcle states The above approach can be generalzed. Consder an N partcle state. Where N modes k, k,... k N are occuped. Then one possble assgnment of partcles to states s to put the frst partcle n k, the second n k, etc. We could, however, equally well consder any other assgnment, e.g. put the st partcle n the mode k 5, the nd s assgned to k 3,... In general, all such permutatons must be consdered. If there are N partcles there are N!permutatons, and the full descrpton of the state s a sum over all permutatons wth approprate weghts, N! SY = S c S = where the sum over s over all permutatons, and S represents a partcular assgnment of partcles to modes, e.g. Sk 5, k 3,. Then the generalzaton of the two partcle states above s that for bosons all permutatons have weght c = ÅÅÅÅÅÅÅÅÅÅÅÅ, whle for fermons the coeffcents have alternatng sgns c = ÅÅÅÅÅÅÅÅÅÅÅÅ - p, where p = 0 f the permutaton s even, N! and p = f the permutaton s odd. The defnton of an even/odd permutaton s that the permutaton can be generated from the dentty permutaton Sk, k, by an even/odd number of two partcle exchanges. N!

10 mar_-8.nb: 8/3/04:::3: 0 Wth these rules, boson states are "totally symmetrc", and fermon states are "totally ant-symmetrc". The fermon rule reproduces the Paul excluson prncple snce exchange of any two partcles n the same mode reproduces the same state, but wth a change n the even/odd nature of the permutaton. Such an exchange produces two components of the state whch are dentcal, but of opposte sgn. The ampltude for the state as a whole s dentcally zero. Two dentcle fermons cannot occupy the same mode. ü Space and Spn Notce that t s the complete state of the system that s ether symetrc or antsymmetrc. It s typcal to wrte the full state of a system as the product of ts spn state and ts space state. In ths case, the full exchange symmetry of the state s the product of the spn exchange symmetry and the space exchange symmetry. For example, denote the full state by Sy, the spn state by S c, and the spatal state by Sj, and denote exchange symmetry by a subscrpt A or S. Then, for fermons, ether Sy A = Sy S = S c A Sj S S c S Sj A S c S Sj S S c A Sj A are vald states, whle for bosons we have the possbltes ü Dscusson n terms of Fock space, creaton and annhlaton operators ü creaton operators and multpartcle states Now consder multpartcle states n terms of occupaton numbers, begnnng wth two partcle bass kets. Suppose the occupaton numbers of the th and j th modes are, and all other n k = 0. Ths ket may be wrtten as S0, 0,, n =,, n j =, ª Sn =, n j = where n the second form modes wth n = 0 are suppressed. A two partcle state, wth the quantum numbers for modes and j wll generally be descrbed by a complex phase tmes the ket. Next, consder the producton of a two partcle state from the ground state by the applcaton of creaton operators. a j a S0 = c j Sn =, n j = where the phase of the ground state has been taken to be. Alternatvely, one could create a two partcle state by a a j S0 = c j Sn =, n j = = l j a j a S0 where l j = c j I c j. One mght ask f c j = c j, or, what are the allowed values of l j? Does the order of the creaton operators matter? Does l j depend on the modes, j? Merzbacher dscusses ths questons n chapter. ere s a smple verson of hs argument. Consder three modes,,3. Then

11 mar_-8.nb: 8/3/04:::3: a a 0( = l a a 0( a a 3 0( = l 3 a 3 a 0( Now, from the pont of formng a set of orthogonal bass kets, there s nothng specal about S and S3. One could just as well use another lnear combnaton of these kets, e.g. S = ÅÅÅÅÅÅÅÅ S S3, whch would be created by a = ÅÅÅÅÅÅÅÅ a a 3. To resolve the queston of operator order, one may now consder a a S0 = l a a S0 = l ÅÅÅÅÅÅÅÅ a 3 a S0 = ÅÅÅÅÅÅÅÅ a l a a l a 3 a S0 Alternatvely, the left sde s a a S0 = ÅÅÅÅÅÅÅÅ a = ÅÅÅÅÅÅÅÅ a = ÅÅÅÅÅÅÅÅ a a 3 S0 a a a 3 S0 l a a l 3 a 3 a S0 Snce a and a 3 create dfferent states, the expressons can be equal only f l = l + = l - = l 3. Generallzng, l j must have a common value for any par of modes, ndependent of and j. Carryng out the exchange operaton twce, a a j S0 = l j a j a S0 = l j l j a a j S0 = l a a j S0 one can see that l =, or there are two possbltes l =. For l =, (bosons) the operators obey the commutator relaton a, a j = 0, and for l = -, (fermons) the operators obey the ant-commutator relaton a, a j = 0. Often to dstngush between the two types of operators, creaton operators for the case l = - are denoted by c or b. I'll use c. The mportance of the sgn of l, arses dramatcally when consderng the case = j. For the ferm case, the spectrum of states cannot nclude n = for any. c c S0 = -c c S0 = 0 In the language of annhlaton and creaton operators t s natural to consder ether bosons or fermons, but not a mxtur of the two, whereas n the tradtonal approach that assumpton has to be ntroduced separately. ü annhlaton operators Takng the adjont of the commutaton relaton for the creaton operators, one fnds that annhlaton operators must also obey ether a, a j = 0 or c, c j = 0, wth the same sgn for l as for the creaton operators.

12 mar_-8.nb: 8/3/04:::3: ü c, c = or a, a =? It remans to show whch statstc s followed when the order of an annhlaton and a creaton operator s changed. Before consderng the more general case of a and a j, start by consderng a sngle degree of freedom ( = j). The dscusson began wth the assumpton that there was a number operator N, the states were egenstates of N, labeled by Sn, and that a and a act as annhlaton and creaton operators. et us make one more assumpton, that a ground state wth no partcles exsts, N S0 = 0. When actng on the ground state aa S0 = S0, assumng an approprate normalzaton. When actng on the ground state, there are two possbltes a, a S0 = aa + a a S0 = + 0 S0 = S0 or a, a S0 = aa - a a S0 = - 0 S0 = S0 Evdently, we have two possbltes, a, a =, or a, a =. Next, consder the operaton of N on the state S = a S0 and study the possblty that a, a =. By defnton N S = S, whle at the same tme N S = a a S = a aa S0 = aa - a, a a S0 = aa - a S0 Agan there are two possbltes. If the annhlaton and creaton operators commute a, a = 0 and a, a = 0 then one has the case of bose or harmonc oscllator statstcs. In ths case aa S = S and everythng s fne. The relaton a, a = s consstant wth a, a = 0 and a, a = 0. On the other hand, f one has ferm statstcs a, a = 0 or a a = 0, then aa S = aa a S0 = 0 and aa - a S0 = -S. Evdently, choosng a, a = and a, a = 0, leads to N S = -S, whch s nconsstant. Smlarly, consder a, a =, n whch case N S = a a S = a aa S0 = a, a - aa a S0 = - aa a S0 In ths case, ferm statstcs gves - aa a S0 = a S0 = S, whch s fne. Now t s bose statstcs whch leads to the nconsstancy. One may conclude that for a consstant defnton of N and rasng and lowerng operators, the choce of statstcs obeyed by annhlaton opertors, by creaton operators, and between annhlaton and creaton opertors must be the same. ü N, c, c Although c and c act to change Sn t remans to evaluate the constants l -, l + n the relatons c Sn = l - Sn - c Sn = l + Sn + For c, proceed as n the Bose case and compare to n N n = n

13 mar_-8.nb: 8/3/04:::3: 3 l - = n c c n = n N n = n For c, reverse the steps and consder l + = n cc n = n - c c n = n - N n = - n Together, these relatons may be combned as c Sn = n Sn c Sn = - n Sn + whch apart from a change of sgn from + n Ø - n s smlar to that for bose operators. All ths may seem a bt fancy. The full set of possbltes for c and c to operate s gven by the short lst c S0 = 0 c S = S0 c S0 = S c S = 0 ü c, c j By an argument smlar to that gven for creaton operators above, for dfferent modes ( j) one has ether a, a j = 0 or c, c j = 0. Takng nto account the possblty that = j, the two cases are bosons: a, a j = d j and fermons: c, c j = d j. ü One partcle operators In a multpartcle system one dstngushes between sngle and multpartcle operators. Sngle partcle operators depend only the state of a sngle partcle. If the partcles are dstnct, ths s a straghtforward sum. For example, neglectng nteractons between partcles, the energy s a straghtforward sum over the ndvdual knetc energes and an external potental = S = S p ÅÅÅÅÅÅÅÅ m + V x Sngle partcle operators should extend to a multpartcle state of dentcal partcles as well. Consder a two partcle system, wthout a consderaton of statstcs + Sk ' k '' = E' + E'' = + Sk '' k ' As dscussed earler, the energes of these two states are degenerate. Accordngly, the symmetrc and ant-symmetrc states have the same sngle partcle energes.

14 mar_-8.nb: 8/3/04:::3: 4 In Fock space, the sngle partcle amltonan takes the form = S = S a a E where the sum s over modes, not partcles. Each mode s weghted by N = a a. For the example at hand, the mode energes are calculated by E = p ÅÅÅÅÅÅÅÅ m + V x. Other sngle partcle operators are gven smlarly by O = S N O ü Interactons and two partcle operators. The nteracton potental between two partcles s the next level of complexty. In the partcle pcture, the nteracton energy would be wrtten as V = Y V x, x Y = x x Y * x, x V x, x Y x, x For non-dentcle partcles one would use sngle partcle wave-functons, e.g. Y j x, x = Y x Y j x f partcle s n the state and partcle s n state j. For dentcle partcles, one needs symmetrc or antsymmetrc wave-functons. For example, for fermons Y j x, x = ÅÅÅÅÅÅÅÅ Y x Y j x - Y j x Y x. Ths leads to two contrbutons to the energy, a drect term V d and an exchange term V x. V d = x x Y * x Y j * x V x, x Y x Y j x V x = x x Y j * x Y * x V x, x Y x Y j x V = V d - V x Usng Fock space, the two body potental s wrtten as V = ÅÅÅÅ S a k a l a a j kl V j jkl where the matrx element k l V j = x x Y k * x Y l * x V x, x Y x Y j x s an ntegral over the mode functons, and the algebra of annhlaton and creaton operators automatcally takes care of the drect and exchange terms. For example, f one wanted the expectaton of the nteracton potental for a partcular two partcle state Sn r =, n s =, then V rs = n r =, n s = ÅÅÅÅ S a k a l a a j kl V j n r =, n s = ) jkl = ÅÅÅÅ S kl V j n r =, n s = a k a l a a j n r =, n s = ( jkl = ÅÅÅÅ S jkl kl V j n r =, n s = a k a l d r d js - d s d jr 0( = ÅÅÅÅ S kl kl V rs - kl V sr n r =, n s = a k a l 0( = ÅÅÅÅ S kl V rs - kl V sr d kr d ls - d ks d lr kl = rs V rs - rs V sr whch has the form of V d - V x.