Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle systems (Boson or Fermons. It also allows one to deal wth systems wth varables number of partcles (e.g. creaton and annhlaton of photon, creaton of electron-postron par. We wll explan later why ths method s called second quantzaton. -> A generc N-dentcal partcle Hamltonan (non-relatvstc: H ˆ = N = h ˆ ( p r, r + 2 N = j V ˆ nt ( r r j (, where h ˆ ( p r, r = p2 2m + V (r r s the sngle-partcle hamltonan and V ˆ nt s the nteracton between partcles (typcally par-wse nteracton. -> Bass set: Let { φ n >} be a complete set of sngle-partcle states where n s a collectve quantum ndex whch ncludes the spn ndex. A convenent set s the egenstates of h ˆ,.e. h ˆ φ n >= ε n φ n >. Any many-body state Ψ > can be expressed as a lnear combnaton of the drect product state of φ's subjected to the restrcton mposed by symmetrzaton (Bosons or ant-symmetrzaton (Fermons. Example: A two-electron state of He-atom where one electron s n S > and one electron s n 2S > s wrtten as Ψ S,2S >= A { S > 2S > } = S > 2S > 2S > S > 2 { } (2 where A s the ant-symmetrzaton operator and we wrote out the ant-symmetrzed state explctly, In ths notaton, t s understood that the frst ket s for electron and the second ket s for electron 2,.e., the wavefuncton corresponds to ths state s gven by Ψ S,2S (,2 < 2 < = { 2 φ (φ (2 φ (φ (2 S 2S 2S S } S > 2S > 2S > S > 2 { } (3 Note: Ths wavefuncton s ant-symmetrc wth respect to nterchange of partcle labels ( & 2, or nterchange of the state labels (S & 2S.
Ths notaton s perfectly fne but t s knd of slly because whch s electron and whch s electron 2 ; they are dentcal! In fact, the many-body state s unquely specfed by the statng whch sngle-partcle state s occuped, one electron s n S > and one electron s n 2S > and whether the partcles are Fermons or Bosons. Ths realzaton leads to a new notaton (second quantzaton. -> Second Quantzaton Notaton Instead of lnng up product state as st partcle, 2nd partcle, etc, one can lne up the sngle-partcle bass states. For example, st poston s S >, 2nd poston s S >, 3rd poston s 2S >, 4th poston s 2S >, etc., up to nfnte postons. The state Ψ S,2S > s denoted by Ψ S,2S >=,0,0,,0,... > (4 where and 0 are the occupaton number of the correspondng sngle-partcle bass state (we wll deal wth symmetrzaton and ant-symmetrzaton later. A general many-body state s denoted by Ψ >= n,n 2,...,n,...> n = 0or n can be 0,,2,3,.. forfermons for Bosons (5 The number of partcles representaton by ths state s N = n, (6 hence ths notaton allows one to represented a state wth varable number of partcles,.e. a lnear combnaton of states whose partcle numbers are dfferent. The Hlbert space spanned by ths set of bass states { n,n 2,...,n,...>} s called the occupaton number space or the Fock space. -> Creaton and annhlaton operators Borrowng the concepts of rasng and lower operators of harmonc oscllator, one can defne the vacuum state, the creaton operator and the annhlaton operator. The vacuum state s defned as a state wth zero partcle: Ψ vac > 0,0,0,...> 0 > (7 where 0> s the shorthand notaton. A state wth one partcle n the th sngle-partcle state can be denoted by the creaton operator, a +, actng on the vacuum state: 0,...,,... >= a + 0 >. 2
One can thnk of the creaton operator creates one partcle (or rasng the occupaton number by one. You may ask, what exactly s ths creaton operator and how does t rase the occupaton number by one?. Remember, we are merely defnton notaton rght now, let s determne what propertes these creaton operators must obey. For example, f the partcles are Fermons, then creaton one more partcle n the th state s not allowed hence a + a + 0 >= 0 (Fermons. (8 Note: The rght-hand sde s the number zero, not the vacuum state, whch s denoted by a ket. One the other hand, f the partcles are Bosons, then a + a + 0 >= a + 0,...,...>= 2 0,...2,... > 0,...2,...>= 2 a + a + 0 > (Bosons (9 The 2 s needed for normalzaton. In general, a + 0,...n,... >= n + 0,...n +,... > (a + n 0 >= n! 0,...n,...> 0,...n,...>= n! (a + n 0 > (0 These formulae are vald for both Bosons and Fermons, one must remember that for Fermons, n can only be 0 or. Lkewse the annhlaton operators have these propertes: a 0,...n,... >= n 0,...n,...> In partcular, a 0 >= 0 = 0 Agan, these formulae are vald for both Bosons and Fermons, one must remember that for Fermons, n can only be 0 or. ( One can easly show that a + a 0,...n,...>= n 0,...n,...> <,...n,...0 a + a 0,...n,...>= n (2 Hence a + a s the (occupaton number operators. You see that these propertes are those of the harmonc oscllators. There are other propertes such as symmetry/ant-symmetry under partcle label exchange (or equvalently state label exchange A state wth one partcles n the th state and one partcles n the jth state s denoted by 3
a j + a + 0 >= 0,...,n =,...,n j =,..0,. > (3 Now how do we embedded n ths notaton that Bosonc state s symmetrc and Fermonc states are ant-symmetrc? ** We wll use the conventon that f <j, that s the th state s order before j n then we choose a postve sgn, as ndcated n Eq.(3 The exchange property s denoted by a + a j + 0 >= a j + a + 0 > (Bosons a + a j + 0 >= a j + a + 0 > (Fermons (4 The amazng thng s that all the above propertes can be easly and automatcally takng care of f we mpose the followng commutaton rules for Bosons and ant-commutaton rules for Fermons (for more rgorously dervaton, see Fetter and Walecka. [a,a j + ] = δ j, [a,a j ] = 0, [a +,a j + ] = 0 (Bosons + { a,a j } = δ j, { a,a j } = 0, a + + {,a j } = 0 (Fermons Cummutaton [A,B] AB- BA Ant- cummutaton { A,B} AB + BA (5 -> Feld Operators One thng about these notatons s that one has to remnd the readers what are the chosen sngle partcle bass states. There s a way to do ths by defnng the so-called feld operator, ( r + ( r φ ( r a φ *( r + a (We have put the ˆ over the ψ's to remnd you that they are operators + ( r ( r d 3 r = a + a = ˆ (We have appled the orthogonalty propertes of the bass states n where φ n ( r s the sngle partcle bass functon, agan n s a collectve quantum ndex ncludng the spn state. The nterpretaton of + ( r ( r d 3 r s an operator whch wll gve you the total number of partcle n the system, hence + ( r ( r s the partcle densty operator. These are operators, you wll get a number only after you compute the expectaton of these operators wth 4 (6
the state whch s expressed n the occupaton number space (Fock space. Often, t s convenent to defne the feld operator for a partcular sngle partcle spn state, ( r ( r φ, ( r a, φ, *( r + a, ( r ( r d 3 r = a + a, = ˆ, n, (Example: For spn / 2 partcles, could be or (7 The feld operator defned n Eq (6 s related to these operators by ( r = ( r Now, ( r ( r s the partcle densty for the spn state. The commutaton/ant commutaton relatons for the feld operators can be derved from the correspondng commutaton/ant commutaton relatons for the a and a+ (plus the completeness property of the sngle partcle bass functons Bosons: ( r, + ( r ' β ( r, β ( r ' ( r, + β ( r ' [ ] = δ β δ( r [ ] = 0 [ ] = 0 Fermons: ( r, + β ( r ' ( r, β ( r ' ( r, + β ( r ' { } = δ β δ( r { } = 0 { } = 0 r r ' r r ' (8 -> Why s t called second quantzaton? Frst quantzaton refers to the quantzaton procedure whch changes the classcal Posson brackets of dynamcal varables to the quantum commutaton relaton, for example 5
[ f,g]posson f g bracket p x f g x p Example: [ p, x]posson = bracket [ p, x] Quantum = h Frst Quantzaton Rule: [ f,g] Quantum = h[ f,g]posson bracket (9 Second quantzaton refers to the commutaton/ant-commutaton rules mposed on the a and a+ as stated n Eq. (5 or on the feld operators as stated n Eq. (8. -> Expressng the many-body hamltonan n second quantzed notaton. To change the Hamltonan from frst operators to second quantzaton operators, one computes the matrx element of H (from Eq. between bass states n the followng manner ˆ H 2nd quantzaton = ( r h ˆ β ( r d 3 r + 2,β,β,',β ' ˆ + ( r ' ( r V nt ( r, r ' β ( r ( β ' ( r 'd 3 r ψ ' ˆ H 2nd quantzaton 2,β,',β ' = a +, < φ, h ˆ φ > a + j,β j,β ',, j, j',β, j a +,'' a +, < φ ',' ( r 'φ, ( r V nt ( r, r ' φ j,β ( r φ j',β ' ( r ' > a j,β a j,β ' Typcally the frst quantzed hamltonan has no spn dependence, then β = ˆ H 2nd quantzaton 2,' ',, j, j' = a +, < φ, h ˆ φ j, > a j, +, j a + ',' a +, < φ ',' ( r 'φ, ( r V nt ( r, r ' φ j, ( r φ j',' ( r ' > a j, a j',' (20 If we assume the sngle - partcle bass states are the egenstates of ˆ h, then ˆ H 2nd quantzaton 2,' ',, j, j' = a +, ε a, + a + ',' a +, < φ ',' ( r 'φ, ( r V nt ( r, r ' φ j, ( r φ j',' ( r ' > a j, a j',' 6
-> Fnd the egenstate of H In general t s not possble to fnd the exact energy egenstate of a many-body Hamltonan. One often guess a tral state, for example of a Fermon system, a tral ground state could be a sngle Slater determnat maded up of the lowest sngle partcle states subjected to the Paul-exlcuson prncple. For example, Be atom has four electrons; one may guess the grond state be (S (S (2S (2S, or n second quantzaton notaton: + Ψ >= a + 2S a + + S 0 > (2 The approxmate ground state energy s gven by E o tral =< Ψ ˆ H 2nd quantzed Ψ >=< 0 a S H ˆ 2nd a + S 0 > (22 quantzed The terms n H whch contrbute are the a s and a+ s whch match the labels n the state. For example, the sngle partcle term < 0 a S a,, + ε =< 0 a S a,, + ε a, + a + 2S a + + S 0 > a + 2S a + 2S a + S a + S a, 0 >= 0 f, S,S,2S,2S that s, f a j, s NOT any one of a + 2S a + 2S a + S a + S, then we can move a j, to the rght (generate a negatve sgn for every nterchange and act on the vacuum whch gves zero. In fact, a +, a, = n ˆ, s the number operator and a sngle Slater determnat ( Ψ > s an egenstate of the number operator,, a +, a, Ψ >= n ˆ, Ψ >= n, Ψ > (the last n, s a number = n ˆ, (23 Hernce, < 0 a S a,, j + ε a j, + a + 2S a + + S 0 >= 2ε S + 2ε 2S. For the par-wse nteracton, here are some of the non-zero contrbutons: 7
S a S <S ' S V nt S S ' > a S (The subscrpt,' s to remnd you that snce V nt has no spn - dependence, the orthogonalty of the spn states the nner states and the outer states must have the same spn state respectvely. Now we remove the subscrbe for smply the notaton. We wth move the to the left two space (every nchange gve one negatve sgn so that we can group the operators nto number operators, lke ths: = 2 <SS V nt SS > (a + S (a + S a S Smlarly for other terms: S <SS V nt SS > a S = 2 <SS V nt SS > (a + S a S (a + S 2S < 2S2S V nt 2S2S > = 2 < 2S2S V nt 2S2S > (a + 2S (a + 2S 2S < 2S2S V nt 2S2S > = 2 < 2S2S V nt 2S2S > (a + 2S (a + 2S For states wth the same spn (t has a drect term and an exchange term S 2S 2S S < 2SS V nt 2SS > = 2 < 2SS V nt 2SS > (a + S ( <S2S V nt 2SS > = 2 <S2S V nt 2SS > (a + S (a + 2S <S2S V nt 2SS > = 2 <S2S V nt 2SS > (a + 2S (a + S < 2SS V nt 2SS > = 2 < 2SS V nt 2SS > (a + 2S (a + S (Why don't we get an exchange term for unlke spns? + 8