Version 1.0 VLADIMIR V. KOROSTELEV. A Primer in Quantum Mechanics for NMR Students

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1 Vrson. VADMR V. KOROSTEEV A Prmr n Quanum Mhans for NMR Sudns

2 Vladmr Koroslv, 8 vladmr.v.koroslv@ramblr.ru Tabl of Conns Conns. nroduon. Quanum Sas of Spn / 3. Opraors for Spn / 6 4. Hamlonan of spn n h sa magn fld 8 5. Obsrvabls and Dns Mar 6. Ensmbl of spns ½ n a sa magn fld 4 7. Pulsd RF magn fld 6 8. Calulaon of NMR sgnal 7 Rfrns

3 . nroduon n lassal phss, h phsal sas of an obj of nrs an b dfnd al o a dgr, whh s manl lmd b prmnal faors suh as random and ssma rrors. Th masurmn of a phsal sa n quanum mhans s dffrn as nluds nrns unran, whh anno b nflund b mprovmns n prmnal hnqus (). Ths onp, orgnall proposd b Hsnbrg (), s alld h Unran Prnpl, whh sas ha h unrans of masurmn of nrg E and nrval of m, durng whh a mrosop parl posssss ha nrg, rla as: E h () whr h s Plank s onsan. Hn, onl h probabls of gng parular rsuls an b oband n quanum mhanal prmns, fundamnall dsngushng hm from h lassal ons. Th unrans n quanum-mhanal masurmns sm from h dsurban whh masurmn slf auss o h masurd sa a a mrosop sal. n NMR, h lf ms of spn sas do no gnrall d h spnla rlaaon m T, and hrfor h half-wdhs of NMR lns n spra mus b a las of h ordr of /T (3). Th unrans faurd n quanum-mhanal masurmns lad o a probabl nrpraon of phnomna, whr h quanum-mhanal sas ar dsrbd b wav funons gvn n parular rprsnaon. n Dra formulsm of quanum mhans (4) usd hroughou hs, a sa of a quanum mhanal ssm s dsrbd b vor alld k and wrn as. Th us of k nsad of wav funon allows h form of analss, whh s ndpndn of h parular rprsnaon hosn. n hs formulsm, dffrn rprsnaons ar rgardd as roaons n vor spa, hn h k rprsns quanum sa no mar wha rprsnaon s hosn for h analss. Th numbr of h k omponns s, whr s spn. Thus, for spn ½ k has wo omponns, ah of whh s a ompl numbr.. Quanum sas of a spn ½ Sngl spn ½ n sa magn fld ang along z-as has wo gnks: (spn along h fld) and (spn s oppos o h fld) dsrbd b olumns: () whn magnar par s zro h numbr s rprsnd b s ral par onl.

4 () n gnral, h sa of spn ½ ma b rprsnd b ombnaon of gnks wrn as h k whr (3), and ar ompl numbrs whh rla o probabls of a spn o b n h parular sa, hr n sa or n sa. Th gnks ar spal ks, whh ar normall orhogonal and rprsn h sas n whh a quanum-mhanal ssm an b found whn s sa s masurd. Th parular gnks an b hosn for dsrpon of a quanum mhanal sa, lk a fram of rfrn. Howvr, an hosn s of gnks mus b normalsd and ompl, n ordr o b appropra for a rprsnaon of a quanum mhanal sa. (Th ondons of orhogonal, normalzaon and omplnss of ks wll b prsnd hr.) A bra vor, wrn as s dfnd n spara spa of bras and whr and ma b rprsnd as: ( ) (4) ( ) (5) ( ) ( ) ( ) ( ) ( ) (6) Thus, wo salar produs an b formd (hr noaon of and s shorn o and rspvl 3 ): salar produs ar usd for spn sas of non-nrang spns. For sas of oupld spns nsor produs ar usd. 3 suprsrp sars dno ompl onjugaon 3

5 4 (7) For ampl: (8) (9) () () Also a mar of an opraor (mor on opraors wll b pland n n son) an b formd: () For ampl: (3) (4) (5) (6) Produs (3) and (6) ar mar rprsnaons of opraors alld projon opraors and dnod as Î and Î, rspvl,.. Î and Î (7-8).

6 Also, produs (4) and (5) ar mar rprsnaons of opraors alld shf opraors dnod as Î and Î, rspvl,.. Î and Î (9-). Opraor φ ϕ (n smlar wh ()) ma a on a bra from h rgh o form a ompl numbr ms h bra ϕ,.. φ ϕ g ϕ () whr g s a ompl numbr. For ampl, for ( k k ) and f φ h f quaon () drmns ompl numbr g as g k f k f, (whr k, k, f, f ar ompl numbrs). Or φ ϕ an a on a k from h lf o form a ompl numbr ms h k φ,.. whr for ( n n ) φ ϕ g φ (), ϕ and r, h ompl numbr g r n r n r (whr n, n, r, r ar ompl numbrs) aordng o h rul gvn b (7). Salar produ of sa wh slf (7) alwas lds a ral numbr and h posv squar roo of hs numbr s alld h norm of h sa vor. Whn sa s normalzd s norm s un,.. (3) Sas and ar normalzd as hr norms, and, rspvl ar qual o un as s shown n (8) and (). 5

7 Sas ar orhogonal whn hr produ gvs zro; for ampl, h sas and ar orhogonal as an b sn from (7) and (8). Whn a quanum mhanal ssm (a spn, for ampl) s ranl prsn n a s of gnks, h s s alld ompl and h sum of dagonal lmns of mar, gvn b produs of bras and ks s un. Ths sum rprsns h oal probabl of fndng a parl n h bass sas. Hn, f s of ks and s ompl hn sa gvn b (3) s ompl as wll and sum of dagonal lmns of mar gvn b () has o b un,.. (4) 3. Opraors for spn ½ A nw onp, spf o quanum mhans, rlas o h wa of alulang h valus of dnamal varabls masurd n prmn. n lassal phss, h dnam varabls ar normall dsrbd b funons, as opposd o quanum mhans, whr dnamal varabls ar rprsnd n rms of paon valus of hr opraors. An opraor s a smbol rprsnaon of a mahmaal opraon aomplshd on on or mor funons. Onl lnar opraors ar usd n quanum mhans and onl Hrman opraors (h ar lnar and qual o hr adjons) o rprsn dnamal varabls. n h mar vrson of quanum mhans, opraors ar rprsnd b squar mars, whos lmns orrspond o h dffrn sas of quanum mhanal ssm. Whn lnar opraor s appld o a k or bra rsuls n a nw k or bra, rspvl. Ths an b wrn for opraor  and ks and n h form: 6

8 A (5) whr and ar h ks, whh rprsn h orgnal and h nw quanummhanal sas, rspvl. For ampl, f opraor  s gvn b mar A: and b olumn vor: a a A (6) a a (7) hn a a a a A (8) a a a a n summar, lnar opraors ar usd for h quanum-mhanal dsrpon of phsal nraons and dnamal varabls, whl quanum-mhanal sas ar rprsnd b wav funons or ks, n Dra formulsm usd hr. A rsul of h masurmn of a dnamal varabl s on of h gnvalus of h opraor, whh rprsns hs dnamal varabl. Th gnvalus of a lnar opraor  an b found solvng s gnvalu quaon: A A (9) n whh lnks h known opraor  and wo unknowns: s gnvalus and hr orrspondng gnks. For ampl, h spn opraors Î, Î, Î z for spn ½ ar rprsnd b spn mars: (3) (3) z (3) 7

9 8 Hn, w an alula how hs opraor hang and sas, for ampl, usng (6): (33) (34) (35) (36) z (37) z (38) 4. Hamlonan of a spn n h sa magn fld Hamlonan opraor rprsns nrg of a quanum mhanal ssm and s usd for dsrpon of h phsal nraons and m voluon of h ssm. s gnvalus ar found b solvng h m ndpndn Shrödngr quaon, gvn b: n E H (39) Th unknowns n (.5) ar h gnvalus n E and h gnfunons. Whn a mndpndn Hamlonan s appld o a k, h lar hangs aordng o h mdpndn Shrödngr quaon: () H ι h (4).

10 Whn k s hr an gnk of h Hamlonan or lnar ombnaon of suh gnks, h soluon s: whr s h k a, and U (4), U H ι h () (4) s h propagaor, whh dsrbs h m voluon of a quanum mhanal ssm. Whn Hamlonan Ĥ s gvn n h uns, lk (3-3), hn (4) smplfs o: U H (43). Tm voluon of k s prssd as hang of vor k s ompons. Whn k dos no hang wh m s omponns ar ompl numbrs (somms ral numbrs, f magnar par of ompl numbr s zro) and f k dos hang wh m hn s omponns bom funons of m. Propagaors ar lnar opraors, and gnrall an b non-hrman. For a spn plad n sa magn fld ang along z-as, h Hamlonan opraor s gvn b (5): H Z (44) z whr (45) s h armor frqun, whh s h frqun of spn prsson n h sa magn fld and rprsns magnud of h sa magn fld. sds rnal sa magn fld spns n moluls prn loal flds, whh alr armor frqun and lad o hmall shfd armor frqun (6). Alhough hs ff s rual for NMR, s bond of sop of hs and hn wll b ngld. 9

11 Eampl: Fnd armor frqun of proon n h sa magn fld. T. For spn ½ mar of Hamlonan opraor gvn b (44) s: H Z (46) As mar of hs Hamlonan s dagonal h mar of orrspondng propagaor s dagonal oo: () U (47) n gnral, whn mar of Hamlonan s non-dagonal h mar of s propagaor gvn b hr (4) or (43) s non-dagonal as wll. For arbrar k gvn b (3) h soluon of m-dpndn Shrödngr quaon (gvn b (4)) an b found as a produ of propagaor mar and k (3): () () U (48) Soluon: T s rad p r hus, s rad , n Hz: [ ] [ ] MHz Hz π π ν.

12 Ths dmonsras ha as a rsul of voluon h omponns of k bom mdpndn hrough ponnals bng mulpld on omponns of nal k. 5. Obsrvabls and dns mar Quanum-mhanal sas and opraors rla o prmnal rsuls hrough obsrvabls. Ths ar dfnd as paon valus of opraors of dnamal varabls whos valus ar masurd n prmns. n quanum mhans, "an rsul of a masurmn of a ral dnamal varabl s on of s gnvalus. Convrsl, vr gnvalu s a possbl rsul of a masurmn of h dnamal varabl for som sa of h ssm"(7). Th paon valu of h obsrvabl for an opraor  an b rprsnd n h form: A A (49). Thus, h paon valu of opraor  gvn b mar a a A (5) a a an b ompud for h as whr k s A ( ) ( a a a a ) A a a as follows: a a a a a a (5). A good wa of sng how hs works s o do a parular ampl. Suppos ha w wan o alula h paon valu of oparaor Πz gvn b z (5)

13 whr a /, a, a and a -/. Thn z z (53). Usng (5) w an also work ou h paon valus of opraors Î and Î (54), (55). Gnrall, a quanum mhanal sa an b dsrbd b h quanum mhanal dns opraor, whh s gvn n mar form b (8): ρ (56), whr h ovrbar ndas akng an nsmbl avrag, whh mans addng up h onrbuons from ah spn n h sam nsmbl and hn dvdng b h numbr of spns. Ths mar s alulad n smlar wh () wha brngs

14 ρ (57). Th dagonal lmns of h dns mar rprsn populaons of h orrspondng quanum sas (9) and h off-dagonal lmns rprsn ohrns bwn orrspondng quanum sas. Ks and dns mars ar boh dsrbng quanummhanal ssms. Th dns mar of a sngl parl provds al h sam nformaon as s k (), and hrfor hr s quall aura for dsrpon of h sngl parl. Howvr, h us of h dns mar s spall onvnn for h dsrpon of nsmbls of man spns. Th m voluon of h dns opraor ρ s dsrbd b h ouvll-von Numann quaon (): [ H, ρ] Whn Ĥ s ndpndn of m, h soluon s gvn b: ρ d ρ (58). d H H ρ (59), whh s a produ of hr mars, H, ρ, H. Thn h paon valu of dnamal varabl for opraor  an b rprsnd usng dns opraor b (): A Tr A ρ (6), n mar rprsnaon Tr rprsns a sum of dagonal lmns of a mar found as a rsul of produ of wo mars,  and ρ. Ths dsrpon of h m voluon of h dns opraor, n whh opraors for dnamal varabls rman m ndpndn (), s rfrrd o as h Shrödngr rprsnaon. 3

15 Th us of dns mar nsad of ks for alulaon of h obsrvabls allows smplfaon of alulaons as for ks w hav o onsdr larg numbr of spns and hs s a man bod problm ha qukl boms unmanagabl du o larg numbr of alulaons rqurd whl onp of dns mar aks a dffrn look a h problm. dos no ask how prsl ah spn bhavs bu dsrbs sasall wha poron of spns whn an nsmbl s prsn n a parular sa (dagonal lmns of dns mar) and wha ohrns dvlop durng spn voluon (off-dagonal lmns). Suh horal smplfaon allows onsdraon of a sngl mar nsad of hug numbr of ks n ordr o fnd h valu of obsrvabl wh prson whh s whn a vr small sasal rror, and hrfor, s normall apabl. 6. Ensmbl of spns ½ n a sa magn fld n h absn of a sa magn fld a hrmal qulbrum, h spns ar soropall ornd. Howvr, whn a sa magn fld s appld (along h z as, b onvnon) mor spns n nsmbl ar ornd along h fld han agans. s onvnn o prss h dns opraor n h form (3): ρ a b z (6), whr h onsan a E, and Ê. Ths rprsns h dns opraor n h absn of h rnal sa magn fld, b z s h par of dns opraor proporonal o h srngh of h appld sa magn fld, gvn b: h b (6), ZkT whr s magnogr rao of nulus, h s Plank s onsan dvdd on π, s h magnud of h sa magn fld, Z s a paron funon whh s qual o for 4

16 spns ½, k s olzmann onsan and T s mpraur (K). Th rao of h spn sa populaons for and sas s dsrbd b (4): n n E kt (63), whr E s h dffrn bwn h nrgs of h sas. Eampl Fnd rao of spn sa populaons for proon nsmbl n h sa magn fld.4 T a h room mpraur (93K). Soluon: Th rao of spn populaons s gvn b: n E kt, whr E s nrg dffrn bwn and sas gvn n r b E h. p Thn E [ J ] 3 and k T [ J ]. Thn 6 E.85 E 6 6 kt and. 79. k T 4.43 Hn, numbr of spns n sa ( m ) s.79 ms largr han numbr of spns n sa ( m ). Sn hs rao hangs as a funon of, h n magnzaon of spn nsmbl bulds up along h appld sa magn fld (5). Ths s alld h qulbrum longudnal spn magnzaon M, whh s gvn for an nsmbl of N spns / b h Cur-Wss aw (6): r r Nµ M (64), kt whr µ s h magnud of h magn momn of a spn and T s h mpraur. 5

17 7. Pulsd RF magn fld Whn a magn fld osllang wh frqun qual o h armor frqun s appld prpndular o, s nrg an b absorbd b spns. Th absorbd nrg hangs h polarzaon of h spns and an ransform spn sas (7). Th armor frquns of nul n NMR prmn ar n h radofrqun (rf) rang, pall ns or hundrds of MHz (or ns and hundrds mllons of prsson ls pr sond). A rf fld, appld along h as n a Carsan fram of rfrn s dsrbd b: r RF os r ( φ ) (65). Th Hamlonan of rf puls, appld along as n h fram of rfrn roang abou h z as of h laboraor fram wh angular frqun n h sam sns as h armor prsson boms m ndpndn and an b dsrbd b: H ro RF h (66), whr onl on roang fld omponn nras wh h spns and s nrg s absorbd b spn nsmbl. Propagaor of hs rf puls appld for a m p s gvn b: U ro p () H RF (67). n mar noaon, whh s ofn usd n alulaons, (67) an b rprsnd as whr os sn () U (68), sn os p (68) s an angl hrough whh h spn magnzaon roas whn an rf puls wh a frqun a h armor frqun of h spns and of magnud s appld o h spns for a m p. 6

18 7 An rf puls whh roas h spns hrough 9 dgrs s alld a 9 dgr puls; s frqun, magnud and duraon dpnd on h nulus nvolvd and on h nsrumns usd n parular prmns. 8. Calulaon of NMR sgnal As NMR sgnal s proporonal o ransvrs spn magnzaon, whh s dd as magnzaon volvng n -plan and hs an b wrn n h form: Tr M ρ (69), whr mar of opraors dos no hang wh m and an b found as (7) Mar of ρ ausd b 9 dgr puls appld along -as, aordng o (59) an b found as a produ of hr mars: os sn sn os os sn sn os 9 ρ b a (7) whr 9. Mar n h mddl rprsns ρ, mars a lf and rgh ar propagaor and s ompl onjuga, rspvl. Afr mulplaon, h dns mar boms: os sn sn os sn os sn os 9 ρ b a (7)

19 8 Sn 9 h dns mar smplfs o: 9 b a ρ wha an b rwrn as: b a 9 ρ (73) Afr h rf puls s urnd off, h spns volv n h sa magn fld. Th Hamlonan of hs nraon s gvn n h uns b: z H (74) Hn, dns opraor volvs afr h rf puls, aordng o: ' ' 9 9 H H ρ ρ Thn dns mar s prssd b: 9 b a b a ρ (75) Afr domposon of ponnal: sn os (76), boms: sn os sn os 9 b a ρ (77). Ths an b rwrn n h form:

20 ρ ε. whr ( ) ( ) a b [ os( ε ( )) ( ε ( ))] (78), 9 sn Aordng o (69), ransvrs spn magnzaon an b found as a rsul of h followng: M Tr Tr () Tr( ρ() ( ) ( os( ε ( )) sn( ε ( ))) ( os( ε ( )) sn( ε ( ))) ( ( ) ( )) ( os( ε ( )) sn( ε ( ))) os ε sn ε Hn, ransvrs spn magnzaon fnall boms: (79) M (8) ε ( ) () ( os( ε ( )) sn( ε ( ))) 9

21 REFERENCES. W.H. ousll, Quanum Sasal Proprs of Radaon, John Wl & Sons,, W. Hsnbrg, Z. Phs. 43, 7 (97). 3. E.D. kr, Hgh Rsoluon NMR, Aadm Prss, 3, P.A.M. Dra, Th Prnpls of Quanum Mhans, Clarndon Prss, M. Goldman, Quanum Dsrpon of Hgh Rsoluon NMR n quds, Clarndon Prss, Oford, M.H. v, Spn Dnams, John Wl and Sons, 96,. 7. P.A.M. Dra, Th Prnpls of Quanum Mhans, Clarndon Prss, 36, M. Munowz, Cohrn and NMR, John Wl & Sons, 8, M.H. v, Spn Dnams, John Wl & Sons, 76,.. A.Whakr, Ensn, ohr and h Quanum Dlmma, Cambrdg Unvrs Prss, 86, R.R. Erns, G. odnhausn and A. Wokaun, Prnpls of nular magn rsonan n on and wo dmnsons, Clarndon Prss, Oford,, R.R. Erns, G. odnhausn and A. Wokaun, Prnpls of nular magn rsonan n on and wo dmnsons, Clarndon Prss, Oford, 3, R. Kmmh, NMR Tomograph, Dffusomr, Rlaomr, Sprngr, 43, E.D. kr, Hgh Rsoluon NMR, Aadm Prss, 9, M.H. v, Spn Dnams, John Wl & Sons, 79,. 6. E.D. kr, Hgh Rsoluon NMR, Aadm Prss,, M.H. v, Spn Dnams, John Wl & Sons, 58,.