Quantum Runge-Lenz ector and the Hydrogen Atom, the hdden SO(4) symmetry Pasca Szrftgser and Edgardo S. Cheb-Terrab () Laboratore PhLAM, UMR CNRS 85, Unversté Le, F-59655, France () Mapesoft Let's consder the Hydrogen atom and ts Hamtonan p H r, where p s the eectron momentum, ts mass, a rea postve constant, and r the dstance of the eectron from the proton ocated at the orgn. We assume that the proton's mass s nfnte. Casscay, from the potenta r, one can derve a centra force F r r that drves the eectron's moton. Introducng the anguar momentum one can further defne the Runge-Lenz vector Z : L r p, Z L p r r. It s we known that Z s a constant of the moton,.e. d dt ths condton reads Z t 0. Swtchng to Quantum Mechancs, H, Z 0. where, for hermtcty purpose, the expresson of Z must be symmetrzed Z L p p L r r. Here, departng from the basc commutaton rues between poston r, momentum nd anguar momentum L a n tensor notaton, we frst derve a usefu coupe of ntermedate denttes, then we demonstrate the foowng commutaton rues between the quantum Hamtonan, anguar momentum and Runge-Lenz vector Z H, L n 0 and H, Z n 0,
L m, Z n m, n, o Z o, Z m, Z n H m m, n, o L o. e Snce H commutes wth both L and Z, defnng M n H Z n, one gets the set of reatons (the frst one s part of the departure pont) L m, L n m, n, o L o, L m, M n m, n, o M o, M m, M n m, n, o L o. Ths set consttutes the Le agebra of the SO(4) group (cosey reated to a Poncaré group n speca reatvty). I Commutaton rues and usefu denttes Quantum commutaton rues bascs and the Hamtonan of the hydrogen atom Set macros for M restart; wth Physcs : wth Lbrary : nterface magnaryunt : macro M : Set the context: Cartesan coordnates, D Eucdean space, owercase etters representng tensor ndces, use automatc smpfcaton (automatcay appy smpfy/sze on everythng, before returnng t on the screen). Here r w represent the potenta of the centra force. Setup coordnates cartesan, hermtanoperators X, reaobjects,,, automatcsmpfcaton true, dmenson, metrc Eucdean, spacetmendces owercaseatn, mathematcanotaton true, quet automatcsmpfcaton true, coordnatesystems X, dmenson, () hermtanoperators X, mathematcanotaton true, metrc,,,,,, reaobjects,,, x, y, z, spacetmendces owercaseatn
Settng quantum (Hermtan) operators and reated commutators: - Z s Hermtan, but we derve that property further beow. - The potenta (X) of the hydrogen atom s assumed to commute wth poston, not wth momentum - the commutaton rue wth p s derved further ahead - The commutator rues for anguar momentum are an easy probem, we take them as the departure pont - The ast two commutators nvovng (X) are for the dfferenta operators approach ony, not reay part of the probem Setuuantumoperators Z, hermtanoperators,, H, L, X, p, agebrarues %Commutator, X 0, %Commutator, p 0, %Commutator, p KroneckerDeta k,, %Commutator L j, L k LevCvta j, k, n L n, %Commutator p j, L k LevCvta j, k, n, %Commutator X j, L k LevCvta j, k, n, %Commutator, X 0, %Commutator X, X 0, %Commutator, X 0 agebrarues X, X 0, L j, L k j, k, n L n, p j, L k j, k, n,, p X 0,, p 0, X j, L k k,,, X j, k, n,, X 0,, 0, hermtanoperators, H, L,, p, x, y, z, quantumoperators, H, L,, Z, p, x, y, z, () Defne the tensors Defne p x, p y, p z, L k L x, L y, L z, Z x, Z y, Z z, quet a, L k, a,,, a, g a, b,, a, b, a, b, c CompactDspay X, X X w now be dspayed as The Hamtonan for the hydrogen atom p H X M X w now be dspayed as H p () (4) (5) Identtes (I): n, X and 0
For more compact cacuus, we use the dmensoness potenta X X sqrt X o X o (6) The gradent of X s d_ n (6) n X o (7) So that subs rhs hs (6), (7) n (8) Equvaenty, X can be wrtten SumOverRepeatedIndces (6) x y z (9) from whch one can deduce X X X, that w often be used afterwards (9) x y z x y z x y z (0) subs rhs hs (9), x y z X, (0) X () And fnay X 0 SumOverRepeatedIndces daembertan (9) 4 x 4 y 4 z x y z 5 4 x y z () Factor () 0 () Identtes (II): the commutaton rues between L, nd the potenta
X One has LevCvta q, m, n m, n, q (4) Commutator (4), X, m, n, q, (5) %Commutator Commutator, X,, (6) %Commutator Commutator, X,, (7) At ths pont, setup dfferentaoperators and set some commutators wth an arbtrary functon (X), to be used n the aternatve demonstratons based on usng dfferentaoperators and to derve the commutaton rues between L, nd (X) Setup dfferentaoperators, x, y, z dfferentaoperators, X (8) Now, appy the dfferenta operators found n the commutators above to a generc functon X that w be removed afterwards hs AppyProductsOfDfferentaOperators@rhs (5) X, m, n, q (9) hs AppyProductsOfDfferentaOperators@rhs (6) X, hs AppyProductsOfDfferentaOperators@rhs (7) X, AppyProductsOfDfferentaOperators p X p X p p (0) () () Defne now the momentum operator as an ndexed procedure p proc oca Ind op procname ; return Physcs:-d_ Ind args ; end: (9), m, n, q n () So that SubsttuteTensor (8), () Inverse X, m, n, q (4) and fnay
Smpfy (4), 0 (5) (0) Inverse X, q (6) SubsttuteTensor (8), (6), (7) () Inverse X, q hs (8) Smpfy SubsttuteTensor (8), rhs (8), 5 q q (8) (9) To remove (), becomes () soate (), X from the equaton above, use (), whch, after defnng s a procedure n X Add these new rues to Setup (5), (7), (9), 0,,,, 5 p p (0) () () Setup () ; agebrarues L j, L k Now undo dfferentaoperators to work usng two dfferent approaches, wth and wthout dfferentaoperators Setup dfferentaoperators none dfferentaoperators none X 0,, p j, k, n L n,, 0, X j, L k k,,, 0,, 0, p j, L k j, k, n,, j, k, n,, p 0,, 5,,,, 0, () (4) II H, L n 0 and H, Z n 0 H, L n 0
Recang the Hamtonan of the hydrogen atom and the defnton of anguar momentum (4) (5); (4); H p m, n, q (5) We get Commutator (5), (4) H, m, n, q p, m (6) Smpfy (6) H, 0 (7) H, Z n 0 Settng up the probem and Z n s hermtan More dffcut. Start from the defnton of the quantum Runge-Lenz vector LevCvta a, b, k L a L b X M a, b, k L a L b (8) From that, one has the hermcty of Dagger (8) (8) a, b, k L a L b L b L a (9) Smpfy (9) 0 (40) Snce the system now knows about the commutaton rue between near and anguar momentum, %Commutator Commutator L a, L a, the expresson for can be smpfed Smpfy (8) a, b, n (4) (4)
a, b, k L b (4) and the anguar momentum removed from the defnng expresson for (4) m, n, q (4) Smpfy SubsttuteTensor (4), (4) (44) Here we set up the probem, to show that H, 0 Smpfy Commutator (5), (44) H, X m k 5 X e X p pk (45) Agebrac approach For the term wth 5 we use the derved dentty () () X (46) X () 5 X Xk (47) Norma (47) 5 X Xk (48) subs (48), (45) H, X p pk (49) Smpfy (49) H, 5 pk (50) Another term wth 5 appeared SubsttuteTensorIndces m, (48) 5 Xk (5) subs (5), (50)
H, pk (5) Make and be contguous to further appy () SortProducts (5),,, usecommutator H, k, m (5) Smpfy (5) H, (54) SubsttuteTensorIndces m, (46) pk (55) subs (55), (54) H, 0 (56) And ths s the resut we wanted to prove. In the next secton there s an aternatve dervaton that coud be seen as more abstract or more drect Aternatve approach usng dfferenta operators As done n the prevous secton when dervng the commutators between near and anguar momentum, on the one hand, and the centra potenta on the other hand, the dea here s agan to use dfferenta operators takng advantage of the abty to compute wth them as operands of a product, that get apped ony when t appears convenent for us Setup dfferentaoperators, x, y, z dfferentaoperators, X (57) So take the startng pont (45) (45) H, X m k 5 X e X p pk (58) and to show that the eft-hand sde s equa to 0, mutpy by a generc functon X foowed by transformng the products nvovng by the appcaton of ths dfferenta operator (45) X H, 5 X X p pk (59)
AppyProductsOfDfferentaOperators (59) H, k k m (60) m m X m k k k k X 5 X Smpfy (60) H, k k X m k k (6) 5 X k k X Use now the derved denttes for the gradents of and and then remove the generc functon from the equaton by mutpyng by the nverse of (8); (); n p (6) Smpfy SubsttuteTensor (8), (), (6) Inverse X H, X k X pk m 5 X (6) To show that the rght-hand sde s actuay 0, recang () () X (64) rhs hs () X pk (65) Ths and usng (48) agan Smpfy SubsttuteTensor (65), (48), (6) H, 0 (66) Reset dfferentaoperators n order to proceed to the next secton workng wthout dfferenta
operators Setup dfferentaoperators none dfferentaoperators none (67) III L m, Z n Strategy: m, n, k.. Express L m and n terms of and from prevous sectons Construct the eft-hand and rght-hand sdes of, prove. Smpfy the resut k, q, u Z, the formua we want to u Step s gven by (4) m, n, q (68) (44) (69) Step. The eft-hand sde of the dentty to be proved s the eft-hand sde of ths commutator Commutator (4), (44), X m m, n, q m e b, m k, n k, n k, m a, n k, m a, m The rght-hand sde of the dentty to be proved s the eft-hand sde of ths equaton LevCvta q, k, u SubsttuteTensorIndces k u, (44) p k, q, u Z u k, q, u u X u p u X u (70) (7) Step. Take one mnus the other one and the rght-hand sde must be equa to 0 (70) (7),, q, u u X m m e k, n m, n, q k, m (7) a, m m, n, q X u k, q, u k, n m, n, q X u k, q, u
b, m m, n, q a, n m, n, q p u k, q, u k, m m, n, q p u k, q, u Smpfy (7), k, q, u Z u 0 (7) I Z m, Z n H m, n, o L o Agebrac approach Here agan the startng pont s (44), the defnton of the quantum Runge-Lenz vector SubsttuteTensorIndces k q, (44) Z q (74) In ths secton the strategy s the same as n the prevous secton: construct the eft-hand and rghthand sdes of the dentty to be proved (n the tte of ths secton), take one mnus the other one, and show that the rght-hand sde s equa to 0. Start wth the eft-hand sde of the dentty to be proved Commutator (44), (74) b, m m, q (75) k, q b, k p n b, k X b X b 5 X b X b a, k a, m n, q p b m, q a, m a, n k, q p n m, q a, k k, q k, q a, k n, q
5 k, q a, k k, q pm m, q Now the rght-hand sde, constructed from the defnton of the anguar momentum (4) m, n, q (76) M H LevCvta q, k, u SubsttuteTensorIndces q u, (4) k, q, u H L u k, q, u m, n, u H (77) Repace the Hamtonan H by ts expresson quadratc n the momentum (5) H p (78) hs (77) SubsttuteTensor (5), rhs (77) k, q, u H L u p m, n, u k, q, u (79) Now set up the probem, takng (75) mnus (79), workng towards havng t n fna form, ready for sovng, n (8) (75) (79) k, q, u H L u (80) m a, n e m p a n, q p b e m, q a, m k, q a, k b, k p n a, k a, m p m, n, u k, q, u k, q a, k b, k k, q pm
m, q p b b, m m, q a, k k, q k, q p n X b X b 5 X b X b n, q 5 m, q k, q So the startng pont to prove that k, q, u H L u s Smpfy (80) k, q, u H L Z u k 5 Xk (8) X Xk a X Xq m g k, q p pm b g k, q p pk a were the proof s acheved showng that the rght-hand sde of ths equaton s ndeed equa to 0. Start checkng the repeated ndces, as we woud do by hand Check (8), a The products n the gven expresson check ok. The repeated ndces per term are:...,...,... ; the free ndces are:... u, k, q a, b, m, n, k, q Check n whch terms - that nvove - are these repeated ndces appearng for term n seect has, map op, ndets (8), ` `, do term Check term, repeated, quet od (8) X Xq m m m X Xk a a
a 5 Xk a (8) By eye, m a resuts n a smpfcaton n that two terms become equa SubsttuteTensorIndces m a, (8) k, q, u H L Z u k 5 X Xk m a e (84) Xk Xq g k, q pa g k, q pk Smpfy (84) k, q, u H L Z u k m e One more round of the same: Check (85), a The products n the gven expresson check ok. SubsttuteTensorIndces a n, (85) k, q, u H L Z u k m e Xk Xq pk The repeated ndces per term are:...,...,... ; the free ndces are:... u, k, q a, n, k, q Xk Xq pk (85) (86) (87) Smpfy (87) k, q, u H L u (88) X Xq p n k X pq n Run now a oop re-sortng products n the key terms of the rght-hand sde above, approxmatng X and (move p to the eft). On the way, create a tabe of hepfu substtuton equatons eq tabe : for term n seect has, map op, ndets (88), ` `, do eq term term SortProducts term,,,,,, usecommutator, totheeft od
eq Xq k, q eq Xk pq n, q k, q X n eq Xk k, q eq Xn Xq k, q X n k, n (89) Substtute now subs convert eq, st, (88) k, q, u H L u There are st terms contanng X to be smpfed. So agan from () X k, n n, q and notng that these two commute k, q X n k, q %Commutator Commutator X, X, X 0 k, q k, q X n (90) (9) (9) rewrte () swtchng the order X X (9) (9) pk (94) Substtute now subs (94), (90) k, q, u H L u k, n n, q k, q X n k, q Smpfy (95) k, q, u H L u k, q X n k, q (95) (96)
Fnay, removng the ast term that nvoves SubsttuteTensorIndces n, () X Xk n (97) Norma (97) pq (98) subs (98), (96) k, q, u H L Z u k 0 (99) whch s the dentty we wanted to prove. In the next secton the same resut s obtaned usng dfferenta operators Aternatve approach usng dfferenta operators The man dea: make be a dfferenta operator, then (8) by a generc functon (X), appy the products of dfferenta operators, then use tensora smpfcatons: (8); (); (); n p X (00) The goa s agan to show that the rght-hand sde of (8) s equa to 0, so set to be a dfferenta operator, mutpy (8) by a generc functon X foowed by appyng where t corresponds Setup dfferentaoperators, x, y, z dfferentaoperators, X (0) (8) X k, q, u H L Z u k 5 Xk (0) Xk Xq g k, q pm g k, q pk Appy now AppyProductsOfDfferentaOperators (0) k, q, u H L Z u k g k, q a (0)
g k, q m q q k k m m m q k Xk q Xq k a a 5 Xk q q k k a a We want to show that the rght-hand sde s equa to 0; start smpfyng wth respect to agebra rues and usng Ensten's sum rue for repeated ndces Smpfy (0) k, q, u H L u m Z e k X (04) m k q e a k k q 5 Xk Xk q Xk q X Xk a q X Xk a q X Xq m k Xq k Xq k Xq k Next, usng (8), the gradent of can be removed (8) n SubsttuteTensor (05), (04) k, q, u H L u q (05) (06) k 5 Xk Xk Xk Xk Xk q Xq Xq Xq Xq k Smpfy (06) k, q, u H L u (07) q k Xk q Xq k
From () the gradent of can be removed () SubsttuteTensor (), (07) k, q, u H L u m Z e k p (08) (09) Xk Xq By eye there are terms that are smar; check the repeated ndces Check (09), a The products n the gven expresson check ok. The repeated ndces per term are:...,...,... ; the free ndces are:... u, k, q a, m, k, q SubsttuteTensorIndces a m, (09) k, q, u H L u m Z e k (0) () Xq pq And there are st two terms of the form X that can be removed usng () SubsttuteTensorIndces m, () () () X ; Xq () rhs hs () X ; Xk Smpfy subs (), (4), () Inverse X k, q, u H L Z u k 0 (4) (5) Whch s aready the resut we wanted to obtan.