The Infinite Square Well Problem in the Standard, Fractional, and Relativistic Quantum Mechanics

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1 Iteatoal Joual of Theoetcal ad Mathematcal Physcs 015, 5(4): DOI: /j.jtmp The Ifte Squae Well Poblem the Stadad, Factoal, ad Relatvstc Quatum Mechacs Yuchua We 1, 1 Iteatoal Cete of Quatum Mechacs, Thee Goges Uvesty, Cha Depatmet of Radato Ocology, Wake Foest Uvesty, NC Abstact Thee has bee a cotuous agumet o the coectess of the Lask s soluto fo the fte squae well poblem the factoal quatum mechacs. I ths pape, we pove that the Lask s fuctos ae ot amathematcal soluto to the factoal Schödge equato ad the equato does ot have ay ozeo solutos at all the sese of mathematcs. As the stadad quatum mechacs, we vew the fte squae well poblem as the lmt of the fte well poblem, ad defe the soluto fo the fte squae well poblem as the lmt of the soluto fo the fte squae well poblem. Usg the smple popety of the fte opeatos, we show that Lask s fucto ca be the lmt soluto fo the fte squae well poblem. The sgle-sded well poblem ad the 3 dmesoal well poblem ae cluded. Ths opeato method woks fo the same poblems the elatvstc quatum mechacsas well. Keywods Factoal quatum mechacs, Relatvstc quatum mechacs, Factoal Schödge equato, Relatvstc Schödge equato 1. Itoducto I 000, Lask toduced the factoal quatum mechacs [1-3]. As a example he solved the fte squae well poblem a pecewse fasho [3]. Howeve, 010 Jeg, et al [4] ctczed that t was meagless to solve a olocal equato a pecewse fasho ad they demostated that t was mpossble fo the goud state fucto to satsfy the factoal Schödge equato ea the bouday sde the well. I a sees of papes [5-8], Bay ssted that he explctly completed the calculato Jeg s pape ad the wave fuctos dd satsfy the factoal Schödge equato sde the well. Hawks ad Schwaz [9] clamed that the Bay s calculato cotaed seous mstakes. Luchko [10] povded some evdece that the soluto dd ot satsfy the equato outsde the well. O the othe had, Dog [11] e-obtaed the Lask s soluto by solvg the factoal Schödge equato wth the path tegal method. It s ot easy fo eades to judge the mathematcal agumet [1, 13], but weagee wth Jeg s opo, cludg that the pecewse way to solve the equato s wog, ad that the soluto does ot satsfy the factoal Schödge equato, sce we wll aguably show that the Lask s fuctos do ot satsfy the factoal Schödge equato ( 1) aywhee o the x-axs. Howeve, fact, ths soluto does ot satsfy the * Coespodg autho: yuchuawe@gmal.com (Yuchua We) Publshed ole at Copyght 015 Scetfc & Academc Publshg. All Rghts Reseved stadad Schödge equato ethe, ad t s othg but the lmt of the soluto to the fte squae poblem. Wthout a mathematcal defto of the egevalues ad egefuctos of a Hamltoa opeato wth local fty, the agumet wll be edless. Theefoe, ths pape we mathematcally defe the fte squae well poblem as a lmt of the fte well. Ths vewpot s also useful fo othe potetals wth fty, such as the coulomb potetal the hydoge atom, the sgle-sded hamoc oscllato, etc. Sce t s dffcult to solve the factoal Schödge s equato wth a fte squae well potetal, we wsh a dect way to fd the soluto of the fte squae well poblem. Fotuately, we ca expess the factoal Hamltoa tems of the stadad Hamltoa. I the same way, we ca also solve the fte squae well poblem the elatvstc quatum mechacs, sce the factoal ad elatvstc quatum mechacs ae closely elated. We ackowledge that these solutos eed to be vefed whe the solutos to the fte squae well poblem ae epoted late. We fst ecall the fte squae well poblem stadad quatum mechacs, ad the solve the poblem the factoal quatum mechacs ad the elatvstc quatum mechacs.. Defto of the Ifte Well Poblem I ths secto we wll ecall the elato betwee the fte ad fte squae well poblems the stadad quatum mechacs, ad accodgly defe the fte squae well

2 Iteatoal Joual of Theoetcal ad Mathematcal Physcs 015, 5(4): poblem the factoal quatum mechacs..1. The Fte ad Ifte Potetal Wells the Stadad Quatum Mechacs I the stadad quatum mechacs [14], the oe dmesoal tme-depedet Schödge equatos H x E x, (1) whee x s a wave fucto defed o the x-axs, ad E s a eegy. The Hamltoa opeato H T V x () s the summato of the ketc eegy ad the potetaleegy of a patcle. The stadad ketc eegy opeato s p d T m m, (3) whee p s the oe dmesoal mometum opeato d p. (4) As usual, m s the mass of the patcle ad ħ s the educed Plak costat. A fte squae well [15] s defed as V f x 0 x a V0 x a whee V0 0 s the depth of the well ad a s the wdth of the well. The subscpt f meas the fte squae well. The egeequato of the fte squae well poblem s (5) d V f x x E x m. (6) Ths equato ca be solved sepaately thee egos fst ad the the pecewse solutos ae coected by the cotuty codto that the wave fucto ad ts devatve must be cotuous. Ths pocess geeates the egevalues Ef ( x ) ad egestates f( x ), wth 1,,3. The state wth =1 s the goud state [15]. Oe ca easly vefy that the explct solutos satsfy the egeequato at evey pot o the whole x-axs d f Vf x f x E f x m, (7) whch covces us that the cotuty codto we use s sutable. I fact ths cotuty codto comes fom the Schödge equato. The fte squae well potetal s defed as V x 0 x a x a At x a, the potetal has two fte jumps. The subscpt meas the fte squae well. Obvously the potetal V s ot eal physcs ad ot well-defed mathematcs, sce we do ot kow how to deteme the value of the multplcato V x() x outsde the well, eve f the wave fucto outsde the well s zeo. (Does 0 equal 0,1, o?) I physcs ths potetal s used as a smplfed model fo a vey deep fte squae well. Covetoally the Schödge equato (8) d V x x E x m (9) s solved a pecewse way aga [15]. The wave fucto outsde the well s zeo because the potetal s fte ad the wavefucto sde the well s a se o cose fucto. The evsed cotuty codto fo ths case s that the wave fuctos must be cotuous but the devatve should ot be. The fte squae well poblem has the smple ad well-kow soluto x 1 s x a x a a a 0 x a, (10) E (11) 8ma fo 1,,3,. Howeve, we emphasze that ths soluto does ot satsfy the Schödge Equatos (9). Take the goud state as a example, we have d m 1 1 V x x E1 1 x x a x a 4ma a ( ) ( ). (1) The wave fucto satsfes the Schödge equato eveywhee except at x=a ad x=-a. Ths s ot a small flaw fo a dffeetal equato, ad we have to say that the wave fuctos (10) ae ot the soluto of the Schödge equato at all. The exta tems the above equato comes fom the dscotuty of the devatve of the wave fucto, but f we keep the devatve of the wavefucto cotuous, we ca oly get a tval soluto x 0, whch s meagless physcs. We have to say that the Schödge equato wth the fte squae well potetal does ot have solutos mathematcs. (See the appedx f oe feels

3 60 Yuchua We: The Ifte Squae Well Poblem the Stadad, Factoal, ad Relatvstc Quatum Mechacs supsed.) Sce the fte squae well s a lmt of the fte squae well, covetoally we call the lmt of the solutos of the fte squae well poblem lm ( x) ( x), lm E ( x) E, (13) f f V0 V0 the soluto of the fte squae well, ad symbolcally wte d V x x E x m. (14) The pecewse way to solve the Schödge equato fo the fte squae well, togethe wth the afoemetoed evsed cotuty codto, s othg but a easy ad dect way to get the lmt of the fte squae well soluto wthout solvg the fte squae well poblem. Ths defto fo the fty potetal s applcable fo seveal mpotat cases quatum mechacs. It helps us to aswe the followg questos: (1) what sese the wave fuctos of the hydoge atom [14]ca satsfy the Schödge equato at the og 0 whee the coulomb potetal V ( ) 1/ s udefed. ( s the dstace of a pot to the og.) () why the wave fuctos fo the potetal V ( ) 1/ / [14], whee 0 s much smalle tha 1, ca be dveget at the og whle we clam that the wave fuctos should be be cotuous ad dffeetable eveyday, (3) why oly the odd (athe tha eve) ege-fuctos of the smple hamoc oscllato [15] ae the solutos of the sgle-sded oscllato V( x) 1 m x x 0. (15) x 0 (4) why half-se fuctos ae the soluto of the sgle-sded fte squae well ( o fte wall) but half-cose fuctos ae ot. We wll expla ths case futhe the secto III. I oe wod, the dffeet cotuty codtos of wave fuctos quatum mechacs ae used ode that oe ca dectly fd the lmt soluto fo a potetal wth fty, stead of solvg the Schödge equato wth a coespodg fte potetal fst ad the calculatg the lmt of the solutos... The Lask s Soluto of the Ifte Squae Well I 000, Lask geealzed the classcal ketc ad mometum elato to 1 p d T D p mc D mc /, (16) whee the coeffcet D mc / ( mc) wth χ a postve eal umbe, ad c s the speed of the lght. Whe, takg 1/ ad hece D 1/ ( m), the factoal ketc eegy ecoves the stadad ketc eegy,.e. T p / ( m) T. Ogally Lask [1-3] equed the factoal paamete 1, but ths pape we allow 0. The factoal Schödge equato s x H x E (17) H T V. (18) Fo the fte squae well poblem, we have d D ( ) ( x) V x x E x. (19) Lask [3] solved the equato above a pecewse fasho, ad got the same wave fuctos ad a ew eegy level x 1 s x a x a a a 0 x a, E (0) D a. (1) Howeve, 010 Jeg, et al [4] ctczed that t was meagless to solve the olocal equato (19) a pecewse fasho ad by cotadcto they demostated that the goud state fucto dd ot satsfy the factoal Schödge equato. We beleve that Jeg et al s coect ad Bas s wog ths agumet. Thee ae two easos. (1) The aguable evdece s that the soluto does ot satsfy the factoal Schödge equato ( 1 ) aywhee o the x-axs. Takg the goud state as a example, we have the fact d 1/ d D ( ) ( x) D ( x) *(1/ x) D x x x S( ) S( ) cos a a a a a 1 D x x x C( ) C( ) s. a a a a a 1 () The above fucto s ot popotoal to 1 ( x) sde the well, t s ot zeo outsde the well, ad t s dveget at the boudaes x a. Hee S ad C ae se ad cose tegal fuctos, espectvely. () I fact, the factoal Schödge equato wth the fte squae well (19) does ot have ay

4 Iteatoal Joual of Theoetcal ad Mathematcal Physcs 015, 5(4): ozeo solutos. Sce the potetal V( x ) s fty, to avod V ( x) ( x) be fty, we have to let ( x) be zeo outsde the well, that s, ( x) s a compactly suppoted fucto. Thuswe have V ( x) ( x) 0 o the x-axs (f we take 0 0). Fo 0, the ketc eegy opeato s olocal, so the esultat fucto D p ( x) wll be exteded outsde the well. Theefoe Equato (19) eques a compacted fucto to equal a o-compacted fucto. Ths s mpossble. Theefoe thee ae o ozeo solutos to the factoal Schödge equato (0 ) wth a fte squae well potetal. By the way, Luchko [10] eve had a cojectue that the soluto fo the fte squae well poblem factoal quatum mechacs should be teated as a ew specal fucto. Now let us defe the fte squae well poblem ad ts soluto the factoal quatum mechacs mathematcally. Defto 1. The soluto of the fte squae well poblem. Suppose the factoal Schödge equato fo the fte squae well potetal d ( ) ( ) f f f f D x V x x E x has the solutos f( x ) ad E f wth 1,,3,. If the lmts exst lm ( x) ( x), lm E E, (3) f f V0 V0 we say that they ae the solutos fo the fte squae well poblem, ad symbolcally wte d ( ) ( ) D x V x x E x. Obvously t wll be dffcult ad complcated to solve the fte squae well poblem accodg to the above defto. Befoe ay stct solutos have bee epoted the publcatos, let us develop a tetatve way to solve the fte squae well poblem based o some tutve popety of the fty opeato. 3. The Ifte Squae Well Poblems Factoal Quatum Mechacs Based o the elato betwee the stadad ad factoal Hamltoas, oe ca costuct a o-pecewse method fo the fte squae well. Thee ae thee cases, (1) oe dmesoal, () thee dmesoal ad (3) oe-sded fte squae well poblems The 1D Ifte Squae Well Poblem Fo the 1D fte squae well poblem, the factoal Hamltoa H T V ( x) (4) ca be expessed tems of the stadad Hamltoa as H T V ( x). (5) / / H ( m) D H. (6) Hee s the deducto of the above opeato elatoshp / / / ( m) D H ( m) D T V x / / ( m) D T x a / / ( m) D V x x a / D ( p ) x a x a T x a x a 0 x a T T x a 0 x a T x a T V H. Theefoe we have / / / / / H ( x) ( m) D H ( x) ( m) D E ( x) D a ( x) (7). (8) We see that the wave fuctos of the factoal Hamltoa ae the same fuctos of the ogal Hamltoa, ad the ew eegy levels ae E D a. (9) That s to say, the Lask s soluto s e-obtaed a o-pecewse fasho. I the above deducto, we use some fomal opeatos of the fty opeato, such as T / T. (30)

5 6 Yuchua We: The Ifte Squae Well Poblem the Stadad, Factoal, ad Relatvstc Quatum Mechacs 3.. The 3D Ifte Squae Well Poblem I quatum mechacs, the 3D fte squae well s defed 0 x a, y a, ad z a V() Vx, y, z (31) othewse. Hee ( x, y, z) s a pot the 3D Eucldea space. Please otce that the 3D fte squae well potetal s the summato of thee oe dmesoal fte squae wells, x y z x y V,, V V V z (3) sce 0 0 0,0, ad. By the way, a 3D fte squae well potetal does ot have such a smple elato, so t s vey dffcult ad complcated to solve. Hee we see that the use of the fty potetal ca geatly smplfy the poblem wth a vey bg fte potetal. The stadad Hamltoa s p H V ( ) V ( x, y, z) (33) m mx x x whee p s the 3D mometum opeato. The Schödge s equato has the soluto H( x, y, z) E ( x, y, z) (34) ( x, y, z) ( x) ( y) ( z) (35) m l ( Elm m l ), (36) 8ma wth, m, l 1,,3. ( x ) s defed (10) The factoal Hamlto s p / H D V () /. (37) D V() x x x Aga, we have the same elato betwee the stadad ad factoal Hamltoa / / H ( m) D H (38) Theefoe, the factoal Schödge equato has the soluto H ( x, y, z) E ( x, y, z) (39) ( x, y, z) ( x) ( y) ( z) (40) m l Eml D ( m l ) a /, (41) Wth, m, l 1,, Oe-sded Ifte Squae Well The 1D oe-sded fte squae well s defed as U x The stadad Hamlto s The Schödge equato has the soluto x 0, (4) 0 x > 0. p H U x. (43) m p H( x) ( x) U x ( x) (44) m 0 x 0 ( x) s( kx) x 0 (45) k E (46) m wth k 0. Hee we epeat that the wave fucto (45) does ot satsfy the Schödge equato (44) at x=0. The oe atual questo s why we do ot call 0 x 0 ( x) cos( kx) x 0 a soluto. I fact, fo a fte sgle sded squae well U f x the soluto to the Schödge equato s f (47) V 0 x 0, (48) 0 x>0, Aexp( x) x 0 ( x) s( kx 0) x 0 Hee the paametes k 0 ad 0 (49) mv / k. The othe paametes A ad 0 ae chose so that ths soluto s cotuous ad dffeetable at x=0. It s easy to vefy that V0 0 x 0, lm f ( x) s( kx) x 0. (50) Ths s the easo why the half-se fucto (45) s a soluto, but the half-cose fucto ( x) (47) s ot. The factoal Hamltoa s H D T U x. (51)

6 Iteatoal Joual of Theoetcal ad Mathematcal Physcs 015, 5(4): Based o (6), the factoal Schödge equato has the soluto wth k 0. H ( x) D T ( x) U x ( x) (5) 0 x 0 ( x) s( kx) x 0 (53) E D k (54) 4. Ifte Squae Well Poblems Relatvstc Quatum Mechacs Accodg to specal elatvty [14], the elato betwee the ketc eegy The subscpt meas specal elatvty. The elatvstc Hamltoa [14] s 4 T ad the 3D mometum p s T p c m c. (55) 4 H p c m c V. (56) I Summefeld s quatum theoy [14], ths Hamltoa leaded to a eegy fomula fo the coulomb potetal, whch accuately matches the hydoge spectum wth the fe stuctue. Accodgly, the elatvstc Schödge equato s 4 (, t) c m c (, t) V (, t) t p. (57) The squae oot opeato above s defed by the Foue Tasfomato of the wave fucto, p c m c (, t) 3 3d exp( / ) c m c 3exp( '/ ) ( ', t) d ' ( ) p p p R p. (58) R Obvously t s dffcult to deal wth ths equato mathematcally [14], so ths equato had bee abadoed utl ecetly whe we dscoveed that ts eegy fomula 5 has a extemely valuable tems, whch s 41% of the expemetal Lamb shft [16, 17]. As the oly excepto ths pape, the otato stads fo the fe stuctue costat athe tha the factoal ode. Fo a low speed moto, the elatvstc equato (57) ecoves the Schödge equato (, t ) mc p (, ) (, ) t m t V t, whose factoal paamete. Fo a vey hgh speed moto, f the est eegy ca be eglected appoxmately, we have (, t) c (, t) V (, t) t p, (59) whch s the factoal Schödge equato wth 1. Geeally speakg, f the speed of a patcle ceases fom low to hgh, the elatvstc Schödge equato (57) wll appoxmately elate to a factoal Schödge equato, whose paametes chages fom to 1. Theefoe the factoal ad the elatvstc Schödge equato should be studed at the same tme as two sste equatos The 1D Ifte Squae Well Fo the oe-dmesoal fte squae well poblem, the elatvstc Hamltoa s 4 H p c m c V x d 4. c m c V x (60) Aga, we ca expess the elatvstc Hamltoa tems of the classcal Hamltoa as sce 4 H mc H m c, (61)

7 64 Yuchua We: The Ifte Squae Well Poblem the Stadad, Factoal, ad Relatvstc Quatum Mechacs 4 mc H m c mc V ( x) m c m d 4 d 4 c m c V ( x) H. (6) See the stepwse deducto (7) fo detals. Theefoe the Lask s wave fuctos ( ) satsfy the elatvstc Schödge equato wth a ew eegy level E x H ( x) E ( x) (63) c 4 m c (64) 4a fo 1,,3,. Fo the exteme elatvstc case, we have 4.. The 3D Ifte Squae Well Poblem E The elatvstc Hamltoa c. (65) a 4 H p c m c V (66) ca be expessed tems of the classcal Hamltoa as p H V m 4 H mc H m c. (67) Theefoe, the elatvstc Schödge equato has the soluto H ( ) E ( ) (68) ( x, y, z) ( x) ( y) ( z) (69) m l c Eml ( m l ) m c a The Oe-sded Ifte Squae Well Poblem. (70) The elatvstc Hamltoa fo the oe-sded squae well s d 4. H c m c U x (71) Based o the elato betwee the elatvstc stadad Hamltoas (61), we kow that the elatvstc Schödge equato has the soluto 5. Coclusos 0 x 0 ( x) s( kx) x 0 4 (7) E k c m c. (73) We agee wth Jeg et al that the factoal Schödge equato caot be solved a pecewse fasho, ad the Lask s fuctos ae ot a mathematcal soluto of ths equato. O the othe had, sce these fuctos ae ot the mathematcal soluto of the stadad Schödge equato ethe ad ae just the lmt of the solutos of the fte squae well, to dspove the Lask s soluto, oe eeds to solve the fte squae well poblem ad take the lmt of the solutos. Befoe the soluto of the fte squae well poblem s epoted, we develop a tetatve dect method to solve the fte squae well poblem thee cases usg the staghtfowad popety of the fty opeato. Meawhle, we pot out that the elatvstc quatum mechacs s a appoxmate ealzato of the factoal quatum mechacs ad the fte squae well poblem ca be teated a same way. Note 1. We just otced that the pape [18] epoted some tal eseach o the oe dmesoal elatvstc fte squae well, but to utlze the exstg mathematcal esults, they used a completely dffeet defto o the fte squae well, whch had o elato to the fte squae well. They defed 0 0 the multplcato V ( x ), ad atfcally demaded the esultat fucto of the Hamltoa opeato to become zeooutsde the well though they ae actually ot. Fom the vewpot of physcs, they chaged the fte squae well poblem to aothe oe. Note. I [4], thee wee two seteces that By asg that Hamltoa to the powe α/, we get a plausble factoal Laplaca ad Eq. (7) s deed a soluto. Howeve, ths s ot the Resz factoal devatve. These wods emd us that Jeg et al kew the method used ths pape but thought t dd ot wok sce the opeato ths method was ot Resz factoal devatve. I fact, ths s the Resz factoal devatve, see Equato (7) [1] fo the defto. ACKNOWLEDGEMENTS Ths wok s suppoted by the Natoal Natual Scece Foudato. The eseach o the elatvstc Schödge equato was suppoted by Gasu Idusty Uvesty dug , wth a subject ttle O the solvablty of a equato wth a squae oot opeato elatvstc quatum mechacs. Thaks fo the suppot fom my famly as well.

8 Iteatoal Joual of Theoetcal ad Mathematcal Physcs 015, 5(4): Appedx 1. Patcle a box Usually quatum mechacs t s also demaded that the devatve of the wavefucto addto to the wavefucto tself be cotuous; hee ths demad would lead to the oly soluto beg the costat zeo fucto, whch s ot what we dese, so we gve up ths demad (as ths system wth fte potetal ca be egaded as a ophyscal abstact lmtg case, we ca teat t as such ad "bed the ules"). Note that gvg up ths demad meas that the wavefucto s ot a dffeetable fucto at the bouday of the box, ad thus t ca be sad that the wavefucto does ot solve the Schödge equato at the bouday pots x = 0 ad x = L (but does solve eveywhee else). a_box Appedx. A Alteatve Defto o the Ifte Squae Well Poblem Defto. The soluto of the fte squae well poblem. We defe a potetal x Vm x V0 a m wth V 0 >0 ad a>0, m=1,,3. Suppose the factoal Schödge equato fo the fte potetal V x m d ( ) ( ) m m m f D x V x x E x has the solutos m( x ) ad E m wth 1,,3,. If the lmts exst lm ( x) ( x), lm E E, m m m m we say that they ae the solutos fo the fte squae well poblem, ad symbolcally wte d ( ) ( ) D x V x x E x. REFERENCES [1] N. Lask, Factoal quatum mechacs, Physcal Revew E 6, pp (000). [3] N. Lask, Factoal ad quatum mechacs, Chaos 10, pp (000). [4] M. Jeg, S.-L.-Y. Xu, E. Hawks, ad J. M. Schwaz, O the olocalty of the factoal Schödge equato, Joual of Mathematcal Physcs 51, 0610 (010). [5] S. S. Bay, O the cosstecy of the solutos of the space factoal Schödge equato, J. Math.Phys. 53, (01). [6] S. S. Bay, Commet O the cosstecy of the solutos of the space factoal Schödge equato, Joual of Mathematcal Physcs 53, (01). [7] S. S. Bay, Commet O the cosstecy of the solutos of the space factoal Schödge equato, Joual of Mathematcal Physcs 54, (013). [8] S. S. Bay, Cosstecy poblem of the solutos of the space factoal Schödge equato, Joual of Mathematcal Physcs 54, 09101(013). [9] E. Hawks ad J. M. Schwaz, Commet O the cosstecy of the solutos of the space factoal Schödge equato, Joual of Mathematcal Physcs 54, (013). [10] Y. Luchko, Factoal Schödge equato fo a patcle movg a potetal well, Joual of Mathematcal Physcs 54, (013). [11] J. Dog, Levy path tegal appoach to the soluto of the factoal Schödge equato wth fte squae well, pept axv: v1 [math-ph] (013). [1] J. Tae ad J. Esguea, Boud states fo multple Dac-δ wells space-factoal quatum mechacs, Joual of Mathematcal Physcs 55, (014). [13] J. Tae ad J. Esguea, Tasmsso though locally peodc potetals space-factoal quatum mechacs, Physca A: Statstcal Mechacs ad ts Applcatos 407(014), pp [14] A. Messah, Quatum Mechacs Vol 1, (Noth Hollad Publshg Compay 1965). [15] M. Belloa ad R.W. Robettb, The fte well ad Dac delta fucto potetals aspedagogcal, mathematcal ad physcal models quatum mechacs, Physcs Repots 540(014), pp5-1. [16] Y. We, The Quatum Mechacs Explaato fo the Lamb Shft, SOP Tasactos o Theoetcal Physcs1 (014), o. 4, pp.1-1. [17] Y. We, O the dvegece dffculty petubato method fo elatvstc coecto of eegy levels of H atom, College Physcs14(1995), No. 9, pp5-9. [18] K. Kaleta, M. Kwasck, ad J. Maleck, Oe-dmesoal quas-elatvstc patcle a box, Revews Mathematcal Physcs5, No. 8 (013) [] N. Lask, Factoal Schödge equato, Physcal Revew E66, (00).