The quantum mechanics based on a general kinetic energy

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1 The quantum mechancs based on a general knetc energy Yuchuan We * Internatonal Center of Quantum Mechancs, Three Gorges Unversty, Chna, 4400 Department of adaton Oncology, Wake Forest Unversty, NC, 7157 *Correspondng author: yuchuanwe@gmal.com Keywords probablty teleportaton, Hermtan operator, relatvstc Schrödnger equaton, Klen-Gordon equaton, Abstract In ths paper, we ntroduce the Schrödnger equaton wth a general knetc energy operator. The conservaton law s proved and the probablty contnuty equaton s deducted n a general sense. Examples wth a Hermtan knetc energy operator nclude the standard Schrödnger equaton, the relatvstc Schrödnger equaton, the fractonal Schrödnger equaton, the Drac equaton, and the deformed Schrödnger equaton. We reveal that the Klen-Gordon equaton has a hdden non- Hermtan knetc energy operator. The probablty contnuty equaton wth sources ndcates that there exsts a dfferent way of probablty transportaton, whch s probablty teleportaton. An average formula s deducted from the relatvstc Schrödnger equaton, the Drac equaton, and the K-G equaton. 1

2 I. Introducton Knetc energy orgnally appeared n Newton s classcal mechancs, and was later revsed by Ensten n specal relatvty. In 196 [1], Schrödnger vewed the classcal knetc energy as an operator, and dscovered that the energy levels of the Hydrogen atom were egenvalues of the Hamltonan operator. In 199 [], Drac expressed the relatvstc knetc energy as a 4- dmesonal lnear operator, and obtaned the energy levels wth fne structure of the H atom. ecently [,4], based on the relatvstc knetc energy operator tself, we obtaned a new energy 5 formula wth an α term. Together wth the effect of spn-orbt couplng, the new energy formula can explan the Lamb shft half-quanttatvely. Addtonally, the fractonal knetc energy appears n the fractonal quantum mechancs [5-8], and the knetc energy wth a poston-dependent mass appears n the deformed quantum mechancs [9,10]. In order to buld a unfed framework for dfferent equatons n quantum mechancs, we wll ntroduce the Schrödnger equaton wth a general knetc energy operator. In a general sense, the conservaton law s proved and the probablty contnuty equaton s derved. The standard Schrödnger equaton, the fractonal Schrödnger equaton, the relatvstc Schrödnger equaton, the reformed Schrödnger equatons, the Drac equaton, and the K-G equaton are consdered as concrete examples. A new way of probablty transportaton, probablty teleportaton, s proposed. II. Varous knetc energy operators n quantum mechancs We frst recall the varous knetc energy operators used n quantum theory through the hstory [1,], and explan the fractonal knetc energy operator n the fractonal quantum mechancs [5-6] and the knetc energy operator wth a poston-dependent mass n the deformed quantum mechancs[9,10]. 1. The formulas for the energy levels of the hydrogen atom and the related knetc energy operators Hstorcally, the explanaton of the hydrogen spectrum has been the target of quantum theory and a better formula for the hydrogen energy levels has been the purpose of theoretcal physcsts [1,]. We wll see that dfferent knetc energy operators lead to dfferent formulas for the energy levels of the H atom. In classcal mechancs, the total energy H (the Hamltonan) of the hydrogen atom was the summaton of the classcal knetc energy T and the Coulomb potental energy V of the electron p e H T V, T, V. (1) m r

3 As usual, m and e are the mass and charge of the electron, and p s the momentum of the electron. Under a new assumpton that the electron ran only along crcular orbts wth angular moment n, n 191 Bohr obtaned the frst formula of the energy levels E 1 n mc α, n=1,,.. () As usual, c s the speed of lght, and α s the fne structure constant. In order to further explan the fne structure of the hydrogen spectrum, Sommerfeld ponted out that the relatvstc knetc energy should be used n the Hamltonan e p () r 4 H T V, T c m c, V and that ellptc orbts should be allowed as well. The revsed energy level formula became E mc 1 n r k 1/ (4) wth n 0,1,,, k 1,,. Ths formula matched the fne structure of the hydrogen r spectrum very well. In 196, Schrödnger dscovered that the Bohr energy levels can be obtaned by solvng the egenequaton of the Hamltonan operator e H E, H. (5) m r Ths was the begnnng of quantum mechancs. It was a natural wsh that we could get the energy level formula for the fne structure from the egenequaton 4 e H E, H p c m c. (6) r However, t was not easy to solve ths equaton, so usually the Hamltonan was approxmated as H p 1 p e m 8m c r 4 4 m c. (7) Based on the perturbaton theory, the energy levels of the frst order approxmaton were

4 1 1 n 4 E mc 1 α ( )α 4 n n l 1/ 4, (8) whch unfortunately dd not match the fne structure of the Hydrogen spectrum. Here n1,,, l 0,1,,. Consderng the relatvstc covarance, the Klen-Gordon equaton was proposed n 196 as well. From the K-G equaton [1,] we got the energy level formula e r 4 ( E ) ( c m c ) p (9) 1/ α E mc 1 1 nr ( l1/ ) α 1 1 n 4 mc 1 α ( )α 4 n n l 1/ 4 (10) 4 Ths formula had the same α term as the result of the perturbaton method (8), whch dd not match the fne structure of the Hydrogen spectrum ether. In secton V, we wll reveal the knetc energy operator hdden n the K-G equaton. In 199 [], usng a 4-dmensonal lnear knetc energy operator, Drac proposed hs equaton where, are 4 4 matrces. H E, H c mc e / r p, (11) The new energy formula was the same as Sommerfeld s formula (4), whch matches the fne structure of the hydrogen spectrum very well. In 1947 [1,], Lamb dscovered that the states S 1/ and P 1/ of the H atom, whch should have the same energy accordng to Drac s equaton, actually had an about 1000Mhz mcrowave radaton between them. Theoretcal physcsts beleved that the Lamb shft was beyond the scope of quantum mechancs and they developed quantum electrodynamcs to explan t. ecently we revsted the popular perturbaton method to deal wth the knetc relatvstc correcton and found some basc problems [,4]. If we keep the 4 p term only,.e., 4

5 H p 1 p e m 8m c r 4 4 m c (1) ts egenvalues do not exst, snce the knetc energy becomes negatve as the momentum magntude ncreases very large. On the other hand, f we nclude matrx elements of the 6 p term as well, some of the 6 p term are nfnty. Thus, t was mpossble to fnd the relatvstc correcton of the energy levels of the H atom by usng the power seres of the relatvstc knetc energy and the relatvstc correcton of the knetc energy of the H atom has never been found n a reasonable way. Therefore, we developed a perturbaton method based on the square root operator drectly, rather than ts power expanson, and calculated the energy levels carefully. The new energy levels contan a valuable 5 α term. Together wth the L-S couplng effect, the new energy formula 1 1 n 64 1 E mc 1 α α α O α n n 1 j 4 15 n l (1) can half-quanttatvely explan the Lamb shft []. Here n 1,,, j 1/ 1,,, l 0,1,, 00 1, In one word, the hstory of quantum theory shows us that varous knetc energy operators lead to dfferent formulas for energy levels of the H atom, and the relatvstc knetc energy operator plays a specal role.. Fractonal knetc energy In the fractonal quantum mechancs [5-6], the fractonal knetc energy operator s expressed as / T D p D, (14) where the fractonal parameter 1, and D s a coeffcent dependent on. Please be remnded of the dfferent meanngs of the three symbols, (the fractonal parameter), α (the fne structure constant), and (4 4 matrces). In the case, takng D 1/ ( m), the fractonal knetc energy s the classcal knetc energy T p m. (15) 5

6 . The knetc energy wth a poston-dependent mass. In the deformed Schrödnger equaton [9,10], the mass m() r s a poston dependent functon and the knetc energy operator s T p 1 p. (16) m( r) Now that there are varous types of quantum mechancs based on dfferent knetc energy operators, t s necessary to ntroduce the quantum mechancs based on a general knetc energy. III. Quantum mechancs based on a general knetc energy In ths secton we propose the Schrödnger equaton wth a general knetc energy, prove the conservaton law, and derve the probablty contnuty equatons n a general sense. 1. The Schrödnger equaton wth a general knetc energy operator All the square ntegrable functons defned on the three-dmensonal Eucldan space form a Hlbert space wth an nner product *, ( r) ( r) d r, (17) where, are any two square ntegrable functons. quantum mechancs, square ntegrable functons are called wavefunctons[1]. We call r s a three-dmensonal vector. In r, t H r, t (18) t the general Schrödnger equaton, where r,t s a tme-dependent wavefuncton. Here the Hamltonan operator H T V T( r, P, t) V( r, t) (19) s a summaton of a general knetc energy operator T and a potental energy operator V. As a real-valued functon, the potental energy s always a Hermtan operator, V V. Usually the knetc energy operator s Hermtan,.e., T T, so that the Hamltonan operator H s Hermtan as well. The quantum system wth a Hermtan Hamltonan s called a Hermtan system. In a Hermtan system, the egenvalues of the total energy are real and the wavefunctons related to dfferent energes are orthogonal. 6

7 Occasonally, the knetc energy operator s non-hermtan, e.g., we wll reveal that the Klen- Gordon equaton has a hdden non-hermtan Hamltonan operator. The quantum system wth a non-hermtan Hamltonan s called a non-hermtan system.. The conservaton law Suppose that A s a tme-ndependent operator and that () r s a square ntegrable functon, we call the ntegral A A A A d *, ( r) ( r) r (0) the value of the operator A on the wavefuncton () r. If A s a Hermtan operator and () r s normal, A s the well-known average. If ( r, t) equaton, then s a soluton of the general Schrödnger A( t) (, t) A (, t) d * r r r (1) s a functon of tme t. Theorem 1. The quantty A s conserved when H AAH 0. () Proof. From (18), we know that the dervatve of the quantty A s d d * A( t) (, t) A (, t) d dt dt r r r d * * d ( r, t) A ( r, t) d r (, t) A (, t) d dt r r r dt 1 * * 1 ( H ) ( r, t) A ( r, t) d r (, t) A H (, t) d r r r 1 * * 1 ( r, t) H A ( r, t) d r (, t) A H (, t) d r r r ) (, ). * ( r, t)( H A AH r t d r () Obvously, we have d A 0 when H A AH 0 (4) dt 7

8 Ths completes the proof. Theorem. For a Hermtan Hamltonan, H and the Hamltonan commutes Proof. Ths s a straghtforward result of the theorem 1. H, the quantty A s conserved f ts operator [ H, A] 0. (5) * Theorem. The ntegral ( r, t) ( r, t) d r s conserved f H H. Proof. The dentty operator s defned as I. (6) Takng A=I n Theorem 1, we get ths theorem at once. Ths completes the proof. Accordng to ths theorem, n a Hermtan system, the probablty s conserved. wavefuncton s normalzed If at t=0, the ( r,0) ( r,0) r 1, (7) * d then the wavefuncton s normalzed at any tme t,.e. ( r, ) ( r, ) r 1. (8) * t t d eversely, n a non-hermtan system, t s useless to normalze a wavefuncton, snce the normalzed functon wll become non-normalzed n the next moment.. The probablty contnuty equatons wth and wthout sources From the general Schrödnger equaton (18) we have * * * T V, t * * * ( T ) V. t (9) Addng the above two equatons, we have * * * ( T ) ( T ) t, (0) * * ( T ) ( T ) * 0 t. (1) 8

9 The probablty contnuty equaton can be wrtten as 0 t j () where the probablty densty and the current densty s defned as * * * j ( T ) ( T ). () However, the form of the probablty contnuty equaton s not unque. Generally we can wrte the probablty contnuty equaton as I t j (4) where the current densty and the source term can be defned n several ways j j ( T ) ( T ) 1 1 * * * * I ( T ) ( T ). ( )( T ) ( )( T ) * * I T T * * ( )( ) ( ), j 0 * * I ( T ) ( T ). (5) (6) (7) If I( r, t) 0, there s a source at poston r and tme t, whch generates the probablty; when I( r, t) 0, there s a snk at poston r and tme t, whch destroys the probablty. Here are the dfferent pctures of the probablty transportaton n the dfferent systems. In a non-hermtan system, the total probablty s not conserved: the contnuty equaton wthout sources () tells us that the probablty can generated or destroyed n nfntely far places, but they are conserved n any fntely far place; the contnuty equaton wth sources (4) tell us that the probablty can be generated or destroyed everywhere. It s lucky that we seldom encounter the non-hermtan systems. 9

10 In a Hermtan system, the total probablty s conserved. (1) The contnuty equaton wthout sources () tells us that the probablty cannot be generated or destroyed anywhere, and t moves from one place to another place. Ths s the popular pcture n our mnd on the probablty transportaton, and the scatterng experments are based on ths understandng. () However, the contnuty equaton wth sources (4) tells us that the probablty can dsappear n one place and smultaneously appear n other places, but the total probablty does not change. In other words, some probabltes can be teleported from one place to another. Furthermore, f the partcle has mass and charge, probablty teleportaton wll mply mass teleportaton and charge teleportaton. Ths s a new pcture of the probablty transportaton. We must further study the dfferent effects caused by dfferent contnuty equatons and desgn sutable experments to decde whch pcture on the probablty transportaton s correct. IV. Examples of the Hermtan knetc energy operator In ths secton we study the concrete examples of the Hermtan knetcs energy operator, ncludng the standard Schrödnger equaton, the relatvstc Schrödnger equaton, the fractonal Schrödnger equaton, the Schrödnger equaton wth a poston dependent mass, and the Drac equaton. 1. The standard Schrödnger equaton In quantum mechancs [1], the standard Schrödnger equaton s where the Hamltonan operator s ( r, t) H ( r, t), (8) t H T V V t ( r, ). (9) m The probablty contnuty equaton wthout source s 0 t j (40) wth * j m * *. (41) Ths well-known result s consstent to the defnton () as well as the defnton (5). 10

11 When a spnless partcle wth mass m and charge q moves n an electromagnetc feld, ts nonrelatvstc Hamltonan [1] s 1 q ( ) H T V p q m A c (4) where A A( r, t) and ( r, t) are the electromagnetc potental. The Paul s equaton s wrtten as 1 ( r, t ) ( / ) (, ) (, ) t m q A c r t q r t. (4) The probablty contnuty equaton wthout source remans Eq. (40) wth a new current densty 1 m j * ( p q A / c) cc, (44) where letters cc s the complex conjuncton of the term n front of t. Here s the deducton from the defnton () j ( T ) ( T ) * * * * ( p qa / c) ( p qa / c) m * ( p qa p / c q A / c ) ( p qa p / c q A / c ) m * * ( p qa p / c) ( p qa p / c) m * * q * * p p A p A p m m c * * q * * A A m mc * * q * A m mc * * q * A m mc 1 * * q * p p A m mc 1 * ( p qa/ c) cc. m * (45) Here we have used Coulomb gauge of the electromagnetc feld 11

12 A 0, p A A p. (46) On the other hand, accordng to the defnton (5), the contnuty equaton can be wrtten as where 1 I1 t j (47) j 1 * 1 ( q / c) cc, m p A (48) I q A (49) mc * 1. As we sad above, the dfference between the two contnuty equatons should be studed carefully n theory and experment.. elatvstc Schrödnger equaton. 1) Wthout magnetc feld Accordng to specal relatvty, the knetc energy s T c m c 4 p. (50) For a partcle movng n a potental feld V, the relatvstc Hamltonan functon s [11,1] 4 H c m c V () p r. (51) The relatvstc Schrödnger equaton s r, t H r, t. (5) t We can easly prove that f the potental s even, V( r, t) V( r, t), then the party s conserved and that f the potental s central V( r, t) V( r, t), then the angular momentum s conserved. As usual, the party operator P and the angular momentum L are defned as P r r, L r p. (5) 1

13 Formula 1. If an egenvalue and an egenfuncton of the Hamltonan operator (51) are E and r,.e. then we have H( r) E( r ), (54) E V c m c V 4 p V. (55) Proof. From (51), we have H V c m c 4 p (56) 4 ( H V ) p c m c (57) 4 H HV VH V p c m c. (58) * Takng the ntegral ( r) ( r) d r of the above equaton, we have * * 4 ( r)( H HV VH V ) ( r) d r ( r)( p c m c ) ( r) d r (59) Usng the Drac s notatons, we have H HV VH V p c m c 4 E E V V p c m c 4 ( E V ) p c V V m c 4 (60) E V c V V m c 4 p. (61) We keep the postve soluton only n the above equaton accordng to the meanng n physcs. Ths completes the proof. Snce E V T, we have the average of the knetc energy T c V V m c 4 p. (6) Ths relaton s smlar to the Vral theorem n the standard quantum mechancs [1]. ) Wth magnetc felds 1

14 More generally, f a relatvstc partcle wth mass m and charge q moves n an electromagnetc feld, ts Hamltonan s 4 H( rp,, t) pc qa( r, t) m c q( r, t) T( rp,, t) V ( r, t). (6) One may defne the knetc energy operator T( rp,, t) as follows. At any gven tme t, suppose that the egenequaton of the operator pc qa( r, t) pc q r t ( t) ( r) ( t) ( t) A (, ) ( r) (64) has been solved,.e., the egenfunctons () () r and egenvalues () t are known. Snce the operator c q (, t) t p A r s Hermtan, c q (, t) p A r s a nonnegatve-defnte operator, and () t s a nonnegatve number. As a smple example, the egenfunctons and egenvalues of (64) for a statc unform magnetc feld can be seen n [1]. Also we suppose that a wavefuncton r,t can be expressed n terms of the egenfunctons () r wth the coeffcent c () t. Then the knetc operator T( rp,, t) s defned as r, t c() tr (65) 4,, c ( t () T( r, p, t) r, t T( r p t) ) r ( t) m c c t r. (66) Based on ths defnton, one can easly verfy that T( rp,, t) s Hermtan, T T. The Schrödnger equaton s,t H(,, t), t t r r p r, (67) or, t c q (, t) m c 4, t q (, t), t t r p A r r r r. (68) 14

15 Hstorcally [11], due to the complcaton and dffculty of the square root operator, ths equaton has been abandoned, and the Klen-Gordon equaton and the Drac equaton were developed to avod the square root operator. However, we dscovered that the square root operator was necessary to avod the dvergence n the perturbaton method for the calculaton on the relatvstc correcton of the hydrogen energy levels. What s more, the egenenergy of the square 5 root equaton (54) has a specal α term, whch s 41% of the measured Lamb shft [,4], whle the energy levels from the Schrödnger equaton, the Klen-Gordon equaton, and the Drac 5 equaton do not have an term at all. eaders can explctly wrte out the probablty contnuty equaton wth or wthout sources accordng to the defntons n Sec. III.. Fractonal Schrödnger equaton The fractonal Schrödnger equaton [5,6] s r, t H r, t, (69) t wth the fractonal Hamltonan operator H T V () r. (70) We can prove that f the potental s even, V( r, t) V( r, t), then the party s conserved and that f the potental s central, V( r, t) V( r, t), then the angular momentum s conserved. In the fractonal quantum mechancs, the probablty contnuty equaton can be wrtten as wth the probablty densty and the current densty 0 t j (71) * j ( T ) ( T ), * * (7) or j 1 I1, (7) t where the current densty and the source term are defned as 15

16 j D 1 * /1 /1 *, (74) 1 ( ) ( ) I 1 * / 1 / 1 * 1 D ( ) ( ). (75) In [6], t was a mstake that the source term I 1 was mssng [14]. 4. The Schrödnger equaton wth a poston-dependent mass The deformed Schrödnger equaton s [9,10] r, t H r, t, (76) t where the deformed Hamltonan operator has a poston-dependent mass m() r. H 1 p p V() m( ) r r (77) We can prove that f the mass functon and the potental are even, m(- r)=m( r) and V( r) V( r ), then the party s conserved and that f the mass functon and the potental s central, m(- r )=m(r) and V( r ) V( r), then the angular momentum s conserved. Accordng to Sec III, the probablty contnuty equaton wthout sources s where the current densty s 0 t j (78) j ( T ) ( T ) * * * 1 1 * p p p p m( ) m( ) r r * 1 1 * p p m( ) m( ) r r * 1 1 * p p. m( r) m( r) (79) eaders can further study the probablty contnuty equaton wth sources. 5. The Drac equaton. 16

17 The Drac equaton s r, t H r, t, (80) t where the Hamltonan operator H c mc V r Please notce that now the wavefuncton has 4 components. p (). (81) Formula. If an egenvalue and an egenfuncton of the Drac s Hamltonan operator are E and r,.e. then we have H( r) E( r ), (8) E V c m c V 4 p V. (8) Proof. The proof s smlar to the proof of Formula 1, snce from (81), agan we have H V c mc p (84) 4 ( H V ) p c m c (85) 4 H HV VH V p c m c. (86) eaders can complete the proof easly. Agan the average of the knetc energy s c c m c V V 4 p p. (87) It s well know that the party and the angular momentum are not conserved. When there s the electromagnetc feld, we have H c ( p qa / c) mc V( r). Based on the defnton (), the probablty contnuty equaton wthout sources s 0 t j (88) where the probablty and current densty are defned as 17

18 j ( c ( p qa / c) mc ) [( c ( p qa / c) mc ) ] c p p c (89) c c. Ths s the popular form of the probablty contnuty equaton. Based on the defnton (5), we have wth 1 1 I1 t j (90) j c (91) I c. V. An example of the non-hermtan knetc energy operator. We wll see that a non-hermtan system s qute dfferent from the well-known Hermtan system. Our dscusson s based on an example, K-G equaton wthout magnetc feld. The K-G equaton for a spnless partcle of mass m and charge q movng n an electromagnetc feld s 4 q r, t pc qa r, t m c r, t. (9) t In a statc pure electrcal feld, A 0, q V( r ), we have 4 V r, t p c r, t m c r, t. (9) t 1. The Schrödnger equaton form of the Klen-Gordon We report that the K-G equaton (9) can be expressed as a Schrödnger equaton wth a non- Hermtan knetc energy operator, 18

19 r, t H r, t, t 4 H T V p c m c V H V,. (94) If V, H s zero, or so small that t can be neglected, ths Hamltonan and the relatvstc Hamltonan (51) s the same. Therefore, we need to pay more attenton on the other cases. Suppose the egenequaton has egenfunctons n H r E r (95) r and egenvalues E n, where n s an ndex. Then the general soluton of the Schrödnger equaton (94) s wth the arbtrary coeffcents c n. r, t c r exp( E t), (96) n n n n Proposton 1. The Schrödnger equaton (94) mples the K-G equaton (9). Proof. The deducton from (94) to (9) s as follows 4 r, t p c m c V, H r, t V r, t t 4 V r, t p c m c V, H r, t t 4 4 p c m c V, H V r, t p c m c V, H r, t t 4 4 V p c m c V, H r, t p c m c r, t t 4 V r, t p c m c r, t. t (97) Ths completes the proof.. The non-hermtcty of the Hamltonan Proposton. The Hamltonan operator s Hermtan f and only f V() r V0 a constant,.e. H H V V. 0 (98) 19

20 Proof. The proof has two steps. (1) On. Snce V() r V0, we have [ VH, ] 0, and H T V c m c 4 V. 0 p (99) Ths s a Hermtan operator. () On. Now H s a Hermtan operator,.e., H H. From (94) we have 4 H V c m c V, H p (100) 4 ( ), Takng the Hermtan adjont of the two sdes, we have Takng (10) - (104), we have Therefore, we have H V p c m c V H (101) 4 H VH V p c m c. (10) 4 H H V V p c m c (10) 4 H HV V p c m c. (104) HV VH, [ V, H]=0. (105) 4 H c m c V, p (106) and 4 [ H, V ] [ c m c, V ] 0. p (107) Further, we have 4 [ p c m c, V ],, 0, p c m c p c m c V p c m c V p c m c (108) 0

21 Therefore, the potental energy s a constant Ths completes the proof. In fact, based on Eq. (10), we have [, V ] 0. p (109) V() r V. (110) 0 4 H T V p c m c H, V V 4 p c m c V, H V. (111). The average formula Formula. If an egenvalue and an egenfuncton of the Hamltonan operator are E and r,.e. H( r) E( r ), (11) then we have E V c m c V 4 p V. (11) Proof. * Takng the ntegral ( r) ( r) d r of the two sdes of Eq. (10), we have * * 4 ( r)( H VH V ) ( r) d r ( r)( p c m c ) ( r) d r (114) Usng the Drac s notatons, we have H VH V p c m c 4 E E V V p c m c 4 4 ( E V ) p c V V m c (115) Usually the rght-hand sde of Eq. (115) s greater than zero for a partcle wth a low speed. Takng the postve soluton, we have the energy formula E V c V V m c 4 p. (116) 1

22 If the rght-hand sde of Eq. (115) s smaller than zero, we see that the energy E wll become a complex number. Agan, the value of the knetc energy operator s p c m c 4 V, H p c m c 4 V V. (117) Notce that Formulas 1,, and have the same form and can be consdered as one formula. 4. The non-orthogonalty of the egenfunctons Proposton. Two egenfunctons 1( r), ( r ), whose egenenerges E 1, E satsfy E * E1, are orthogonal f and only f * 1( r) V( r) d r 0. (118) Proof. From Eq. (10) we have Smlarly, we have Takng the complex conjugate, we have Takng Eq. (119) - Eq. (11), we have H VH V p c m c 4 E E V V p c m c 4 E E V V p c m c. (119) * * * * 4 * E E V V p c m c. (10) * * * * 4 * E E V V p c m c. (11) * * * * * * 4 * ( E E ) ( E E ) V p c p c. (1) * * * * * * * Takng the ntegral d r, we have ( E E ) d r ( E E ) V d r 0 (1) * * * * * * * ( E E ) d r V d r 0 (14)

23 d r V d r. (15) * * 1 * 1 E E1 It s easy to see that 1( r), ( r ) are orthogonal f and only f * 1Vd r 0. (16) Ths completes the proof. 5. The Non-conservaton of physcal quanttes Based on Theorem 1, the quantty * ( r, t) H( r, t) d r s not conserved generally. Snce H s not Hermtan (except the trval case V() r V0 ), there must exst some wavefunctons ( r ) such that ths ntegral s a complex number [11]. For these two reasons we know that H s not an operator assocated to the total energy. Strctly speakng, the physcs meanngs of the knetc energy operator and the Hamltonan operator are not clear. It s not sutable to call the egenvalues of H (e.g. Eq. (10) for the Coulomb potental) the energy levels; they are just a parameters wth the dmenson of the energy appearng n the soluton (96). * Based on Theorem the ntegral ( r, t) ( r, t) d r s not necessarly conserved. Stll, readers can study the probablty contnuty equaton accordng to the formulas n Secton III. We can say that f the potental s even, V( r) V( r ), then snce [ H, P ] 0, (17) H H V ( p, ( r)), [ p, P] 0, [ V ( r), P] 0. (18) If the potental s central, V( r ) V( r), then [ H, L ] 0 (19) snce

24 H H V r ( p, ( )), [ p, L] 0, [ V( r), L] 0. (10) From (19), we know that [ H, ] 0, [ H, Lz] 0, [, Lz] 0 L L. (11) Thus n the central potental, one can fnd the common egenfunctons of the sets H, L, Lz example the soluton to the K-G equaton wth a Coulomb potental (9) can be seen n [1,]. However, the party and the angular momentum are not conserved anymore.. For We see that the quantum mechancs based on K-G equaton s qute dfferent from the standard quantum mechancs. For these reasons, we thnk t would be better to use the relatvstc Schrödnger equaton (5) than the K-G equaton (9) to descrbe the spnless partcle, e.g. ponc hydrogen atom (a pon movng n the Coulomb feld of a proton). 6. Another non-hermtan Hamltonan related to the K-G equaton In addton, we menton another non-hermtan knetc energy operator, whch also relates to the K-G equaton. Proposton 4. The Schrödnger equaton t H t t r, ' r, H T V c m c V H V 4 ' ' ' p ', ' ' (1) mples the K-G equaton (9) as well. Proof. The proof s smlar to that of Proposton 1. Proposton 5. If V' Proof. If V' V V, then H' H., accordng to (1), we have 4 H ' c m c V, H ' V p (1) 4 H ' p c m c V, H ' V (14) Comparng (14) wth the defnton of H n (94), we have H H '. Ths completes the proof. 4

25 Exmaple. The two operators p (15) 4 H c m c e / r, H e / r p (16) 4 H ' c m c e / r, H ' e / r have the same egenfunctons and opposte egenenerges. VI. Concluson The Schrödnger equaton wth a general knetc energy operator s ntroduced. The conservaton law s proved and the probablty contnuty equaton s deducted n a general sense. Examples wth a Hermtan knetc energy operator nclude the standard Schrödnger equaton, the relatvstc Schrödnger equaton, the fractonal Schrödnger equaton, the deformed Schrödnger equaton, and the Drac equaton. We reveal that the Klen-Gordon equaton has a hdden non- Hermtan knetc energy operator. The probablty contnuty equaton wth sources ndcates that there exsts a dfferent way of the probablty transportaton, whch s probablty teleportaton. An average formula s deducted from the relatvstc Schrödnger equaton, the Drac equaton, and the K-G equaton. Acknowledgement The research on the relatvstc Schrödnger equaton was supported by Gansu Industry Unversty (currently Lanzhou Unversty of Technology) durng , wth a project ttle On the solvablty of the square root equaton n the relatvstc quantum mechancs. The author thanks evewer B for hs correcton on Sec III. and remndng me of some related monographs. Appendx. A dscusson on the relatvstc Schrödnger equaton (5). evewer A: The relatvstc Schrödnger equaton (5) s not relatvstcally nvarant at all, and was ruled out n the early days of relatvstc quantum mechancs, see [15]. Why do you pay attenton to such a wrong and useless equaton? esponse. 5

26 Frst of all, ths equaton does not have an offcal name [11,15]. The name relatvstc Schrödnger equaton here means the Schrödnger equaton wth a relatvstc knetc energy operator. I just notced that ths name s used n [16] as well. Based on my own research, there are two new reasons to revve ths equaton. (1) Ths equaton appears naturally when calculatng the relatvstc knetc correcton of the energy levels of the H atom by the perturbaton method [, 4]. () Ths equaton s an approxmate realzaton of the well-known fractonal Schrödnger equaton [7,8]. Addtonally, I want to add the followng two ponts. 1. elatvstc covarance and correctness Indeed ths equaton s not relatvstcally nvarant, but the standard Schrödnger equaton s not ether. Therefore a non-covarant equaton can be valuable f the equaton s practcally useful, snce a non-covarant equaton can become an approxmaton of a covarant equaton somehow n the future, as the Schrödnger equaton becomes an approxmaton of the Drac equaton and the Klen-Gordon equaton.. Expermental crteron. The fnal crteron to judge the value of an equaton s whether the equaton can predct new expermental results. (1)The energy formula for the Hydrogen atom based on ths equaton s even better than that from the Drac equaton. In addton, I wsh that the expermental physcsts can soon judge whch equaton generates the correct energy level formula for the ponc hydrogen atom [1,], the relatvstc Schrödnger equaton or the K-G equaton []. () The relatvstc Schrödnger equaton ndcates the possblty of probablty teleportaton, qute dfferent from the Schrödnger equaton, the K-G equaton, and the Drac equaton. We are desgnng related experments to observe ths new relatvstc quantum mechancs phenomenon. If the two expermental results dd not support the relatvstc Schrödnger equaton, I would admt that ths equaton s wrong both theoretcally and expermentally. 1 D. Y. Wu, Quantum Mechancs (World Scentfc, Sngapore,1986). D. Y. Wu and W. Hwang, elatvstc Quantum Mechancs and Quantum Felds (World Scentfc, Sngapore,1991). Y. We, The quantum mechancs explanaton for the Lamb shft, SOP Transactons on Theoretcal Physcs 1(014), no. 4, pp.1-1, 6

27 4 Y. We, On the dvergence dffculty n perturbaton method for relatvstc correcton of energy levels of H atom, College Physcs 14(1995), No. 9, pp5-9 (In Chnese wth an Englsh Abstract) 5 N. Laskn, Fractonal quantum mechancs, Phys. ev. E 6, 15 (000). 6 N. Laskn, Fractonal Schrödnger equaton, Phys. ev. E 66, (00). 7 Y. We, The nfnte square well problem n the standard, fractonal, and relatvstc quantum mechancs, Internatonal Journal of Theoretcal and Mathematcal Physcs 5 (015), No.4, pp Y. We, Some solutons to the fractonal and relatvstc Schrödnger equatons, Internatonal Journal of Theoretcal and Mathematcal Physcs, Vol. 5 No. 5, 015, pp O. V. oos, Poston-dependent effectve masses n semconductor theory, Phys. ev. B 7, 7547 (198). 10 M. Chabab, et al, Exact solutons of deformed Schrödnger equaton wth a class of noncentral physcal potentals, Journal of Mathematcal Physcs 56, (015) 11 A. Messah, Quantum Mechancs vol. 1, (North Holland Publshng Company 1965) 1 K. Kaleta, M. Kwasnck, and J. Maleck One-dmensonal quas-relatvstc partcle n a box, evews n Mathematcal Physcs 5, No. 8 (01) L. D. Landau and E. M. Lfshtz, Quantum Mechancs Non-relatvstc Theory, Course of Theoretcal Physcs, Vol., Pergamon, Y. We, Comment on Fractonal quantum Mechancs and Fractonal Schrödnger equaton, Phys. ev. E 9 (016) (to be publshed). 15 J Bjorken and S Drell, elatvstc Quantum Mechancs, McGraw-Hll, 1964, p5. 16 Laudau, Quantum Mechancs II, Wley,