Surprises with Logarithm Potential

Transcription

1 Supises with Logaithm Potetial Debaaya Jaa Dept. of Physics, Uivesity College of Sciece ad Techology 9 A P C Road, Kolkata W.B. djphy@caluiv.ac.i Abstact The oigi of logaithmic potetial is ivestigated though a simple dimesioal aalysis ad its physical sigificace has bee bought out i coectio with dimesioal egulaizatio of field theoy. Besides, we would like to poit out the boud state eegy levels of logaithmic potetial by ucetaity piciple, phase space quatizatio ad Hellma-Feyma (H-F) theoem. Although the eegy levels do deped o the mass of paticle, howeve, it tus out that the eegy level sepaatio betwee ay two levels is idepedet of mass tem as well as Plack s costat. We also poit out the impotace of this potetial i d iteactig systems i codesed matte physics.. Itoductio The logaithmic potetial i physics foms a iteestig oe as it povides some uusual pedictio about the system. Moeove, this potetial ca be used suitably to illustate some of the impotat featues of field theoy such as dimesioal egulaizatio ad eomalizatio. I most of ou text books, this potetial is ot discussed at detail; although the calculatios ae quite simple to demostate some of its uique featues. We have obtaied the boud state eegy of this logaithmic potetial though ucetaity piciple, phase space quatizatio ad Hellma-Feyma (H-F) theoem. The pape is ogaized as follows. With a bief discussio of the oigi of this logaithmic potetial though dimesioal aalysis, we diectly go to calculate the boud state eegy levels by ucetaity piciple, phase space quatizatio ad Hellma- Feyma theoem i sectio 3. Fially, we give ou coclusios i sectio 4.

2 . Oigi of Logaithmic potetial i a electostatic poblem: The dimesioal aalysis (DA) [-3] ad scalig agumets [4] fom a itegal pat i theoetical physics to solve some impotat poblems without doig much calculatio. The dimesios of ay physical quatity ca be expessed i tems of thee paametes mass-legth-time (MLT). Evey equatio i sciece must be dimesioally homogeeous; o i othe wods, the left had side (lhs) ad the ight had side (hs) of the equatios must have same powe of M, L ad T. Wheeve, thee ae moe tha oe tems i a equatio, it is evidet that evey tem i such a equatio must have the same dimesio. This immediately idicates that the coectess of a equatio ca be veified by this appoach. Howeve, thee exist some situatios whee this aïve appoach may fail. I this wok, we would like to discuss the implicatios i logaithmic fuctio. Just like agles, expoetials ad othe tigoometic fuctios, logaithmic also falls ito the categoy of dimesioless fuctios. Natually, thee will be some legth scales o eegy scales ae built i these fuctios. Let us evisit a simple poblem fom electostatics whee we fid the logaithmic vaiatio [5,6] of the potetial with distace. Suppose a ifiitely log wie (o equivaletly log chaged ods i 3 dimesios) is cayig a (liea) lie chage desity λ. We ae iteested to fid the electic field ad potetial eveywhee due to this chage desity. Because of its obvious cylidical symmety, we costuct cylidical Gaussia suface with the wie as its axis ad apply the Gaussia theoem to compute the electic field as λ E ds = 4πλL; E = () whee is the adius of the cylide ad L is its legth. So, the electic field vaies ivesely with distace. This vaiatio of electic field is also cosistet with dimesioal / aalysis as[ ] / = M L T = M L T / λ ;[ ] / E. Now, the questio is: what is the potetial at ay poit? Fist, we would like to sot the aswe though simple dimesioal aalysis. But we will show below that this appoach fails completely. Note that the poblem has the paamete λ, theefoe, the potetial Φ () ca deped o λ ad oly. Thus, we ca evetually wite Φ ( ) = λ x y. A aïve dimesioal aalysis eveals

3 that x = ad y = 0. Theefoe, Φ () = λ ad is idepedet of. The potetial i this case is completely dictated by the costat liea chage desityλ. But this aswe is meaigless as it gives a zeo electic field. But we have aleady oted that the electic field is ot zeo but vaies ivesely with distace. So, what s goig o hee? Let us do a fist piciple calculatio of the potetial due to this lie chage desity. The potetial ca be computed as: Φ( ) = + λdx + x = λ 0 Moeove, by chage of vaiable q = fom equatio () that x dx + x (), a dimesioless vaiable, it is easy to visualize dq Φ( ) = λ (3) 0 q + The equatio () ad (3) togethe poit out that the potetial is idepedet of distace ad is ifiite at the uppe limit. Theefoe, to avoid the divegece at the uppe limit, we put the uppe limit to Λ ad the we ca coside the ifiite wie (with liea chage desity) as a limitig case Λ >>. Theefoe, the potetial Φ(, λ, Λ) is give by Λ Φ(, λ, Λ) = λ log + λ log + + (4) Λ Thus, the fiiteess of the potetial is edeed by the uppe-cutoff used i the fist piciple calculatio. I the asymptotic limit Λ >>, we fid the potetial fom equatio (4) as (the secod tem is ot sigificat eough apat fom a shift i its magitude as see fom figue ) Λ Φ(, λ, Λ) = λ log (5) Although the potetial is depedet o the uppe cut-off legth scale Λ, howeve, the physically measuable quatity electic field is ot. Moeove, thee is a poit ( 0 < < ), at which the potetial vaishes ( Φ( = Λ) = 0 ). This bouday coditio is quite diffeet fom ou usual oes whee the potetial vaishes oly at the ifiity. Ad, i fact, the electic field ca be computed simply as 3

4 *log(x)+*log(x+(+(/x)^)^(/)) *log(x) *log(x)+log() Φ 0 (y)=φ(y)/λ y 0 =Λ/y Figue : Schematic vaiatio of the dimesioless potetial fuctios as dimesioless cut-off distace y 0. λ E(, λ) = (6) which ca be compaed with equatio (). The depedece of the potetial o the uppe cut-off legth scale Λ ca be elimiated if we ca coside the diffeece of the potetial at two distaces ad a: a Φ(, λ, Λ) Φ( a, λ, Λ) = λ log (7) This logaithmic fom of the potetial ca also be deived simply fom scalig agumets [5] as well as dimesioal egulaizatio [6]. The abitay distace a meely shifts the potetial by a costat. Thus, it is obvious ow that why the aïve agumets fom 4

5 simple dimesioal aalysis fail i the logaithmic potetial. This type featue is vey commo i the discussio of eomalizatio goup study of high eegy physics [7] as well as codesed matte physics [8]. This paticula logaithmic fom of the potetial does appea i computig the voltage (capacitace) diffeece betwee the coaxial cylidical diodes (capacitos) by solvig the Laplace s equatio with appopiate bouday coditios. The idepedece of the potetial with chage of distace i case of ifiite lie chage as evidet fom the aïve dimesioal aalysis ca be demostated i aothe poblem elated with Diac-Delta fuctio i two dimesios. The Schödige equatio fo two dimesioal attactive Diac-Delta fuctios ca be witte as h ψ m - λδ ( ) ψ = Eψ (8) mλ m E I tems of scaled vaiables λ = ad E =, the equatio (8) ca be witte as h h Ψ + λ δ ( ) ψ = Eψ (9) 0 0 I two dimesios, the delta fuctio has dimesio M L T. Thus, λ is 0 0 dimesioless while E has the dimesio of M L T. This idicates that we caot fid the depedece of E o the paamete λ. I othe wods, this paticula fact i two dimesios fo Diac-Delta potetial violates the Hellma-Feyma (H-F) theoem [9]. I quatum mechaics, thee is o othe sigle paticle potetial poblem whee we ca fid the cotadictio of poweful H-F theoem. Aothe way of visualizatio this diectly is though Fouie tasfomatio [5]. Witig ψ ( k ) = ψ ( )exp( ik ) d, we fid fom equatio (9) that k = λ ψ (0 ψ ( ) ) (0) k + E By pefomig the ivese Fouie tasfomatio (itegatig ove k ), we obtai the boud state eegy eigevalue equatio = λ 4π d k d q = + k E q + () 5

6 I tems of chage of vaiables q, we ote that λ is idepedet of E. Moeove, the itegal is diveget; so itoducig a mometum cut-off Λ i the above itegal with the limit Λ >> E, we fially get fom equatio () E = log () λ 4π Λ This immediately idicates the bidig eegy is give by E = Λ exp( 4π / λ ) (3) 3. Boud State Eegy Levels of Logaithmic potetial: The logaithmic potetial has a bach poit sigulaity at = 0. Because of this paticula type of sigulaity, the well-kow powe seies method adopted fo Coulomb potetial (/) ad hamoic oscillato potetial ( ) becomes ieffective i calculatig the eegy eigevalues ad eigestates. Howeve, a shootig method has bee used successfully to obtai the exact the eegy eigevalues [0]. I this sectio, we would like to compute the boud state eegy levels of the logaithmic potetial by ucetaity piciple, phase space quatizatio ad Hellma-Feyma (H-F) theoem. We will explicitly show although the eegy levels deped o the mass tem but the eegy diffeece betwee ay two eegy levels is idepedet of the mass tem. This is quite supisig i a typical sigle body potetial poblem i quatum mechaics. (a). Boud state fom ucetaity piciple: I udegaduate ad post-gaduate classes, although the exact esults ae demostated via Schödige equatio; howeve, befoe the solutios, this simple method witte ca be illustated as follows. The full o-elativistic Hamiltoia of this sigle paticle i logaithmic potetial ca be witte as p H = + λ log (4) m a Thee ae two legth scales built i to the potetial oe isλ ad the othe is a at which the potetial vaishes. Usig ucetaity piciple, the total eegy compisig of kietic eegy ad the potetial at legth scale b ca be witte as 6

7 h b E( b) = Eki ( b) + V ( b) = + λ log mb a (5) Sice the dimesio of λ is of eegy dimesio ( ML T ), it is bette to plot the dimesioless eegy ( ˆ E( b) E = ) as a fuctio of b fo thee sets of values of a as show λ i figue. 4 a=0 a=50 a=00 E(b)/λ b Figue : Schematic vaiatio of the total eegy as a fuctio of distace b fo thee values of a. 7

8 It is see that the miimum of the eegy occus at b mi which depeds o λ but ot o a. The miimum eegy, howeve, depeds o both λ ad a as h Emi ( m, a, λ) = λ + λ log (6) a mλ It is clea fom equatio (6) that the goud state eegy essetially cotais the mass tem. But ude the quatizatio coditio such as a does ot deped at all o the mass tem ad is give by E This immediately idicates that ( λ ) = λ( log ) (7) mλ h, we fid that the eegy Δ E = E+ E = λ log (8) + ad the eegy diffeece E 0 as. The idepedece of the eegy o mass Δ tem ca be easily udestood fom pue dimesioal aalysis as the scale λ sets the eegy i the poblem. Moeove, Boh quatizatio ca be applied to this poblem to udestad the boud state eegy depedece o the quatum umbe as follows. If L is the obital agula mometum, the simple scalig aalysis gives us L m λ log (9) a Usig the Boh quatizatio L = h ad a, we fid the eegy levels E h λ ) = λ log( ) m ( The quatum mechaical atue of the poblem is maifested oly though the discete quatum umbe athe tha Plack s costat h. (0) (b). Boud state fom Hellma-Feyma (HF) theoem: To apply the Hellma-Feyma theoem, we have to ote dow the viial theoem. This appoach elies o computig the chage i eegy with espect to some paamete without explicitly kowig the wave fuctio. Detailed accouts of this appoach with seveal iteestig poblems have bee discussed i this joual []. Although ad p ae idividually hemitea opeatos, howeve the combiatio p is ot. But we ca 8

9 p + p fom the combiatio G = ad use it to obtai the Heisebeg s equatio fo p the Hamiltoia H = + V ( ). A simple calculatio usig the fudametal m commutatio elatio [ x, p] = ih eveals that d G dt = p m V () Now, fo statioay states, the left had side is zeo, we get p m = V () Applyig equatio () to specific logaithmic potetial, we fid that the expectatio value of the kietic eegy i the -th state is set by the scale λ as idicates that the expectatio value of the full Hamiltoia is give by H = λ + ψ V ( ) ψ (3) T = λ. This Now, usig Hellma-Feyma theoem, we ca calculate the vaiatio of eegy levels with espect to mass paamete. Ad it implies fom equatio (3) that m ( E E ) 0 + = It is ow evidet fom equatio (4) that the sepaatio betwee ay two eegy levels (eed ot be cosecutive oes) of such potetial is idepedet of mass. (4) (c). Boud state fom phase space quatizatio: I this sectio, we would like to compute the boud state eegy fom the quatizatio of agula mometum. Howeve, istead of the usual quatizatio fom WKB appoximatio [, 3] fo this logaithmic potetial of age a, a 0 p( ) d = ( / 4) πh (5) we poceed via the elatio as idicated alteatively by the phase space itegatio techique[4] 9

10 I the give situatio, 0 E p λ 4mλ ae e ( p) dp = ( + / )h (6) =, we fid fom equatio (6) E p λ 4mλ ae e dp = ( + / )h (7) 0 Afte simple itegatio, the boud state eegy eige value tus out as ( + / ) h E = λ log (8) a 4πmλ Hece, the eegy level sepaatio betwee ad +states is give by + 3/ E+ E = λ log (9) + / This equatio (9) should be compaed with WKB esult [9]: + 3/ 4 E+ E = λ log (30) / 4 I figue 3, we plot the successive eegy level sepaatio as obtaied i equatio (9 ad (30) i uits of λ as a fuctio of. We have teated as a cotiuous vaiable fo the compaiso of the two esults obtaied i the above equatios. It is evidet fom the above figue that although thee is diffeece i eegy level sepaatio fo low quatum umbes, howeve, the two esults match fo >6. It is evidet fom equatios (9) ad (30) that the level sepaatio betwee ay two eegy levels is idepedet of mass tem m as well as Plack s costat h. I othe wods, whe a paticle dops fom the fist excited state to the goud state, the eegy of the emitted photo is E -E 0. Howeve, supisigly, the fequecy of the emitted photo tus out to be idepedet of the mass of the paticle i this logaithmic potetial. 0

11 Successive Eegy level sepaatio Phase space quatizatio WKB esult Figue 3: Compaiso of successive eegy level sepaatio (measued i uits of λ ) as a fuctio of The effective logaithmic iteactio betwee quasi-paticles i d iteactig system i a stog magetic field was fist demostated by Laughli [5] i coectio with quatum Hall effect [6]. The may itiguig featues of Laughli wave fuctio ad its coectio with Ladau poblem have bee lucidly discussed i the liteatue [7]. 4. Coclusios ad Pespectives To coclude, we have used the dimesioal aalysis as oe of the easiest yet poweful tool i theoetical physics to pedict the depedece of some obsevable quatities o the physical paametes of the boud state eegy levels of logaithmic potetial. The supises ecouteed i the logaithmic potetial have bee demostated fom diffeet poits of view. The sepaatio betwee the eegy levels i such a potetial is show to

12 be idepedet of the mass of the paticle as well as Plack s costat. We hope that studets will beefit fom the appoaches of this aalysis of logaithmic potetial i quatum mechaics. 8. Ackowledgemet I would like to ackowledge my studets of post-gaduate class fo askig seveal questios elated to this matte. Refeeces:. G. I. Baeblatt, Dimesioal Aalysis, (Godo Beach Sciece Publishes, New Yok, 987).. H. L. Laghaa, Dimesioal Aalysis ad theoy of models, (Wiley, New Yok, 95). 3. D. Jaa, Phys. Edu.,5, 35 (008). 4. D. Jaa, Phys. Edu.,9, 67 (00). 5. T. Padmaabha, Resoace, Jue, 50 (008). 6. M. Has, Am. J. Phys., 5, 694 (983). 7. P. Ramod, Field Theoy: A Mode Pime (Bejamis/Cummigs, Readig, MA, 98). 8. N. Goldefeld, Lectues o Phase tasitios ad the Reomalizatio Goup, ( Addiso-Wesley Publishig Compay, New Yok, 99). 9. P. Gosdzisky ad R. Taach, Am. J. Phys., 59, 70 (99). 0. K. Eveke, D. Gow, B. Jost, C. E. Mofot III, K. W. Nelso, C. Stoh ad R. C. Witt, Am. J. Phys., 58, 83 (990) ad efeeces thee i.. D. Jaa, Phys. Edu.,5, 7 (008).. David J. Giffiths, Itoductio to Quatum Mechaics, (Peaso Educatio, Sigapoe, d Editio, 005). 3. David Pak, Itoductio to the Quatum Theoy, (Mcgaw-Hill, 99). 4. S. Nagabhusaa, B. A.Kagali ad S. Vijay, Am. J. Phys., 65, 563 (997). 5. R. B. Laughli, Phys. Rev. Lett., 50, 395 (983). 6. R. E. Page ad S. M. Givi, The Quatum Hall Effect, (Spige, New Yok, 990). 7. D. Jaa, Ladau Poblem ad Laughli Wave fuctio (To appea i IUP. J. Phys. (00)).