ERIC EJ1053857: Eigenenergies of a Relativistic Particle in an Infinite Range Linear Potential Using WKB Method

2011·0.77 MB·English

by  ERIC

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Eigenenergies of a Relativistic Particle in an Infinite Range Linear Potential Using WKB
Quantum-Classical Connection for Hydrogen Atom-like Systems
Method
Trajectory of Charged Particle in Combined Electric and Magnetic Fields
Using Interacti ve Spreadsheets
DebapriyoT S. ySahmi1v aanldin Agrauspw Raomy2y * and B. A. Kagali**
1Department of PhysicsP, Boapraats Sat. GTaomvebrnamdee n t College, 10 KNC
Road, Barasat, Kolkata - 700124, India
Prof. Ramkrishna More College, Akurdi
*Department of PhysicEs-,m Gaoil:v esyranmm.deenbta pCroiylole@ggem faoirl. cWomo men, Mandya-571401, India. [email protected]
2DepartmentP ouf nPeh y4s1ic1s ,0 S4c4o,t tIinshd iCah urch College
**Department of Physicpss, tBaman3@garleodrieff mUaniilv.ceormsi ty, Bangalore-560056, India, [email protected]
Urquhart Square, Kolkata – 700006, India
E-mail: aryscot [email protected]

Abstract

Energy eigenvalues for a non-relativistic par t icle in a linear potential well are available. In this paper we obtain the
eigenenergies for a relativistic spin less particle in a similar potential using an extension of the well-known WKB
Abstract
method treating the potential as the timeA cbosmtrapcot nent of a four-vector potential. Since genuine bound states do not
TTheh eB ooebhxjrei-scStto ivmfeom ro efp rtfhoeilssdi a tqirvtuiaecnl etr uiimss it nothg ge roparpoyh tsiecpnaeltcliiyaf ilieslls,u tsohtreua rrte u clteoas l tcohufe lqsauttuaidnoetninzstas t aitohrnee pfvohray lsciiidcrca ulo lpanhrl eyann ofdmo erel nlifopantii rcolafyl m olroobtwiiotsn- lf yoofri ncagh oa nrlgeee-vde ls. The transcendental
elpeacrttriocenlie gh uyendndrvoeagr eltunhe ea taoecmqtiuo–nali tkoieof snsyi msttheumaltta. nTiesho iuoss ba erttlaiecicnltere iidcll uaisnstdr a smtoeasl vghneoedwti cua fsfioienrldmgs u bMlay casotihmnneuemlcattaiinntiggc taph aesr tpoircfiltnewc miaporaetli oqtnuo ao nngtu eamt c enouimgmpebuneteerr n. ergies. Our results are
‘nD’ iaffnedr etnhtei alle enqgutha tioofn tsh oef mmaojotiro nasx iasr eo fs oalnv eedl laipntaiclyatli coarlblyit amnady p abteh aorfr ipvaerdt icalte s itna rtthinrege -fdroimme tnhseio qnuaal nstpuamce mareec hoabntaicinael d
deusscinrigpc tioinomtne rpaanacdtri evhdeo wswp irniet ahtdh seth hleiometsi. te Sw pfhroeeran d ‘tsnhh’ee ie stn sl oacrnagn-er boeenl aest egitveutipss tttohic es oecxlavpese ecn.tu emdT echrleaic sasrlie cssaoull rulettssiou nlmts. aoyf cfoimnpdl eaxp spylsitcematsi.o Tnhsi sb weislil des having conceptual
Kmeyawkeos rithgdesn :i cQfoiunccaaennptuct,me w., hhiycdhr osgoe fna ar tiosm le, fqtu taon ttuhme ambsetcrhaacnt iicms,a ogninea-teiloenc tcroonm aet oamlivse, cfloars stihcea ls mtuedcehnat,n iacnsd initiate a deeper
underKsteanydwinogr odf sp:a rSticelme mi-octliaosns. i c al theories and applications, Bound states, Relativistic wave equations
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Introduction
According to the non-relativistic version of quantum mechanics, the energy of the electron in a
Introduction
hydrogen-like atom depends only on the principal quantum number n (apart from the atomic
In recent years, physics educators have started to look more closely at what their students

number Z of the nucleus.) Again, in non-relativistic classical mechanics, the energy of such a
understand about physics concepts. A primary goal of physics education research is to identify
particTleh, ew hWich. Kde.Bsc rimbeest hano del litphtaicta lw paatsh ithnadt ekpeeenpsd tehnet lnyu cpleruosp aot soende fboycu sG, d. eWpenednst zoenlly, oHn. A.Kramers and L.
difficulties in learning physics through traditional methods and to develop new instructional
thme osdiBzeesr i(fl2olaro )eu ofiffne tc hitsiev aem alsejeoamrrn aiin xcgisl .a( MsAssci cDoaneler m aepoxtppte,r co2t0xs 0itm1o) r.a etOcioonvene orm ft heteht ehc olbadesss,t i wcianhls trirecushcu tliitos i nnea xlt htmee noladsreigvse ea nllyl ol iwumsiinet gd in quantum theory
of quantum mechanics (‘the correspondence principle’), there ought to be a connection between
for tthoe odebvtealionp mboenutn odf isntdaitveisd uianl esxlopwerileyn cvea irsy cionmg ppuotetre snitmiaullsat i(oBnrsa. Cnsodmepnu taenr sdi mJoulaactihoanisn c,a 2n 004). It can only be
n and 2a. This connection will be examined in this article. We begin by recalling the salient
be the best mode, when the tactile, kinesthetic experience of real objects is impracticable (Arons,
applied to one-dimensional problems and for three-dimensional problems that are reducible to
features of the two approaches:
1987). Computer simulations are also useful for providing more extended practice in thinking
The colanssei cdailm orebnitssi uonnd.e Wr in.vKe.rBse -tseqcuharnei tqyupee fionrvceo l(v‘Keesp pleoriwane ro rsbeitrsi’e):s expansion of the wave function in terms
about a wide variety of examples. It is capable of supplying continual feedback regarding error
Tahned f ooclfol orrrweeicdntngue cqseus daan ntPidtli aersen ircnekfmo'srac iincn ocgon ntshsteea rnvhtea!dn d(.sR Saonmna ooabonsdteh Jr ovmaagtia,o t1nc9sh 9iw1n;h gSe nyo nlfga ettt haenre d h wGavraeifv fceita hrf,r ui1en9d5c 9oti)uo:t .n Fiosr done at the points
p r o p w e r h u e n r d e e r sk t ai nn1d.e i tEnicgn eorefg nype,h rwygshiycic sh oc dof enpctheenpedt ss p tooan rtlthyie co lsnet u thdbeee nsctiszo,e mt eoafec sthh eeer smq cuaajanol r u astxoei sc:op motpeuntetri aslim eunlaetirognys.. This leads to the
Teacqhuear nshtiozualdti odenv ecloopn dciotmiopnu tefrr osimm uwlahtioicnZhs e oa2fp pphryosxicism paroteb leemigse snoe tnheatr giti ebsec comouesld e abseie rd feotre rmined for ground
E =! (1)
studeanntds toe xucnidteerdst asntda tthees p(hSehnaonmkeanar,. 2 0 1 0).2 aSeveral researchers have used this method to obtain bound
Spreadsheets can be a powerful tool in physics teaching-learning. From data analysis and
states in a slowly varying potentials (Trost & Friedrich, 1997) Kagali et.al. used a new approach
g r a p h i n g t o a n i2m. aAtinognu laanr dm soimmeunlatutimon vse, cMtoirc; rosoft Excel(cid:138) is a very versatile. program for the
t e a c ht o e r s c a a n l dc u s t l ua3dt.e eRn utbsn.o gTueh-nLed ea ndzsv tvaanetcteatsog re, u owsfhi nuicsghin ggsi pvae asscp terhe ea iddnsitrheeecgetrti oainnti ootefn at chhmei nmegta-hjloeorad ran xii(nsKg. apgroacleis se its. athl.a,t 1997). The WKB
the ptrroegartammemnitn gto i sr setlraeatimvliisnteidc awnda vlees se tqimuaet iiso nnese dise dn toot efnatmer itlhiea rn.e Icnes sraerlya tciovdies.t iTch qe ustarnontug m theory, for many
Hfyedartuoprgreeosn bo-llfie ksmep raset aoodmfs hipneh eqty uasarinect authlm ei inmr tececerhlela sbntai csesex d(aG scthtra ustcaotkul uraent idaon Lndos tk haaern eas tihmnaopntl,e a1 i9vn8ate4irl)fa:a bcele t heavt eisn eians yo tnoe u dsei mension. Therefore
Tfhoer wnieta wvie s u fsuhenricsgt hiaollnsyo ( .i igTmnhoper oipnrogtw asenprit n a)tn ood f sitinhmev peslysicsttiietgyma o t:ef aa snpdre aadpshpeleyt itsh teh amt theeth doadtas moafn ipauplpatrioonxsi marae tion, which readily
held in front of the user in a very direct and accessible manner. In addition, the spreadsheet
works in non-relativistic cases. It is found that the extension of well known non-relativistic
program itself provides for# scre(re,n" ,g!r)ap=hNicRs, c(rh)aYrts,(" a,!nd) e a s y(2-d) ata manipulation using large
numabperp roof xfuimncatitoinosn, so nm-sicgrheetnn lmb neu mpoersisciabl laen dnilf vtihsuelma lr efelaedtibvaicskt,i ca nwd afavset ceaqlcuualtaitoionnss a(Wrea grneedru, ced to Schrödinger
like form. In this article we discus relativistic approach to WKB method and we apply the same
60
to obtain eigenenergies of a spin less particle in an infinite range linear potential.
49

WKB Method for Relativistic Bound States

It is of great practical importance to extend approximate methods to relativistic problems, since
a very few eigenvalue problems could be solved exactly in relativistic quantum theory. In this
section we extend WKB method to obtain bound states of relativistic spin zero particles. We
follow the formal method of non-relativistic quantum mechanics to choose a simple well-shaped
effective potential with two classical turning points x and x as shown in figure 1. It is shown
1 2
that WKB approximation could be used in the regions for which E is greater or less than V ,
eff eff
72
EEuurrooppeeaann J oJuorunranl aolf oPfh yPshicyss (iSccsi eEndcue)c Eatdiuocna t i o n V o lV.2o lN.2o N.2o . 2 2 0 21011 1 I S ISSSNN 1 1330099 77220022 S ySahmiv aanldi nRgoays wamy and Kagali
!"#$%&'()*$"#('+)$,)-./0120)!3"2'41$()))))))5$+6)7)8$6)7)))))))))))))))))))79::))))))))))))))))))))))))))));<<8):=9>)?797)) ) @'AB'3&)

but fails at classical turning points. However connection formulae would serve at these points. In
Quantum-Classical Connection for Hydrogen Atom-like Systems
relativistic quantum theory, it is well known fact that the one-dimensional relativistic equations
Trajectory of Charged Particle in Combined Electric and Magnetic Fields
could be transformed into Schrödinger form as:
Using Interacti ve Spreadsheets
Debapriyo Syam1 and Arup Roy2
1Departmednt2 "of Physic2sP,m Boapra[ats Sat. GTaomvebrnamdee n t C]ollege, 10 KNC
ProRfo. aR+da, mBakrraisshatn,E aK Molkoartea! C- Vo70ll0e1g2e4, ,A "Inkduirad= i 0 (1)
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[email protected]
Urquhart Square, Kolkata – 700006, India
Where effective energy anEd-m eaifl:f earcytsicvote ti [email protected] for a general potential V(x) in vector coupling

scheme could be written as,

E = E2 !m2c4 ;VAAbbssttrr=aacc2tt E V(x)!V2(x) (2)
TTheh eB oobhjre-cStoivmem oef rtfheilsd a qrtuiacnlet uims t oteh fgefroarpyh sicpae2lclmiyf iicells2u tshtrea rteu lteoes f tofhfe qsutuadnetinztas2t itmohnec pf2ohry sciicrcaul lpahr eannodm eelnliopnti coafl morobtiiotsn f oofr cah oanrgee-d
elpeacrttriocnle h uynddroegr etnhe a taocmtio–nli koef ssyimsteumlta. nTehoiuss a ertliecclter iicll uasntdra mteas ghnoewti ca ffioerldmsu blay csoimnnuelcattiinngg tphaer tpircilnec mipoatli oqnu aonntu am c noummpbueterr .
‘nD’ iaffnedr etnhtei alle enqgutha tioofn tsh oef mmaojotiro nasx iasr eo fs oalnv eedl laipntaiclyatli coarlblyit amnady p abteh aorfr ipvaerdt icalt e s itna rtthinrege -fdroimme tnhseio qnuaal nstpuamce mareec hoabntaicinael d
deusscinrigpT tiinohtneu raasnc dtei vhqeou wsap irtnei aotdhnseh lei(me1ti.) t S wcphroeeaund l‘sdnh’e biest esl a crwagner boietn tese egnteut psa ttsoh es oexlvpee cntuemd ecrlaicsasli csaoll ruetsiounlts. of complex systems. This will
Kmeyawkeo rthdes : cQouncaenptut,m w, hhiycdhr osgoe fna ar tiosm le, fqtu taon ttuhme ambsetcrhaacnt iicms,a ogninea-teiloenc tcroonm aet oamlivse, cfloars stihcea ls mtuedcehnat,n iacnsd initiate a deeper
understanding of particle motion.
InKteryowdourdcst:i oPnh y s ic s Educationd, e2l!ectromaPgn2etic fields, spreadsheets, simulation.
+ R!=0 (3)
Introduction dx2 !2
According to the non-relativistic version of quantum mechanics, the energy of the electron in a
hydrogen-like atom depends only on the principaVl qu(axnt)um number n (apart from the atomic
In recent years, physics educators have started teoff look more closely at what their students
nuunmdbeerrs taZn odf atbhoeu tn upchlyesuisc.s) cAognacienp, tsin. Ano pnr-irmelaartyiv gisotiacl oclfa psshi ycsailc ms eeEdchuacnatiicosn, trhees eeanrcehrg iys otof siduecnht iafy
pdariftifcicleu,l twiehsi cihn dleeascrnriibnegs pahny eslilcisp ttihcarol upgahth ttrhaadti tkioeneapls mtheet hnoudcsle uans da te tfoof n dee fvoecluosp, dneepwe nidnss torunclyti oonna l
the size (2a) of the major axis. As one expects to recover the classical result in the large n limit
modes for effective learning (McDermott, 2001). One of the best instxructional modes allowing
of quantum mechanics (‘the correspondence principle’), there ought to be a connection between
for the development of individual experience is computer simulations. Computer simulations can
nfeb aaetn utdhre e2s baoe.f s Ttth hmei sot wdceoo, n awnpehpcertnoio atcnhh ewe tsai:lc lt iblee, ekxinaemstihneetdi ci ne xtphexisr1i eanrctiec loe fx. r2We ael obbejgeicnt sb iys irmecparlalicntigc atbhlee s(aAlireonnts ,
1987). Computer simulations are also useful for providing more extended prac tice in thinking
The classical orbits under inverse-square type force (‘Keplerian orbits’):
abouFt iag uwried e1 .v Sarciheteym oaft iecx armepprleess.e Intt aist icoanp aobfl ew oelfl ssuhpapplyeidn ge fcfoencttiinvuea pl ofeteendbtiaaclk w reitgha rtdwiong c elarrsosri cal turning points x
The following quantities remain conserved (Rana and Joag, 1991; Synge and Griffith, 1959): 1
and correctness and reinforcing the hands on observations when latter have carried out. For
a n d x 1. Energy, which depends only on the size of the major axis:
proper unde2r standing of physics concepts to the students, teachers can use computer simulations.
Teac her should develop computer simulationZs eo2f physics problems so that it becomes easier for
E =! (1)
studWenths eton uenfdfeercsttaivnde tphoe tpehnetnioaml evnaar.i e s s m2oaothly, the wave function may be written as,
Spreadsheets can be a powerful tool in physics teaching-learning. From data analysis and

g r a p h i n g t o a n i2m. aAtinognu laanr dm soimmeunlatutimon vse, cMtoirc; rosoft Excel(cid:138) is a very versatile program for the
tt eh a e c ph re o r gs r aa nm d m s it n u3gd. eRinsu tssn.t rgTeeah-!mLe elainnd(zev dxva neact)natogd=re ,l eows"fsh uitcsim(hin egxg i ivas)e ssnep etrhe"#$ede±a edd!disi rhte!oece tPeti onRintne( rotx eft) ahtcdhehe xni %&'nmegca-ejlsoesraa rarnyxi inscg.o dper.o cTehses sistr othnagt
Hfyedartuorgeesn o-lfi ksep raetaodms hine eqt uaarnet uthme imr eccehlla bnaicsse d(G sthrautcatku raen dan Ldo tkhaen astihmapnl,e 1 i9n8te4 r )f:a ce that is ea sy to use( 4)
Tfhoer wneawve ufsuenrcst iaolnso (.i gTnhoer ipnogw sepri na)n odf stihme pslyicstietym o :f a spreadsheet is that the data manipulations are

held in front of the user in a very direct and accessible manner. In addition, the spreadsheet
progWramh eirtese l!f (pxro)v iidse ss lfoowr# lsycr e(vrea,n"r y,g!irn)apg=h NifcuRs,n cc(rthi)aoYrnts.,(" Ta,!nhd)i s e a i ss y( 2-td)h aeta bmasainsip oulfa tWionK uBsi nmg elathrgoed . Carrying out the
numdbeert aoifl sf,u wnceti oonbst, aoinn- stchree enqnlmu naunmtiezraictailo nan cdnlo vnisduilmatil ofnee adsb ack, and fast calculations (Wagner,

60
x2 4'9 1 $
! P (x)dx =%n + "(!
R (5)
2
& #
x
1

Application of WKB Method to Obtain Bound States of A Particle In An Infinite Range
Linear Potential.

For a relativistic spin zero particle in an infinite range linear potential defined by
73
EEuurrooppeeaann J oJuorunranl aolf oPfh yPshicyss (iSccsi eEndcue)c Eatdiuocna t i o n V o lV.2o lN.2o N.2o . 2 2 0 21011 1 I S ISSSNN 1 1330099 77220022 S ySahmiv aanldi nRgoays wamy and Kagali
!"#$%&'()*$"#('+)$,)-./0120)!3"2'41$()))))))5$+6)7)8$6)7)))))))))))))))))))79::))))))))))))))))))))))))))));<<8):=9>)?797)) ) @'AB'3&)

Quantum-Classical Connection for Hydrogen Atom-like Systems
Trajectory of ChaVrge(dx P)ar=tickle xin Combined Electric and Magnetic Fields
(6)
Using Interacti ve Spreadsheets
Debapriyo Syam1 and Arup Roy2
T h e s c h e m a t i c1D peploartt moefn te offf PehcytsiicvseP, Bo paproaatts Seat.n GTtiaoamvelbr wnamdieet nh t Ceoflfleegce,t i1v0 eK NeCn ergy would be as shown in figure 2.
Road, Barasat, Kolkata - 700124, India
Prof. Ramkrishna More College, Akurdi
E-mail: [email protected]
2DepartmentP ouf nPeh y4s1ic1s ,0 S4c4o,t tIinshd iCah urch College
[email protected]
Urquhart Square, Kolkata – 700006, India
E-mail: aryscot [email protected]

Abstract
Abstract
TTheh eB oobhjre-cStoivmem oef rtfheilsd a qrtuiacnlet uims t oth geroarpyh sicpaelcliyf iiellsu tshtrea rteu lteos tohfe qsutuadnetinztast itohne pfohry sciicrcaul lpahr eannodm eelnliopnti coafl morobtiiotsn f oofr cah oanrgee-d
elpeacrttriocnle h uynddroegr etnhe a taocmtio–nli koef ssyimsteumlta. nTehoiuss a ertliecclter iicll uasntdra mteas ghnoewti ca ffioerldmsu blay csoimnnuelcattiinngg tphaer tpircilnec mipoatli oqnu aonntu am c noummpbueterr .
‘nD’ iaffnedr etnhtei alle enqgutha tioofn tsh oef mmaojotiro nasx iasr eo fs oalnv eedl laipntaiclyatli coarlblyit amnady p abteh aorfr ipvaerdt icalte s itna rtthinrege -fdroimme tnhseio qnuaal nstpuamce mareec hoabntaicinael d
deusscinrigp tiinotne raancdti vheo wsp irne atdhseh leimeti. t Swphreeand ‘snh’e iest sl acragne boen es egteutps ttoh es oexlvpee cntuemd ecrlaicsasli csaoll ruetsiounlts. of complex s ystems. This will
KmeyawkeoF rthidges :u cQrouneca 2enpt:ut ,mS w,c hhhiycedhmr osgaoe tfniac ar triosem lpe, rfqteu tasone ttnuhmeta amtbiseotcrnha acont ific mse,af ogfenineca-tteiilovence t crpoonom taeet oanmltivisae, clfl oawrs sitihtchea ls emtuffedecehcnatt,ni viacnesd e innietiragtey a f doere paenr infinite range linear
underpsotatnednintgi aolf. p article motion.
InKteryow dourdcst:i oPnh y s ic s Education, electromagnetic fields, spreadsheets, simulation.

The momentum of a relativistic particle with a general potential V(x) in vector coupling scheme
Introduction
According to the non-relativistic version of quantum mechanics, the energy of the electron in a
could be written as,
hydrogen-like atom depends only on the principal quantum number n (apart from the atomic
In recent years, physics educators have started to look more closely at what their students
number Z of the nucleus.) Again, in non-relativistic classical mechanics, the energy of such a
understand about physics concepts. A primary goal of physics education research is to identify
particle, which describes an elliptical path that keeps the nucleus at one focus, depends only on
difficulties in learning physics thro&ug(hE tr'adViti(oxna)l) 2me'thmods2 can4d# to develop new instructional
thme osdizees (f2oar )e offfe tchtiev em alejoarrn Painxgis .(= MAsc Done$erm eoxtpte, c2t0s 0t1o) r. eOconvee or ft hteh ec lbaesss!ti c ianls trreuscutlito inna lt hme oladregse a nll oliwmiint g( 7)
of quantum mechanics (‘theR correspondence principcle2’), there ought to be a connection between
for the development of individual ex%perience is computer simulati"ons. Computer simulations can
n and 2a. This connection will be examined in this article. We begin by recalling the salient
be the best mode, when the tactile, kinesthetic experience of real objects is impracticable (Arons,

features of the two approaches:
1987). Computer simulations are also useful for providing more extended practice in thinking
The classical orbits under inverse-square type force (‘KepleErian orbits’):
abouAt ta twuirdnei nvgar ipeotyi notfs eExampl!esV. It is= c0ap, atbalke ionfg s uxpp=lyingy c, owntein ouablt afeiend back regarding error
Tahned f oclolorrweicntnge qsus aanntidti erse irnefmoerafcifinn cgo nthseeef rfvheadn d(sR aonna oabnsde Jrovaagti,o k1n9s 9w1;h Seny nlgaett aenr dh Gavreif fcitahr,r i1e9d5 9o)u:t . For
1. Energy, which depends only on the size of the major axis:
proper understanding of physics concepts to the students, teach ers can use computer simulations.
Teacher should develop computer simulationZs eo2f physics problems so that it becomes easier for
students to understand thex phe=no!m$&e1na'. E m = c !2 !#2a (1) (8)
Spreadsheets can b1e, 2a pow$erful toEol in! physics teaching-learning. From data analysis and
% "
g r a p h i n g t o a n i2m. aAtinognu laanr dm soimmeunlatutimon vse, cMtoirc; rosoft Excel(cid:138) is a very versatile program for the

t e a c h e r s a n d s t u3d. eRnutsn. gTeh-Le eandzv vanectatogre, owfh uicshin ggi vae ssp trheea ddsirheecetti oinn otef atchhei nmga-jloera ranxiinsg. process is that
the pTrohgursa mbmy iungsi insg s tqreuaamnltiinzeadt iaonnd lreuslse t,i mwee i sg enet eded to enter the necessary code. The strong
Hfyedartuorgeesn o-lfi ksep raetaodms hine eqt uaarnet uthme imr eccehlla bnaicsse d(G sthrautcatku raen dan Ldo tkhaen as tihmapnl,e 1 i9n8te4r)f:a ce that is easy to use
Tfhoer wneawve ufsuenrcst iaolnso (.i gTnhoer ipnogw sepri na)n odmf scti2hme pslyicstietym o :f a spreadsheet is that the data manipulations are
held in front of the user 2inE a v1e"ry Edir[ect and accessible manne]r1. In addi(tion, the s)preadsheet
pnruomgbraemr oift sfeulnf cptiroonvsi,d eosn -sfkoccrr# eesnnclmr e(n!reu,n"m ,eg!rriE)acp=ah2l Ni(ca1Rsn,dn" l c(vrhyi)asY)rult2masl,(" "fa,e!nmed)d b2 e ac ac sk4 y(,2- da) an2tdad fymas=at nciapnluclu+altaiot1ino2 nus#s i(nW!g a glanregre,( 9)
0

60
On simplifying the above integral w4e9 g et,

( )
E E2 !m2c4 ! n+ 1 "!kc
2
(10)
[ ( ) ]
=m2c4 Log E+ E2 !m2c4 !Log(mc2)

The above transcendental equation is solved graphically using Mathematica (Wolfram, 1996).
To obtain numerical solution we take m =c =! = k =1. The graphical solution for
74
EEuurrooppeeaann J oJuorunranl aolf oPfh yPshicyss (iSccsi eEndcue)c Eatdiuocna t i o n V o lV.2o lN.2o N.2o . 2 2 0 21011 1 I S ISSSNN 1 1330099 77220022 S ySahmiv aanldi nRgoays wamy and Kagali
!"#$%&'()*$"#('+)$,)-./0120)!3"2'41$()))))))5$+6)7)8$6)7)))))))))))))))))))79::))))))))))))))))))))))))))));<<8):=9>)?797)) ) @'AB'3&)

eigenenergies is obtained by plotting LHS of equation (10) versus its RHS. It is as shown in
Quantum-Classical Connection for Hydrogen Atom-like Systems
figure: 3.
Trajectory of Charged Particle in Combined Electric and Magnetic Fields

Using Interacti ve Spreadsheets
Debapriyo Syam1 and Arup Roy2
1Department of PhysicsP, Boapraats Sat. GTaomvebrnamdee n t College, 10 KNC
Road, Barasat, Kolkata - 700124, India
Prof. Ramkrishna More College, Akurdi
E-mail: [email protected]
2DepartmentP ouf nPeh y4s1ic1s ,0 S4c4o,t tIinshd iCah urch College
[email protected]
Urquhart Square, Kolkata – 700006, India
E-mail: aryscot [email protected]

Abstract
Abstract
TTheh eB oobhjre-cStoivmem oef rtfheilsd a qrtuiacnlet uims t oth geroarpyh sicpaelcliyf iiellsu tshtrea rteu lteos tohfe qsutuadnetinztast itohne pfohry sciicrcaul lpahr eannodm eelnliopnti coafl morobtiiotsn f oofr cah oanrgee-d
elpeacrttriocnle h uynddroegr etnhe a taocmtio–nli koef ssyimsteumlta. nTehoiuss a ertliecclter iicll uasntdra mteas ghnoewti ca ffioerldmsu blay csoimnnuelcattiinngg tphaer tpircilnec mipoatli oqnu aonntu am c noummpbueterr .
‘nD’ iaffnedr etnhtei alle enqgutha tioofn tsh oef mmaojotiro nasx iasr eo fs oalnv eedl laipntaiclyatli coarlblyit amnady p abteh aorfr ipvaerdt icalte s itna rtthinrege -fdroimme tnhseio qnuaal nstpuamce mareec hoabntaicinael d
deusscinrigp tiinotne raancdti vheo wsp irne atdhseh leimeti. t Swphreeand ‘snh’e iest sl acragne boen es egteutps ttoh es oexlvpee cntuemd ecrlaicsasli csaoll ruetsiounlts. of complex systems. This will
Kmeyawkeo rthdes : cQouncaenptut,m w, hhiycdhr osgoe fna ar tiosm le, fqtu taon ttuhme ambsetcrhaacnt iicms,a ogninea-teiloenc tcroonm aet oamlivse, cfloars stihcea ls mtuedcehnat,n iacnsd initiate a deeper
understanding of particle motion.
InKteryowdourdcst:i oPnh y s ic s Education, electromagnetic fields, spreadsheets, simulation.

Introduction
According to the non-relativistic version of quantum mechanics, the energy of the electron in a
hydrogen-like atom depends only on the principal quantum number n (apart from the atomic
In recent years, physics educators have started to look more closely at what their students
number Z of the nucleus.) Again, in non-relativistic classical mechanics, the energy of such a
understand about physics concepts. A primary goal of physics education research is to identify
particle, which describes an elliptical path that keeps the nucleus at one focus, depends only on
difficulties in learning physics through traditional methods and to develop new instructional
the size (2a) of the major axis. As one expects to recover the classical result in the large n limit
modes for effective learning (McDermott, 2001). One of the best instructional modes allowing
offo qru tFahneitg udumerv eem l3oe.pc Gmharenanitpc soh fi( c‘itnahdle i svcoiodlruurteaislo penoxsnp dteoeri neecniegc eep nriisen nccoeipmrlgepy’u) ,ot etfrh aes irrmee uolaluatgtiihvotin ststoi. cCb eso pmai npc uoztneenrre osc iptmiaounrlt aibtceioltewn sien ec naan n infinite range linear
nb aen tdhp eo2 batee. nsTtt ihmailso. d ceo,n wnehcetnio tnh ew tailclt iblee, ekxinaemstihneetdi ci ne xtpheisri eanrctiec loef. rWeael obbejgeicnt sb iys irmecparlalicntigc atbhlee s(aAlireonnts ,
features of the two approaches:
1987 ). Computer simulations are also useful for providing more extended practice in thinking
The classical orbits under inverse-square type force (‘Keplerian orbits’):
abouItn a twheid ef ovlalroiewtyi nogf etaxabmlep,l etsh. eI t eiisg ceanpeabnleer ogfi essu popfly ninogn c-orentliantuivali sfetiecd b(aSchk arnegkaarrd,i n2g0 e1r0ro)r, relativistic particle
The following quantities remain conserved (Rana and Joag, 1991; Synge and Griffith, 1959):
and correctness and reinforcing the hands on observations when latter have carried out. For
evaluated by WKB method are compared with that of exact value (Casaubon, 2007) for n = 1
1. Energy, which depends only on the size of the major axis:
proper understanding of physics concepts to the students, teachers can use computer simulations.
Teacahnedr snh o=u l3d. develop computer simulationZs eo2f physics problems so that it becomes easier for
E =! (1)
stud ents to understand the phenomena. 2a
Spreadsheets can be a powerful tool in physics teaching-learning. From data analysis and
g r a p h i n g t o a n i2m. aAtinognu laanr dm soimmeunlatutimnon vse, cMtoirc; rosEoft Excel(cid:138) is a veEry vers atile program Efo r the
t e a c h e r s a n d s t u3d. eRnutsn. gTeh-Le eandzv vanectatogre, owfh uicshin ggi vae ssp WtrhKeeaB dd sirheecetti oinn otef atchheWi nmKga-Bjloera ranxiinsg. process is that
non-relativistic relativistic exact
the programming is streamlined and less time is needed to enter the necessary code. The strong
Hfyedartuorgeesn o-lfi ksep raetaodms hine eqt uaarnet uthme imr eccehlla bnaicsse d(G sthrautcatku rae n dan Ldo tkhaen astihmapnl,e 1 i9n8te4r)f:a ce that is easy to use
Tfhoer wneawve ufsuenrcst iaolnso (.i gTnhoer ipnogw sepri na)n odf st1ihm e pslyicstietym o :f 1 a. 8sp4r2e adsheet is th1at. 8th9e3 d ata manipu1l.a8ti5o5ns7 5ar e
held in front of the user in a very d3i rect and a3cc.2es4s0ib le manner. 2 I.n8 2ad7d ition, the s3p.r2e4ad4s5h7ee t
pnruomgbraemr oift sfeulnf cptiroonvsi,d eosn -sfocrr# eesnnclmr e(nreu,n"m ,eg!rri)acp=ahl NicaRsn,dn l c(vrhi)asYrultmasl,(" fa,e!ned)d b e a ac sk y(,2- da) antda fmasat nciapluclualtaiotino nuss i(nWg aglanregre,

60
49

Results and Discussions

We have extended WKB method to relativistic situation (Klein Gordon approach) and we have
applied the method to obtain eigenenergies of a relativistic spin zero particle in an infinite range
linear potential well. Also we have compared these eigenenergies with that of exact values. We
found that the eigenenergies obtained are well suited for relatively lower values of n, but with
increase in n, the difference between eigenenergies estimated from WKB and exact value
75
EEuurrooppeeaann J oJuorunranl aolf oPfh yPshicyss (iSccsi eEndcue)c Eatdiuocna t i o n V o lV.2o lN.2o N.2o . 2 2 0 21011 1 I S ISSSNN 1 1330099 77220022 S ySahmiv aanldi nRgoays wamy and Kagali
!"#$%&'()*$"#('+)$,)-./0120)!3"2'41$()))))))5$+6)7)8$6)7)))))))))))))))))))79::))))))))))))))))))))))))))));<<8):=9>)?797)) ) @'AB'3&)

increases due to variation in the effective potential. Further one should be careful that very high
Quantum-Classical Connection for Hydrogen Atom-like Systems
value of n for the potential of this kind may lead to a scattering situation instead of yielding
Trajectory of Charged Particle in Combined Electric and Magnetic Fields
bound states. Further the relativistic version of WKB method surely serves as an important
Using Interacti ve Spreadsheets
technique to obtain eigeDneebnaperriygoi eSysa mo1f a npda Artruicp lReosy 2i n specific potentials for which solutions are not
a v a i l a b l e i n r e1Dlaeptiavrtimsetnict o fr PehgyismicseP, B.o apIratat s wSat. oGTuaomvledbr nabmdeee n ti nCotellregees, t1i0n KgN tCo apply the method for relativistic spin
Road, Barasat, Kolkata - 700124, India
half particles. Prof. Ramkrishna More College, Akurdi
E-mail: [email protected]
2DepartmentP ouf nPeh y4s1ic1s ,0 S4c4o,t tIinshd iCah urch College
[email protected]
Urquhart Square, Kolkata – 700006, India
Acknowledgements E-mail: aryscot [email protected]

T. Shivalingaswamy would like to th ank University Grants Commission for granting financial

assistance to carryout minor research project. MRP(S)-810/10-11/KAMY056/UGC-SWRO.
Abstract
Abstract
TTheh eB oobhjre-cStoivmem oef rtfheilsd a qrtuiacnlet uims t oth geroarpyh sicpaelcliyf iiellsu tshtrea rteu lteos tohfe qsutuadnetinztast itohne pfohry sciicrcaul lpahr eannodm eelnliopnti coafl morobtiiotsn f oofr cah oanrgee-d

elpeacrttriocnle h uynddroegr etnhe a taocmtio–nli koef ssyimsteumlta. nTehoiuss a ertliecclter iicll uasntdra mteas ghnoewti ca ffioerldmsu blay csoimnnuelcattiinngg tphaer tpircilnec mipoatli oqnu aonntu am c noummpbueterr .
‘nD’ iaffnedr etnhtei alle enqgutha tioofn tsh oef mmaojotiro nasx iasr eo fs oalnv eedl laipntaiclyatli coarlblyit amnady p abteh aorfr ipvaerdt icalte s itna rtthinrege -fdroimme tnhseio qnuaal nstpuamce mareec hoabntaicinael d
deusscinrigpR tiinoetnef raeancrdtei vhneo wcsp eirnse a tdhseh leimeti. t Swphreeand ‘snh’e iest sl acragne boen es egteutps ttoh es oexlvpee cntuemd ecrlaicsasli csaoll ruetsiounlts. of complex systems. This will
Kmeyawkeo rthdes : cQouncaenptut,m w, hhiycdhr osgoe fna ar tiosm le, fqtu taon ttuhme ambsetcrhaacnt iicms,a ogninea-teiloenc tcroonm aet oamlivse, cfloars stihcea ls mtuedcehnat,n iacnsd initiate a deeper
under standing of particle motion.
InKteryowBdourrdcastn:i osPndh ye s n ic ,s BEd.uHca.t,io &n, eJleoctarocmhaaginnet,i cC fi.eJld. s(, 2sp0re0a4ds)h.e eQts,u saimnutluatmion M. echanics, 2nd ed., Pearson Education:

I n d i a.
Introduction
Accor ding to the non-relativistic version of quantum mechanics, the energy of the electron in a
hydrogen-like atom depends only on the principal quantum number n (apart from the atomic
In reCcaensat uybeaorns,, Rph.y (s2ic0s0 e7d)u. cVataorrsi ahtiaovne Pstrairntecdi ptloe lfooork am Loirne ecalro sPeolyt eant tiwahl.a tT uthrekir. Jst.u Pdehnytss . 31, 117–121.
number Z of the nucleus.) Again, in non-relativistic classical mechanics, the energy of such a
understand about physics concepts. A primary goal of physics education research is to identify

particle, which describes an elliptical path that keeps the nucleus at one focus, depends only on
difficulties in learning physics through traditional methods and to develop new instructional
the siKzea (g2aal)i o, fB t.h eA m.,a jSorh aaxriasd. aA,s Non.,e &ex pVeicjtas yto, Sre.c (o1v9er9 t7h)e. cPlahsasiscea ls rpeasuclet iinn tteheg rlaartgioe nn mlimeitth od for
modes for effective learning (McDermott, 2001). One of the best instructional modes allowing
of qu an tum mebcohuannidcs s (t‘athtees c. oArrmes.p Jon. dPehncyes .p 6ri5nc, i5pl6e3’)–, 5th6e4re. ought to be a connection between
for the development of individual experience is computer simulations. Computer simulations can
nb aen tdh e2 bae. sTt hmiso dceo,n wnehcetnio tnh ew tailclt iblee, ekxinaemstihneetdi ci ne xtpheisri eanrctiec loef. rWeael obbejgeicnt sb iys irmecparlalicntigc atbhlee s(aAlireonnts ,
features of the two approaches:
1987S)h. aCnokmapru, teRr . s(im20u1la0ti)o.n Ps rairne cailpsole us soeffu ql ufoarn pturomvi dminegc hmaonreic esx. t2enndde de dpirtaicotinc,e Sinp rthiningkeinr,g Third Indian
The classical orbits under inverse-square type force (‘Keplerian orbits’):
about a wide variety of examples. It is capable of supplying continual feedback regarding error
reprint.
The following quantities remain conserved (Rana and Joag, 1991; Synge and Griffith, 1959):
and correctness and reinforcing the hands on observations when latter have carried out. For
1. Energy, which depends only on the size of the major axis:
proper understanding of physics concepts to the students, teachers can use computer simulations.
TeacThreor ssth,o Ju.l,d & de Fverlioepd rciocmhp, uHte.r (s1im9u9l7at)i.o WnZs eKo2fB p haynsdic se pxraocbtl ewmas vseo fthuant citt iboencso mfoers ienasvieerr sfeo rp ower law
E =! (1)
stud en ts to unpdoetrestnantida lths.e Pphhein. oLmeetn. aA. . 2 2 8,2 1a27–133.
Spreadsheets can be a powerful tool in physics teaching-learning. From data analysis and
g r a p W h i n o g l f t ro a ma n , i2 mS. aA. t(ino1gn9u l9aan6r d)m . soTimmheuenl atMutimoan tvshe, ecMmtoirac; triocsaof tb oEoxcke.l (cid:138)3 rids ead vne.r yW voelrfsraatimle pMroegdraiam / fCora mtheb ridge University
t e a c h e r s a n d s t u3d. eRnutsn. gTeh-Le eandzv vanectatogre, owfh uicshin ggi vae ssp trheea ddsirheecetti oinn otef atchhei nmga-jloera ranxiinsg. process is that
Press, USA.
the programming is streamlined and less time is needed to enter the necessary code. The strong
Hfyedartuo rgeesn o-lfi ksep raetaodms hine eqt uaarnet uthme imr eccehlla bnaicsse d(G sthrautcatku raen dan Ldo tkhaen astihmapnl,e 1 i9n8te4r)f:a ce that is easy to use
Tfhoer wneawve ufsuenrcst iaolnso (.i gTnhoer ipnogw sepri na)n odf stihme pslyicstietym o :f a spreadsheet is that the data manipulations are
held in front of the user in a very direct and accessible manner. In addition, the spreadsheet
program itself provides for# scre(re,n" ,g!r)ap=hNicRs, c(rh)aYrts,(" a,!nd) e a s y(2-d) ata manipulation using large
number of functions, on-screennlm numerical andnl visulmal feedback, and fast calculations (Wagner,
60
49

76