Problems on Harmonic Oscillator, Assignments of Quantum Mechanics

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Harmonic oscillator Consider the ob : in Fig Harmonic oscillatoy sf mac (mn! as shoen fa tet [o> 14> 12Y,--Inm> be the Ground, fisst Second - and \n? excibed state. Consider lowering and Yaising operator @ = oh ey Asi, Jamie Om +P.) ane 2mhe Cmo3-iP a) The (!' 4), State express ag in terms of o> is 3) Ind= “4 (at) Jos b) Inds = (at) LoS ) Inez BE by 3) In = ory (ot)°” [op Bind= Aln-aAS = of ind = AInedS aln>= (nIn-4dy of iny = SneT Int dS al = Ids > (| = al [o> Put Nea a4 =Vzay > Ros Zot lay > I = a at ot lay “12> = EO)? pul neo a*)> = BIS 4 I = \E a [ay ln bot (EN > Id = Eo) > In = = 7 ? Yn! (<*) 2 Consider simple Harmonic escillates of mass = = {nat [nvignlny 4amw~ > ex = 4m Pasa GP 4en43) : GS zn be the 7 | position. The oncertanity 1” we P© &c00n is | & 3k d) [2h 0 (ES ame 9 amr dames | > we know = [Be * a ti (a*+a) S> = kniainy = (5 Cn! attal(n> = 0 <> = |B Scan (a°b +b¥a) >) a Wt Rls + A las n=4 = ak me 3 <> > [2% (20h) = a2 Gh = 3 2 Su S “mw wd =? 0b <2 = amu [cen lal 4 candy) [b1? | dx = _k A ee Zz 2m L@ 3 + (9). = a A, aa a1 | 25 = Bb /as _ ae = amin (SE) = Bo He. oak 608 Tees Gy? = |e pS 2h \ mw 25 Mw DX = [ seek -t@bh =| 12a Aagmw wma AMe Qt. Consider of mass'm! and Angular Frequency 'uo' The Osallater is known 40 be a state which Is in linear Gmbination of basis State. The OSCillator fanckion has the Harmonic oscillator one and ree nodes. The probability of finding the oscillakor in the given Slade are & Ss and = vespechvely, The expectation value = [Eocen (4- (a-b)*) are the slate with tn! is Integer The Probability of Binding the oSCillator in lad Is Vs VD A O23 QS a ‘ lator > Gwen oscillator js agsymmetne oS ci (lato Steonly odd Values are allowed SUD = Eis {El i» = (214 EID PUD =4 | Cnader ¢ Classical Hatmonio OScillatoy Wh bh Maximum amplitude “A’. The osail iS Symmetric about origin, e} the Oscitiadoy ob a [email protected]. ich ° SKlla tec (obd¥ potentia] The Pw babi |ity density Point oh cre amplitude 15 =a of A from mean position. 24 Ba vy —_ . Gia > Sia 9 tA 7) TR Se > Hormonic ogeiilater. doe 2 —— 4080 =0 ak > x: Agsnwt > = =sinw kt ve de FET AW Cosw t = AWL\-srtel - 2 Vz Aw are = tos Great Prrly = Fav outcames ib abled od Total outcomes eens xm=V b or =, Oz) Consider the wavefunction of the Hatmon jo Oscillator _2 4 git EY, The &x Dectation valye of the Position at bing £16 (m &eisthe mass € o 7/8 Frequenr.” of oscillator ) Y a) pet _cosw t b) ZER coswt \ N 1 oc) x‘. swt d) Ist _@swt San, lemw \ w= 4 4 WS GI>t-ED . -iE2t bgt Winds Bek ip 44 Vs Ve & lay iGat leat t <4lxlven> = 2 eh falt & ~iG2t igs Re ast ala| 2% Se Flay tlh Bra, TE Ws thy + bY nay Fig ath real ae sae Cate a atta ih) <*) Pont!) 2ab swt Here a= 2 gy Fa a Fe. = thea, % 2 “7 ¢= Bae aE Resot ~ fee Bh swt ° omno 5 SO a = pa cosw t Som Ww = [ak . Zengeset Given Seiliaka Is Truncoded Hoo Conty odd Valves allocwes) Gz ntl) hide 4 Cry +L) hwy Bet &) Bet mye L) bcaus) G-= (nace +4ny +2) Bex = Gat any +S/,) 60 &16 The Harmonie ogei\ \oLor potential for 2D ig given by Vet.) = mux + > muy? to SE M4W are the mass and Freruencs of the Scillater. The ground state eneray of this oscillafar ® to by ho Vv | ee Se => VGuy\e moe 4 Bmody? Lm eyes 4 . a EDS 25 Sm (Be) y* L 22 TNwGy 4 paar Lr pos QMS y “x= NEw ' wy [Ew Q es Ox+ 1) boy Cry + 4) hey . = (mart) § Gu + Cnye lL) + [Ro = (Vans 4 v5 St. (Ey ee Sy + 2)3 AN) for Ge i Jan avi) = Py so e€- 4 4. (a+ FB)se a cet of particles ahich Interac by A pair Vr) = arf, Where + I's S€paration and ad0 such) particles has Yatio of the the inher pasticle is q@ Cont. Vf the system, of yeached \itial equilibyum, the kinetic 45 the total eneray of the Syslen| ic Eas oy Come oat 2 VM= ay? wee Kz = 3

7= 04,019 F ») ema (2P+3) 5 M2 4:3/8,4 | Le 3) | Bas Canes) ; Peeves | pa Sra (2n + £) , => 4 2 Vix) = Tmax fy x56 a Por Xo for this only odd Values of hh’ exist Na 1,3,5 Can . For Symmetno oscillator Kx2®> = she NAVE Replace “YH with Can+0) (edd aum) >) = Ba Canto) (M=0,1,2,---) — G22. Consider a parle of mass in moving in to Harmonic potential Vee) = Ee » lxl slate are K25= ©,

co then expectation Value of <2 PES [6 a zero WR vE th yg -k Oi Q ‘ay. > n Lo oan 4 ‘ Tears (mex +b Px.) Q24) The Hamilton for a simple Harmonic OFC lla ay kg bs a 2 ' He _ + Lise . introduce the Compley q G Noa a PE xt (2 )) §, [me [me (= mi Mus hich of the Folloeing yepresents Solution Rauations af motion for @ and a* S op a) a- ae iv . ate a0 £ 1 arwt by = Heh ya Ko dae| { -<) A= Ay gt & ; af = ax eivt | - iw ‘ _ $) aA=Q @! ee aieit Js > ae! of motion d L mT Te ¥) da * Et hWarx dt 1K da = -twa® dt a da = -iwa at Similar da : A= ASG iat * * dow. Wa d | ei more. @.28) (Consider the dimensionless Harmonic oscillators Hamiltonian 7 dl We Page ith P= 7! ay . WoCa) = oxi? and ‘@,¢x)= C1 +ax> e «fa CigenFunction of th. The Value of the Be fficrent 1 @ such that — apacsx) = Citaxd)g 2h is otthogonal are to ox) is PR eer on ott d) - eae D) Pal {ategy ic. soln: Gadition is ~ j hs Cayce) doe <0 7? a 2 2/ = fa -X/2 Ve (4+ a2 e daxso -~7 2 - (nyt >) S pestuibalion EC . Ens Zac - HS