UNIT II QUANTUM MECHANICS

Transcription

1 UNIT II QUANTUM MECHANICS Introduction: At te beginning of te 0 t century, Newton s lws of motion were ble to successfully describe te motion of te prticles in clssicl mecnics (te world of lrge, evy nd slow bodies) nd Mxwell s equtions explined penomen in clssicl electromgnetism. However te clssicl teory does not old in te region of tomic dimensions. It could not explin te stbility of toms, energy distribution in te blck body rdition spectrum, origin of discrete spectr of toms, etc. It lso fils to explin te lrge number of observed penomen like potoelectric effect, Compton Effect, Rmn Effect, Quntum Hll effect, superconductivity etc. Te insufficiency of clssicl mecnics led to te development of quntum mecnics (QM). Quntum mecnics gives te description of motion nd interction of prticles in te smll scle tomic system were te discrete nture of te pysicl world becomes importnt. Wit te ppliction of quntum mecnics, most of te outstnding problems ve been solved. Te direct implictions of QM toug very subtle ve drmtic effect in our dy to dy life. For exmple, QM successfully introduced te concept of discrete energy wic led to te conception nd dvncements in quntum computing, it lso pved wy for efficient long distnce trnsfer of lrge quntities of dt electroniclly. By understnding te electronic spin nd relted tomic properties, QM s ided in relizing energy efficient mterils tt cn be pplied to consumble electronic equipment nd in commercil trnsport veicles like Hyper loop, Mglev trins mong oters. Blck Body Rdition Blck-body rdition is te type of electromgnetic rdition emitted by blck body (n opque nd non-reflective body) eld t constnt, uniform temperture. In nture tere re no perfect blck bodies. Blck Body Spectrum: It is grp sowing te vrition of te energy of te blck body rditions s function of teir wvelengts or frequencies. Te energy distribution in te blck body spectrum is explined by Wien s distribution lw in te lower wvelengt region nd Ryleig Jens lw explins te energy distribution in te lrger wvelengt region. Wien s lw: E d 5 A e B T d

2 Ryleig Jens lw E d 8k 4 T d Fig.: Blckbody Rdition spectrum Neiter Wien s lw nor Ryleig- jen s lw could explin te energy distribution in te entire blckbody spectrum. Te energy distribution in te entire blckbody spectrum ws successfully explined by Mx. Plnck by quntum Teory. Plnck s quntum teory Te energy distribution in te blck body rdition spectrum ws successfully explined by Mx Plnck in te yer 900. According to Plnck s quntum teory terml energy is not emitted or bsorbed continuously, but it is emitted or bsorbed in discrete quntities clled qunt. Ec qunt s n energy ν were is te Plnck s constnt. Applying te Plnck s quntum teory n expression for te energy distribution in te blck body spectrum ws obtined nd it is clled Plnck s formul. Te Plnck s formul is s follows 8c Ed 5 e ( c ) kt d Were k is te Boltzmnn s constnt; - Plnck s constnt nd c is te velocity of ligt, λ is te wvelengt of te blck-body rdition nd ω is te ngulr frequency of ligt.

3 Potoelectric effect: Wen te ligt of suitble wvelengt sines on certin mterils, ten electrons re spontneously emitted from te surfce of mteril. It cn be observed in ny mteril but most redily in metls nd good conductors. Tis penomenon is known s te potoelectric effect. Te mterils tt exibit potoelectric effect re clled potosensitive mterils nd te emitted electrons re clled potoelectrons. Heinric Hertz first observed tis penomenon in 887. Te electrons re emitted only wen te potons rec or exceed tresold frequency (energy) nd below tt tresold, no electrons re emitted from te metl regrdless of te ligt intensity or te lengt of time of exposure to te ligt. To explin tis penomenon, Albert Einstein proposed tt ligt be seen s collection of discrete bundles of energy (potons), ec wit energy υ, were υ is te frequency of te ligt tt is being quntized nd is known s te Plnck constnt. Einstein s potoelectric eqution: Einstein, in 905, proposed tt te ligt energy is loclized in smll pckets similr to te Plnck s ide of qunt, nd nmed suc pckets s potons. According to Einstein, in potoelectric effect one poton is completely bsorbed by one electron, wic tereby gins te quntum of energy nd my be emitted from te metl. Tus te poton energy is used in te following two prts: i). A prt of its energy is used to free te electron from te toms of te metl surfce. Tis energy is known s potoelectric work function of metl (W o ) ii) Te oter prt is used in giving kinetic energy (½ mv ) to te electron. Tus W mv o were v is te velocity of te emitted electron. Tis eqution is known s Einstein s potoelectric eqution. Wen te poton s energy is of suc vlue tt it cn just liberte te electron from metl, ten te kinetic energy of te electron will be zero. Ten te bove eqution reduces to o W o, were o is clled te tresold frequency. Tresold frequency is defined s te minimum frequency wic cn cuse potoelectric emission. Below tis frequency no emission of electron tkes plce.

4 Compton Effect: Wen monocromtic bem of ig frequency rdition (X rys, γ rys, etc.) is scttered by substnce, ten te scttered rdition contins two components - one ving lower frequency or greter wvelengt clled s modified rdition nd te oter ving te sme frequency or wvelengt clled s unmodified rdition. Tis penomenon is known s Compton effect nd ws discovered by Prof. A.H. Compton in 9. Te process of recoiling of electron nd scttering of poton is s sown in te following figure: Scttered poton Φ----- is te recoil ngle θ is te scttering ngle. Incident poton Electron t rest θ φ Recoil electron Scemtic digrm of Compton Effect According to te quntum concept of rdition, te rdition is constituted by energy pckets clled potons. Te energy of poton is ν, were is Plnck s constnt nd ν is te frequency of rdition. Te potons move wit velocity of ligt c, possess momentum ν/c nd obey ll te lws of conservtion of energy nd momentum. According to Compton, te penomenon of scttering is due to n elstic collision between two prticles, te poton of incident rdition nd te electron of te sctterer. Wen te poton of energy ν collides wit te electron of te sctterer t rest, it trnsfers some energy to te electron, i.e., it loses te energy. Te scttered poton will terefore ve smller energy nd consequently lower frequency (ν ) or greter wvelengt (λ ) tn tt of te incident poton. Te observed cnge in frequency or wvelengt of te scttered rdition is known s Compton Effect. In te scttering process, te electron gins kinetic energy nd tus recoils wit velocity v. Te cnged wvelengt of te poton scttered troug n ngle θ is given by ' ( ) [ cos ] mc 0

5 Wve nd prticle dulity of rdition: To understnd te wve nd prticle dulity, it is necessry to know wt prticle is nd wt wve is. A prticle is loclized mss nd it is specified by its mss, velocity, momentum, energy, etc. In contrst wve is spred out disturbnce. A wve is crcterised by its wvelengt, frequency, velocity, mplitude, intensity, etc. It is rd to tink mss being ssocited wit wve. Considering te bove fcts, it ppers difficult to ccept te conflicting ides tt rdition s wve prticle dulity. However tis cceptnce is essentil becuse te rdition exibits penomen like interference, diffrction, polriztion, etc., nd sows te wve nture nd it lso exibits te prticle nture in te blck-body rdition effect, potoelectric effect, Compton Effect etc. Rdition, tus, sometimes beve s wve nd t some oter time s prticle, tis is te wve prticle dulity of rdition. De Broglie s concept of mtter wves. Louis de-broglie in 94 extended te wve prticle dulism of rdition to fundmentl entities of pysics, suc s electrons, protons, neutrons, toms, molecules, etc. de-broglie put bold suggestion tt like rdition, mtter lso s dul crcteristic, t time wen tere ws bsolutely no experimentl evidence for wvelike properties of mtter wves. de Broglie Hypotesis of mtter wves is s follows. In nture energy mnifests itself in two forms, nmely mtter nd rdition. Nture loves symmetry. As rdition cn ct like bot wve nd prticle, mteril prticles(like electrons, protons, etc.) in motion sould exibit te property of wves. Tese wves due to moving mtter re clled mtter wves or de-broglie wves or pilot wves. Wvelengt of mtter wves: Te concept of mtter wves is well understood by combining Plnck s quntum teory nd Einstein s teory. Consider poton of energy E, frequency γ nd wvelengt λ. By Plnck s teory c E By Einstein s mss-energy reltion E mc By equting nd rerrnging te bove equtions, we get

6 c mc mc Were, p is te momentum of te poton nd is Plnck s constnt. p Now consider prticle of mss m moving wit velocity v nd momentum p. According to te do-broglie ypotesis mtter lso s dul nture. Hence te wvelengt λ of mtter wves is given by mv Tis is te eqution for te de-broglie wvelengt. p De Broglie wvelengt of n electron Consider n electron of mss m ccelerted from rest by n electric potentil V. Te electricl work done (ev) is equl to te kinetic energy E gined by te electron. E ev E mv m v me mv me Terefore wvelengt of electron wve But E ev p mv me mev me V Substituting te vlues of e, m nd, we get.8a me o

7 .8 A o V Note: Insted of n electron, if prticle of crge q is ccelerted troug potentil difference V, ten mqv Properties of mtter wves nd ow tey re different from electromgnetic wves. Ligter te prticle, greter would be te wvelengt of mtter wves ssocited wit it.. Smller te velocity of te prticle, greter would be te wvelengt.. For p = 0, λ is infinity ie., te wve becomes indeterminte. Tis mens tt mtter wves re ssocited wit moving prticles only. 4. Mtter wves re produced by crged or uncrged prticles in motion. Weres electromgnetic wves re produced only by moving crged prticle. Hence mtter wves re non electromgnetic wves. 5. In n isotropic medium te wvelengt of n electromgnetic wve is constnt, weres wvelengt of mtter wve cnges wit te velocity of te prticle. Hence mtter wves re non- electromgnetic wves. 6. A prticle is loclized mss nd wve is spred out disturbnce. So, te wve nture of mtter introduces certin uncertinty in te position of te prticle. 7. Mtter wves re probbility wves becuse wves represent te probbility of finding prticle in spce. Heisenberg s uncertinty principle Heisenberg s uncertinty principle is direct consequence of te dul nture of mtter. In clssicl mecnics, moving prticle t ny instnt s fixed position in spce nd definite momentum wic cn be determined if te initil vlues re known (we cn know te future if we know te present) In wve mecnics moving prticle is described in terms of wve group or wve pcket. According to Mx Born s probbility interprettion te prticle my be present nywere inside te wve pcket. Wen te wve pcket is lrge, te momentum cn be fixed, but tere is lrge uncertinty in its position. On te oter nd, if te wve pcket is smll te position of te prticle my be fixed, but te prticle will spred rpidly nd ence te momentum (or velocity) becomes indeterminte. In tis wy certinty in momentum

8 Intensity involves uncertinty in position nd te certinty in position involves uncertinty in momentum. Hence it is impossible to know witin te wve pcket were te prticle is nd wt is its exct momentum. (We cnnot know te future becuse we cnnot know te present). Tus we ve Heisenberg s uncertinty principle. According to te Heisenberg s uncertinty principle It is impossible to specify precisely nd simultneously certin pirs of pysicl quntities like position nd momentum tt describe te bevior of n tomic system. Qulittively, tis principle sttes tt in ny simultneous mesurement te product of te mgnitudes of te uncertinties of te pirs of pysicl quntities is equl to or greter tn /4π (or of te order of ) Considering te pir of pysicl quntities suc s position nd momentum, we ve ΔpΔx /4π.. Were Δp nd Δx re te uncertinties in determining te momentum nd te position of te prticle. Similrly, we ve oter cnonicl forms s ΔEΔt /4π ΔJΔθ /4π Were ΔE nd Δt re uncertinties in determining energy nd time wile ΔJ nd Δθ re uncertinties in determining ngulr momentum nd ngulr position. Illustrtion of Heisenberg s uncertinty principle Brodening of spectrl lines Wen n tom bsorbs poton, it rises to te excited stte nd it will sty in te excited stte for certin time clled te lifetime. Lifetime of toms in te excited levels is of te order of 0-8 s. Wen te tom comes to te ground stte it emits poton of energy exctly equl to te energy difference between te two levels s sown in te figure.4. λ ΔE Δλ E =ν E =ν Fig..7. Line widt for emitted potons λ

9 Te energy of te emitted poton is given by c E...(). were is Plnck s constnt, ν is te frequency, c is te velocity of ligt nd λ is te wvelengt. Differentiting eqution () wit respect to wvelengt (λ), we get c E c...() According to Heisenberg s uncertinty principle, te finite lifetime Δt of te excited stte mens tere will be n uncertinty in te energy of te emitted poton is given by 4t Substituting for ΔE from () nd pplying te condition of minimum uncertinty, we get or c 4t 4 ct Tis sows tt for finite lifetime of te excited stte, te mesured vlue of te emitted poton wvelengt will ve spred of wvelengts round te men vlue λ. Tis uncertinty in te mesured vlue of wvelengt demnds for very nrrow spred, te lifetime of te excited stte must be very ig (of te order of 0 - s). Suc excited levels re clled Metstble sttes. Tis concept is dopted in te production of lser ligt. SCHRÖDINGER S WAVE EQUATION In96 Scrödinger strting wit de-broglie eqution (λ = /mv) developed it into n importnt mtemticl teory clled wve mecnics wic found remrkble success in explining te bevior of te tomic system nd teir interction wit electromgnetic rdition nd oter prticles. In wter wves, te quntity tt vries periodiclly is te eigt of wter surfce. In sound wves it is pressure. In ligt wves, electric nd mgnetic fields vry. Te quntity wose vrition gives mtter wves is clled wve function (ψ). Te vlue of wve function ssocited wit moving body t prticulr point x in spce t time t is relted to te likeliood of finding te body tere t time. A wve function ψ(x,t) tt

10 describes prticle wit certin uncertinty in position, moving in positive x-direction wit precisely known momentum nd kinetic energy my ssume ny one of te following forms: Sin( ωt - kx), cos( ωt - kx), e i(ωt kx), e -i(ωt kx) or some liner combintions of tem. Scrödinger wve eqution is te wve eqution of wic te wve functions re te solutions. It cnnot be derived from ny bsic principles, but cn be rrived t, by using te de-broglie ypotesis in conjunction wit te clssicl wve eqution. Time Independent one dimensionl Scrödinger wve eqution(tise) In mny situtions te potentil energy of te prticle does not depend on time explicitly; te force tt cts on it, nd ence te potentil energy vry wit te position of te prticle only. Te Scrödinger wve eqution for suc prticle is time independent wve eqution. Let ψ(x,t) be te wve function of te mtter wve ssocite wit prticle of mss m moving wit velocity v. Te differentil eqution of te wve motion is s follows....() x v t Te solution of te Eq.() s periodic displcement of time t is ψ(x,t) =ψ 0 (x) e -iωt..() Were ψ 0 (x) is te mplitude of te mtter wve. Differentiting Eq. prtilly twice w.r.t. to t, we get i ψ 0 (x) e -iωt t i t ψ 0 (x) e -iωt ψ 0 (x) e -iωt t - ψ.() t Substituting Eq. in Eq. x v..(4) 4 We ve k v Substituting tis in Eq4, we get

11 4 x 4 x...(5) 0...(6) Substituting te wvelengt of te mtter wves λ=/mv in Eq.6 we get x m v (7) If E nd V re te totl nd potentil energies of te prticle respectively, ten te kinetic energy of te prticle E m v mv Substituting tis in Eq.7, we get E V m( E V) 8 m ( E V ) 0...(8) x Hence ψ is function of x lone nd is independent of time. Tis eqution is clled te Scrödinger time- independent one dimensionl wve eqution. Pysicl significnce of te wve function Te wve function ψ(x, t) is te solution of Scrödinger wve eqution. It gives quntummecniclly complete description of te bevior of moving prticle. Te wve function ψ cnnot be mesured directly by ny pysicl experiment. However, for given ψ, knowledge of usul dynmic vribles, suc s position, momentum, ngulr momentum, etc., of te prticle is obtined by performing suitble mtemticl opertions on it. Te most importnt property of ψ is tt it gives mesure of te probbility of finding prticle t prticulr position. ψ is lso clled te probbility mplitude. In generl ψ is complex quntity, weres te probbility must be rel nd positive. Terefore term clled probbility density is defined. Te probbility density P (x,t) is product of te wve function ψ nd its complex conjugte ψ *. * P( x, t) ( x, t)

12 Normliztion of wve function If ψ is wve function ssocited wit prticle, ten d is te probbility of finding te prticle in smll volume dτ. If it is certin tt te prticle is present in volume τ ten te totl probbility in te volume τ is unity i.e., d. Tis is te normliztion condition. In one dimension te normliztion condition is dx Note: Wen te prticle is bound to limited region te probbility of finding te prticle t * infinity is zero i.e., t x is zero. x Properties of wve function: Te wve function ψ sould stisfy te following properties to describe te crcteristics of mtter wves.. ψ must be solution of Scrödinger wve eqution.. Te wve function ψ sould be continuous nd single vlued everywere. Becuse it is relted to te probbility of finding prticle t given position t given time, wic will ve only one vlue.. Te first derivtive of ψ wit respect to x sould be continuous nd single vlued everywere, since it is relted to te momentum of te prticle wic sould be finite. 4. Ψ must be normlized so tt ψ must go to 0 s x, so tt d over ll te spce be finite constnt. Eigen functions nd Eigen vlues Te Scrödinger wve eqution is second order prtil differentil eqution; it will ve mny mtemticlly possible solutions (ψ). All mtemticlly possible solutions re not pysiclly cceptble solutions. Te pysiclly cceptble solutions re clled Eigen functions (ψ). Te pysiclly cceptble wve functions ψ s to stisfy te following conditions:. ψ is single vlued.. ψ nd its first derivtive wit respect to its vrible re continuous everywere.. ψ is finite everywere. Once te Eigen functions re known, tey cn be used in Scrödinger wve eqution to evlute te pysiclly mesurble quntities like energy, momentum, etc., tese vlues re

13 clled Eigen vlues. In n opertor eqution ^ O were ^ O is n opertor for te pysicl quntity nd ψ is n Eigen function nd λ is te Eigen vlue. For exmple : H E Were H is te totl energy (Hmiltonin) opertor, ψ is te Eigen function nd E is te totl energy in te system. We cn ve similr equtions for te momentum P p Were P is te momentum opertor nd p denotes te momentum eigen vlues. Anoter exmple would be : L Z m Were L z is te z-component of ngulr momentum opertor nd m is te zimutl quntum number. Applictions of Scrodinger s wve eqution. For Prticle in n one-dimensionl potentil well of infinite dept (Prticle in box) Consider prticle of mss m moving freely in x- direction in te region from x=0 to x=. Outside tis region potentil energy V is infinity nd witin tis region V=0. Outside te box Scrodinger s wve eqution is 8 m x E 0...() Tis eqution olds good only if =0 for ll points outside te box i.e., 0, wic mens tt te prticle cnnot be found t ll outside te box. Inside te box V = 0, ence te Scrodinger s eqution is given by, 8 m E 0 x k 0...() x 8m E V= were, k...() V=0 x=0 x x=

14 Discussion of te solution Te solution of te bove eqution is given by Acos kx Bsin kx...(4) were A & B re constnts wic depending on te boundry conditions of te well. Now pply boundry conditions for tis, Condition: I t x =0, = 0. Substituting te condition I in te eqution 4, we get A =0 nd B 0. (If B is lso zero for ll vlues of x, ψ is zero. Tis mens tt te prticle is not present in te well.) Now te eqution cn be written s Bsin kx...(5) Condition: II t x =, = 0 Substituting te condition II in eqution 5 we get 0= B sin(k) Since B 0, sin k 0 k n n k n k Substitute te vlue of k in eqution (). were, n =,,. 8m E n n E...(6) 8m Te eqution (6) gives energy vlues or Eigen vlue of te prticle in te well. Wen n=0, n = 0. Tis mens to sy tt te electron is not present inside te box, wic is not true. Hence te lowest vlue of n is. Te lowest energy corresponds to n = is clled te zero-point energy or Ground stte energy. Ezero point 8m All te sttes of n re clled excited sttes. To evlute B in eqution (), one s to perform normliztion of wve function.

15 Normliztion of wve function: Consider te eqution, x kx B n Bsin sin Te integrl of te wve function over te entire spce in te box must be equl to unity becuse tere is only one prticle witin te box, te probbility of finding te prticle is. dx 0 0 sin n B xdx But cos sin B B n n B x n n x B dx x n dx B dx x n B 0 sin sin cos ` cos Tus te normlized wve function of prticle in one-dimensionl box is given by, x n n sin were, n=,, Tis eqution gives te Eigen functions of te prticle in te box. Te Eigen functions for n=,,.. re s follows. x x x sin sin sin Since te prticle in box is quntum mecnicl problem we need to evlute te most probble loction of te prticle in box nd its energies t different permitted stte.

16 Let us consider te tree lowest energy solutions: Cse (): n= Tis is te ground stt nd te prticle is normlly found in tis stte. For n=, te Eigen function is sin x In te bove eqution =0 for bot x=0 & x=. But s mximum vlue for x=/. sin nd Ψ x=0 x=/ x= x=0 x=/ x= A plot of te probbility density versus x is s sown. From te figure, it indictes te probbility of finding te prticle t different loctions inside te box. =0 t x = 0 nd x =, lso is mximum t x = (/). Tis mens tt in te ground stte te prticle cnnot be found t te wlls of te box nd te probbility of finding te prticle is mximum t te centrl region. Te Energy in te ground stte is given by E 8m. Cse : n = Tis is te first excited stte. Te Eigen function for tis stte is given by sin x Now, =0 for te vlues x 0, nd nd reces mximum for te vlues x, 4 4 Tese fcts re seen in te following plot.

17 x 0 x 0 4 From te figure it cn be seen tt 0 t x = 0,, nd / t x, 4 4 Tis mens tt in te first excited stte te prticle cnnot be observed eiter t te wlls or t te center. Te energy is E 4E. Tus te energy in te first excited stte is 4 times te zero point energy. Cse : n = Tis is te second excited stte nd te Eigen function for tis stte is given by sin x now, =0 for te vlues nd reces mximum x 0,,, t x,, x 0 x x 0 x =0 for te vlues x, 0,, nd reces mximum / t x,, 6 5 t wic te prticle is most likely to be found. 6 Te energy corresponds to second excited stte is given by E 9E.

18 . Free Prticle: Free prticle mens, it is not under te influence of ny kind of field or force. Tus, it s zero potentil, i.e., V=0 over te entire spce. Hence Scrodinger s eqution becomes, 8 m E 0 x E m x 8 Te bove eqution olds good for prticle for wic te potentil V=0 over te entire spce (no boundries t ll). Since, for free prticle, V=0 olds good everywere, we cn extend te cse of prticle in n infinite potentil well to te free prticle cse, by treting te widt of te well to be infinity, i.e., by llowing =. We ve te eqution for energy Eigen vlues for prticle in n infinite potentil well s, n E 8m Were n =,,. Rerrnging, we ve, n Em Here, we see tt, for prticle wit constnt energy E but confined in te well n depends solely on. In te limiting cse wen =, it lso follows tt n =, wic essentilly mens tt free prticle cn ve ny energy Eigen vlues or possible vlues of energy re infinite in number. Hence s,, n. Keeping in mind te energy level representtion, we sy tt te permitted energy vlues re continuous not discrete.. Tree Dimensionl Well of infinite dept(prticle in D infinite well) Te generliztion of te one dimensionl prticle in box result to te tree dimensionl cse is quite strigt forwrd. Te Time Independent Scrodinger Eqution is now (-ђ /m)[ ψ/ x + ψ/ y + ψ/ z ] + V(x,y,z)ψ = (E x + E y + E z )ψ Te potentil V(x,y,z) = 0 witin te well tt is 0 < x,y,z < nd V(x,y,z) = outside te well. Tus, witin te well, te differentil eqution cn be esily broken down into independent differentil equtions long te xes wic re :

19 ψ/ x + (me x /ђ )ψ = 0, ψ/ y + (me y /ђ )ψ = 0, ψ/ z + (me z /ђ )ψ = 0 () Te solutions for ec of te tree equtions bove re similr giving us finlly : E = E x + E y + E z = ( /8m )[n x + n y + n z ]..() 4. Squre Well of finite dept in one dimensions. Motivtion : Infinte squre well potentil ssumes electrons never leve te well. So, V(0) = V() = nd ψ(0) = ψ() = 0. A more relistic description of te confining potentil is wen dept of well is finite. Tis ALLOWS to be outside te well )box) lso wit finite probbility(quntum Tunneling). Wt would ppen if te prticle s kinetic energy is iger tn te confining potentil, so totl energy is positive? Tis cn lso be modelled only by well of finite dept. Time Independent Scrodinger Eqution: (-ђ /m) ψ/ x + V(x)ψ = Eψ () Were ψ is te wve function we wnt to find nd E is te energy, lso clled Eigen energy. Te Squre well potentil cn now be described by different regions: REGION I x < 0 V(x) = V o REGION II 0 < x < V(x) = 0 REGION III < x V(x) = V o Rewriting Equ tion we get: ψ/ x = (m/ђ )(V(x) E)ψ.() In region II, V(x) E < 0, ence ψ/ x = -k ψ..() were k = (m/ђ )(E - V(x)) As V(x) = 0 in region II, k = me/ђ ψ II (x) = Acos(kx) + Bsin(kx) OR EQUIVALENTLY ψ II (x) = Ae ikx + Bexp -ikx.(4) In regions I nd III, E < V(x) (Tis is te clssiclly forbidden region, s kinetic energy is negtive, wic is not possible in clssicl mecnics, but possible in Quntum mecnics.) Terefore ψ/ x = α ψ..(5) were α is rel, α = (m/ђ )(V o E) (6) Te solution in tese two regions is given by : Ψ III (x) = Ce αx + De -αx..(7) Ψ I (x) = Ee αx + Fe -αx...(8) As x, Ce αx will diverge, unless C = 0. Terefore C = 0..(9)

20 As x -, Fe -αx will diverge, unless F = 0. Tus F = 0 (0) Tese two conditions, given by equtions (9) nd (0) re required s oterwise te solutions will blow up (diverge) nd will terefore not be normlizble. So, Ψ I (x) = Ee αx, ψ II (x) = Acos(kx) + Bsin(kx), Ψ III (x) = De -αx.() Now, we ve to mtc te wve-functions s well s teir derivtives t te boundries of tese different regions. Tus: Ψ I (0) = Ψ II (0) nd (dψ I /dx) x=0 = (dψ II /dx) x=0..() Ψ II () = Ψ III () nd (dψ II /dx) x= = (dψ III /dx) x=.() We will not work out te mts(look t te Appendix for detils), but just nlyse te results. ) Inside te well picking Cosine solution mens B = 0 nd E = D. Te corresponding trnscendentl eqution to be solved becomes α = k tn(k/). On te oter nd picking te Sine solution mens A = 0 nd E = -D nd te trnscendentl eqution is α = -k cot(k/) ) Te energy equtions tt emerge from Equtions nd cnnot be solved nlyticlly, but by metod of. Recll tt bot k nd α depend on E. Tus te continuity eqution cnnot be stisfied for ny rbitrry vlue of E, but only finite discrete set of vlues. Te corresponding eigen function re te bound stte solutions. By contrst, te energy levels bove V o is continuous. )Te potentil well is not infinitely deep but finite, so te prticles witin it re not strictly confined but cn now extend into te clssiclly forbidden region(region were totl energy is less tn te potentil energy) 4) As te probbility of finding te prticle outside te well is finite, tere is Quntum Tunneling of te prticles from witin te well to outside te well. 5) Te energies of te prticle in infinite well re iger tn te corresponding energy levels in finite well. (Tis cn be rougly understood s te finite well problem s te effective widt of te well iger tn te infinite well, ence lower energies s per te formul E n = n /8m )

21 Te concept of Quntum Tunneling s understood from te Squre Well Potentil elps us to understnd te fields of Rdictive decy, Scnning Tunneling Microscopy wic is used to study surfces nd te binding of molecules. Q.No Smple Questions CO. Stte De Broglie ypotesis.. Wt is wve function? Give its pysicl significnce nd properties.. Stte Heisenberg s uncertinty principle. By pplying Heisenberg s uncertinty & principle, illustrte te brodening of spectrl lines. 4. Using Heisenberg s uncertinty principle explin te brodening of spectrl lines. 5. Wt re Eigen functions? Mention teir properties. 6. Setup time independent one-dimensionl Scrodinger s wve eqution for mtter wve. 7 Apply te time independent Scrodinger s wve eqution to find te solutions for & prticle in n infinite potentil well of widt. Hence obtin normlized wve function 8 Solve te Scrodinger s wve eqution for free prticle. & 9 Point out te difference between prticle in n infinite well for D nd D nd rrive t te solution of te problem. 0. Set up te differentil eqution for prticle in D well of finite dept nd rrive t te solutions inside nd outside te well.. Explin te concept of tunneling by nlyzing te solution of prticle in finite well in dimensions.

22 PNo. Problems CO. Clculte te de Broglie wvelengt ssocited wit proton moving wit velocity equl to (/0) t of te velocity of ligt. To be found: de Broglie wvelengt, λ Solution: mv.670 (/ 0) 0 4 m. An electron nd proton re ccelerted troug te sme potentil difference. Find te rtio of teir de Broglie wvelengts. To be found: Rtio of de Broglie wvelengt, λ Solution: De Broglie Wvelengt, λ, me m For electron, For proton, e m e p m p e Rtio of De Broglie Wvelengts, p m m p e. Compre te energy of poton wit tt of neutron wen bot re ssocited wit wvelengt of A o. Given tt te mss of neutron is kg. To be found: Comprison of energy of poton wit tt of neutron Solution: Energy of neutron, E n 4 P (6.650 ) m m (0 ).67 0 J =0.08eV Energy of poton, Rtio of energies, E p 4 8 c Ep E n 5 6 J =4.9eV 4. An electron s speed of 4.8 x 0 5 m/s ccurte to 0.0 %. Wit wt ccurcy wit wic its position cn be locted. To be found: Uncertinty in position, Δx

23 Solution: Uncertinty principle is given by, xp 4 Uncertinty in speed, Δv = 4.8 x 0 5 x = 57.6m/s Uncertinty in position, x 0 4mv Te inerent uncertinty in te mesurement of time spent by Iridium-9 nuclei in te excited stte is found to be.4x0-0 s. Estimte te uncertinty tt results in its energy in te excited stte. To be found: Uncertinty in energy, ΔE Solution: Uncertinty principle is given by, Et E t Te position nd momentum of kev electron re simultneously determined nd if its position is locted witin Å. Wt is te percentge of uncertinty in its momentum? 5 J 4 To be found: Percentge of uncertinty in momentum of electron, Δp Solution: Uncertinty principle is given by, xp Uncertinty in momentum, p 0 4x m / kg m s Momentum, p = 6 me / kgm s p Percentge of uncertinty in momentum of electron, 00. p 7. Sow tt te energy Eigen vlue of prticle in second excited stte is equl to 9 times te zero point energy. To be found: Energy Eigen vlue for second excited stte is equl to 9 times te zero point energy. n Solution: Energy Eigen vlue eqution is given by, E 8m n=, zero-point energy stte, E 8m

24 n=, second excited stte, E 9 8m 9E 8. An electron is bound in one-dimensionl potentil well of widt Å, but of infinite eigt. Find te energy vlue for te electron in te ground stte. To be found: Energy Eigen vlue Solution: n Energy Eigen vlue eqution is given by, E 8m For n=, ground stte energy, E eV 9. An electron is bound in one dimensionl potentil well of infinite potentil of widt 0. nm. Find te energy vlues in te ground stte nd lso te first two exited stte. To be found: Energy Eigen vlue Solution: n Energy Eigen vlue eqution is given by, E 8m For n=, ground stte energy, E eV For n=,first excited stte, For n=,second excited stte, E 4E 04.6eV E 9E 5.44eV 0. An electron is trpped in potentil well of widt 0.5nm. If trnsition tkes plce from te first excited stte to te ground stte find te wvelengt of te poton emitted. To be found: Wvelengt of te poton emitted, λ Solution: For n=, ground stte energy, E.40 J.507eV For n=,first excited stte, E E J ev Energy difference,

25 E E E J ev Wvelengt of te poton emitted, 4 8 c nm 9 E 7. 0 J Tings to tink bout! (CO). Tking Plnck s lw s te strting point derive Ryleig Jens lw in te limit of ig λ.. How cn we nturlly reconcile te dul Wve nd Prticle nture of mtter?. Is tere ny connection between te position momentum nd energy-time uncertinty reltionsip? Discuss. 4. Is te derivtion for prticle in box done bove vlid for igly energetic reltivistic prticles? Discuss. 5. Wit respect to te prticle in box problem, find out wt is tunneling. Wt is its utility in different engineering domins?