Copyright c 8 by Austin J. Hedemn Physics 7A - Quntum Mechnics - Spring 8 Midterm Mondy, Februry 6, 6:-8: PM You hve two hours, thirty minutes for the exm. All nswers should be written in blue book. You re llowed one side of one pge of hndwritten notes. Plese explin your work. Answers with no explntion will not receive full credit. In the problem sttements, the phrse time-dependent Schrödinger eqution is bbrevited TDSE nd the phrse time-independent Schrödinger eqution is bbrevited TISE. Mthemticl Formuls Trigonometry Euler s Formul: e ix = cos x + i sin x =) cos x = eix + e ix, sin x = eix e ix. i Sum nd Di erence Formuls: sin( ± b) =sin() cos(b) ± cos()sin(b), cos( ± b) = cos() cos(b) sin()sin(b). Integrls Trigonometric: Exponentil: Gussin: Z sin(nu)sin(mu) du = nm, n,m Z, Z x n e x/ dx = n! n+, n Z +, sin(nu)e i u du = e x dx = n x e x dx = e x +bx dx = n e i (n+ ), n Z; R,, R+,, R+, e b /4, R + ; b C, Integrtion by Prts: Z b udv= uv b Z b vdu. Simple Hrmonic Oscilltor Eigenfunctions ( r m! ~ x) m! /4 (x) = e / m! /4p, (x) = e /, m! /4 (x) = p ( ) e / m! /4, (x) = p ( ) e /.
Postultes nd the SHO [8 points] Consider prticle of mss m in simple hrmonic oscilltor of frequency!, V (x) = m! x. You my use the expression n (x) to stnd for the n-th energy eigenfunction in ny expression, unless we explicitly sk for the function. Consider the following two wve functions t t =describingtwo (di erent) sttes of prticle in the simple hrmonic oscilltor, (x) = 5 (x)+ (x)+ +i 5 (x), () (x) =Ae x /. (b) where C is complex constnt nd A, R re both rel constnts. Postulte (Sttes): There re certin conditions we need to plce on the wve function in order for us to properly interpret them s probbility mplitudes. () [6 points] Wht is the most generl form tht the complex number cn tke in order for the wve function (x) ineq. to describe the physicl stte of prticle? Wht restriction must we plce on the vlue of in order for the wve function (x) ineq.b to describe the physicl stte of prticle? Wht is possible vlue tht A cn be in order for (x) to describe physicl stte? Postulte (Observbles): The energy of the simple hrmonic oscilltor my clssiclly be split into the kinetic energy T nd the potentil energy V. We know by Postulte tht there will be quntum mechnicl opertor tht corresponds to these clssicl observbles. (b) [6 points] Find ˆT (x) nd ˆV (x), the ctions of the kinetic energy opertor ˆT nd potentil energy opertor ˆV on wve function (x). Use this to find the expecttion vlue of the potentil energy hv i for the stte (x) ineq.b. Postulte (Mesurement Possibilities): Postulte sid tht the only possible results of mesurement re the eigenvlues of the opertor. (c) [6 points] Given wht you know bout the simple hrmonic oscilltor system, wht re the possible results of n energy mesurement on n rbitrry stte? By plugging into the TISE, find the vlue of tht mkes n eigenfunction. Wht is the ssocited energy eigenvlue of the stte? Postulte 4 (Mesurement Probbilities): Suppose we prepre simple hrmonic oscilltor system described by the stte (x) from Eq. t t =, At time t = the energy of the system is mesured. (d) [8 points] Wht re the possible results, wht re the probbilities of getting ech result, nd wht is the expecttion vlue of the energy, hei?
Postulte 5 (Collpse): Suppose we strt with the stte (x, ) = p m! /4 e /. At t = we mesure the energy of the stte nd get the result ~!. After this energy mesurement we mesure the momentum of the prticle. (e) [6 points] Wht is the wve function immeditely fter the energy mesurement? Explin. Wht is the probbility tht the subsequent momentum mesurement gets within dp of the result p = p m~!? Postulte 6 (Time Evolution): At t = we prepre the sttes (x) nd (x) from Eqs. nd b t time t = nd let them evolve in time. (f) [6 points] Wht is the wve function (x, t) t lter time? Describe (in two or three sentences) how to determine the time-evolution of the stte (x, t). If you need ny integrls, set them up but don t bother explicitly evluting them. Alph Decy, Gmow s Model nd Delt Scttering [6 points] Figure : The potentil in the Gmow model. In 98, George Gmow proposed model of lph decy wherein n lph prticle is considered prticle in the potentil 8 ><, x pple V (x) = V, <x<x, () >: C x, x <x s shown in Fig.. When the lph prticle is inside the nucleus ( <x<x ), the potentil is constnt negtive number, V, representing the nucler binding forces. When the prticle is outside the nucleus (x <x), it experiences repulsive Coulomb force (both the lph prticle nd the nucleus re positively chrged) nd thus sees the potentil C/x, wherec is some constnt. We set the potentil to be when x pple. Looking t this exm in the future for study mteril? Here s n extr prt: Check tht this does indeed hve the pproprite units to be momentum. Tht s right. I hve footnotes nd non-for-credit prts in exms too! Let s hope I don t hve ny puns... Our theme for this problem is the greek lphbet. Gmow hd sense of humor bout his nme. Look up the Alpher-Bethe-Gmow pper when you get chnce.
( ) [4 points] Qulittively describe the energy spectrum of n lph prticle in this potentil. Tht is, for wht energy rnge(s) will the energy eigenvlues hve discrete spectrum nd for wht energy rnge(s) will the energy eigenvlues hve continuous spectrum? For wht energy rnges cn we gurntee tht there re no llowed energy eigenvlues? Let the lower of the two dshed lines in Fig. represent the second excited bound stte energy. ( ) [8 points] Mke qulittive sketches for the wve function for the two energies represented by the two dshed lines. On ech sketch lbel the clssiclly llowed region(s) (CARs), the clssiclly forbidden region(s) (CFRs), nd the clssicl turning point(s) (CTPs). Provide brief explntion of the qulittive fetures of the wve functions (these fetures include whether the function is oscilltory or exponentil, the behvior of the nodes, ny symmetries the wve function my or my not hve, how the locl wvelength nd mplitude vry where pproprite, etc.). Wht is the qulittive behvior of ech wve function s x! + (does it symptote to zero, oscillte, or blow up)? Suppose we wnted to pproximte the eigenfunctions nd energies of this potentil by the eigenfunctions nd energies of n infinite squre well from <x<x, with the potentil being V inside the well. (Remember from Problem Set 5 tht ll this does is shift the infinite squre well energies down by V.) ( ) [6 points] Wht is the mximum vlue of n for the infinite squre well sttes so tht n infinite squre well eigenfunction could still be bound (rther thn tunneling or scttering) stte in the Gmow model? Mke n rgument bsed on the qulittive fetures of the energy eigenfunctions tht the infinite squre well bound stte energies should be slightly higher thn the corresponding bound stte energies for the Gmow model. Now consider n lph prticle with energy E incident from the left on delt brrier of strength t x =. V (x) =+ (x). Let Region I be the region x< nd let Region II be the region x>. ( ) [6 points] Find the generl solution to the TISE for this energy in ech of the two regions, I(x) nd II (x). Indicte whether ech term represents left- or right- trveling wve nd whether it is n incident piece, reflected piece, trnsmitted piece, or not present in our experiment. [Note: It my help to define some constnts in terms of the energy E nd the prmeters of the problem s we did in lecture. Mke sure you clerly define wht these constnts re before you use them, though!] Recll tht when we hve delt function in our potentil, the first derivtive of the wve function hs discontinuity, () = m (). () ~ (") [6 points] Integrte the TISE in smll intervl from to + nd tke the limit s! to show Eq.. Write down the equtions tht result from ll of the boundry conditions tht pply in this problem. ( ) [6 points] Find the trnsmission coefficient T. 4
The Infinite Squre Well nd Free Prticle [6 points] Consider prticle in n infinite squre well stretching from to. Let n (x) nd E n be the n-th energy eigenfunction nd n-th energy eigenvlue, respectively. Since this potentil is symmetric bout the point x = /, our energy eigenfunctions will be either symmetric or ntisymmetric bout x = /. () [8 points] Use the TISE to solve for the normlized energy eigenfunctions nd energy eigenvlues from scrtch, explining your steps. For which vlues of n re the eigenfunctions symmetric bout the center of the well nd for which vlues re they ntisymmetric? Suppose we prepre the system so the wve function t t = is given by superposition of the first nd third excited sttes (wht n odd stte to choose...), (x, ) = r (x) i p 4(x). (b) [5 points] Find the expecttion vlue of position hxi. Wht is the probbility of finding the prticle within smll intervl dx of x = /? We cn write generl infinite squre well wve function s liner combintion of the energy eigenfunctions, X (x, ) = c n n (x). (c) [5 points] Show tht if (x, ) is normlized then P c n =. n= A prticle is prepred in the ground stte of the infinite squre well. At time t = the wlls of the well re suddenly removed, leving free prticle t t = in the stte 8r < (x, ) = sin x, pple x pple, :, else. (d) [8 points] Find the momentum spce wve function (p, ) t t =. Use this to find hpi(), the expecttion vlue of momentum t time t = nd (p, t), the momentum spce wve function t lter time. [Note: Short on time or stuck on finding (p, )? At lest set up the expressions for hpi nd (p, t) in terms of the unknown (p, ) for prtil credit!] 5