Quali cation Exam: Quantum Mechanics

1 Qualification Exam: Quantum MechanicsName:, QEID#43228029:July, 2019 Qualification ExamQEID#4322802921 Undergraduate levelProblem :QM-U-2 Consider two spin 1/2 particles interacting with one another and with an externaluniform magnetic field~Bdirected along thez-axis. The Hamiltonian is given byH= A~S1 ~S2 B(g1~S1+g2~S2) ~Bwhere Bis the Bohr magneton,g1andg2are theg-factors, andAis a In the large field limit, what are the eigenvectors and eigenvalues ofHin the spin-space in the basis of eigenstates ofS1zandS2z?2. In the limit when|~B| 0, what are the eigenvectors and eigenvalues ofHinthe same basis?3. In the Intermediate regime, what are the eigenvectors and eigenvalues ofHinthe spin space? Show that you obtain the results of the previous two parts inthe appropriate :QM-U-201. Show that, for an arbitrary normalized function| , |H| > E0, whereE0is the lowest eigenvalue A particle of massmmoves in a potentialV(x) ={12kx2, x 0+ , x <0(1)Find the trial state of the lowest energy among those parameterized by (x) =Axe x22 does the first part tell you aboutE0?}

2 (Give your answers in terms ofk,m, and = k/m).Problem :QM-U-44 Consider two identical particles of spin zero, each having a massm, that are con-strained to rotate in a plane with separationr. Bearing in mind that the wave-function ( ) must be symmetric with respect to the interchange of these bosons,determine the allowed energy levels of this system. (Give the answer in terms ofm,r, and an integern.) Quantum MechanicsQEID#43228029 July, 2019 Qualification ExamQEID#432280293 Problem :QM-U-54A spinless particle of massmmoves non-relativistically in one dimension in the po-tentialV(x) = V0, d/2 x d/2V(x) = 0,elsewhereThis particle is incident with energyEon the potential well fromx= , movingtowardx= + .1. What is the probability that the particle will, sooner or later, reachx= 100d?2.

3 What is the most likely time interval between when the particle passesx= 100d, and when the particle arrives atx= 100d?3. Compare your answer to the previous part to the corresponding answer fromclassical :QM-U-72A spinless particle of massmmoves non-relativistically in one dimension in the po-tential wellV(~r) ={ V0|~r| a= 1 A = 10 The potential has just one bound state. From this fact, derive upper and lowerbounds onV0(for fixeda).2. Given that the particle is in its bound state, find the probability that it is inthe classically forbidden Given that the particle is in its bound state, find the probability that its mo-mentum is betweenpandp+dp, wheredpis very :QM-U-95An electron (massme, intrinsic spin~2) moves non-relativistically in 3 dimensions inthe potentialV(~r) =12me 2|~r|21. Find a complete set of commuting observables and describe their eigenfunctionsand Show that the total angular momentumJis The energy of the electron is52~.}

4 A measurement ofJis performed. Whatare the possible results?4. List, in the basis of part first part, all the wavefunctions corresponding to eachpossible eigenvalue of J in the third MechanicsQEID#43228029 July, 2019 Qualification ExamQEID#4322802945. What is the degeneracy of the ground state of two non-interacting electrons inthis potential? What are the corresponding wave functions?Problem :QM-U-117 Let us apply Bohr s ideas to a nonrelativistic electron moving in a constant magneticfield~B. The electron s orbit is a circle of radiusrin thexyplane, and~Bpoints alongthezaxis. The angular momentum~L=~r ~pis to be quantized just as in Bohr stheory of the hydrogen atom, where~pis the canonical momentum. Now, however,~p=m~v+qc~A, q= e,wherem~vis the mechanical momentum and~Ais the vector Show that we can choose~A= 12~r ~ Using the fact that the centripetal force is the force due to the magnetic field,obtain the allowed values ofrn.

5 , obtainrnin terms of~,c,e,B, and thequantum Determine the allowed energiesEn. How does your result compare with theexact result, n= (n+ 1/2)~ c, where cis the cyclotron frequency?Problem :QM-U-137 Let us perform a proper Quantum -mechanical calculation for the problem of a nonrel-ativistic electron moving in a constant uniform magnetic field~Bdirected along thezaxis. The classical Hamiltonian isH=12m(~p qc~A)2,wheremis the electron s mass,q= eis the electron s charge,~p=m~vis theelectron s mechanical momentum, and~Ais the vector potential. It is convenient tochoose the Landau gauge:~A=Bx y,where yis the unit vector in theydirection. Following Landau, let us look for asolution of the form (~r) = (x)ei(kyy+kzz).1. Show that, ifkz= 0, (x) satisfies the Schr odinger equation for a one-dimensionalharmonic What are the angular frequency and the equilibrium positionx0for thiseffective harmonic oscillator, in terms ofe,B,m,c, andpy=~ky?

6 3. Ifkz6= 0, what are the allowed energiesEn(kz)? Quantum MechanicsQEID#43228029 July, 2019 Qualification ExamQEID#432280295 Problem :QM-U-164 Consider a particle of massmin the 1-dimensional potentialV(x) = , x 0,region I0,0< x a,region IIV0, a < x,region III1. Write down the generalsolution to the time independent Schrodinger equationin each of the above three Derive an equation which, at least formally, determines the energy IfV0a2= 4 2~2/m, how many bound levels does the potential have?Problem :QM-U-185 Consider an electron moving in a deformable medium (in one dimension). The coor-dinate of the electron isx, and the deformation of the medium isX. The classicalHamiltonian is modeled byHcl=p22m+P22M+12KX2+ApX,whereMandKare parameters describing the medium (which is thus equivalent toa harmonic oscillator with massMand force constantK).

7 After quantization,H= ~22m 2 x2 ~22M 2 X2+12KX2+A( i~ x) solutions of the form kn(x,X) =ceikx n(X),wherecis a normalization Find the energy eigenvaluesEn(k) whenA= Find the energy eigenvalues whenA6= Find the effective massm of the electron, whenA6= 0. (The electron hasbeen renormalized and is a model polaron . The effective mass is defined by~2k/m =dEn(k)/dk.) Quantum MechanicsQEID#43228029 July, 2019 Qualification ExamQEID#432280296 Problem :QM-U-211 The Schr odinger equation for a simple harmonic oscillator is( 12d2dx2+12x2) n= n that if nis a solution then so are a (ddx+x) nand b ( ddx+x) nFind the eigenvalues of aand bin terms of n. By consider1ng 0=e x2/2findwhat :QM-U-2241. For a one-dimensional single-particle system, prove that any two nondegenerateeigenfunctions Eand E ofH=p22m+V(x) must be orthogonal.

8 (You mayassume that Eand E go to zero exponentially asx . You must provethat the energy eigenvaluesEandE are real, if that is required by your : Consider the time -independent Schr odinger equation for Eand E . ThepotentialV(x) is real.)2. Prove thatddt |x| =1m |px| for a single particle three dimensional sys-tem, where the only condition imposed on is that it satisfies the time -dependentSchr odinger equation. (This is part of Ehrenfest s theorem. For simplicity as-sume that goes to zero exponentially asr .)3. Consider an infinite well of width 2a,V(x) ={ ,for|x| a0,for a < x < timet= 0 the wavefunction of a particle of massmin this well is (x,0) ={Nsin( x/a),for a x 00,forx < aandx >0,whereNis a constant. At a later timetwhat is the probability that a mea-surement of the energy will yield the valueE=4 2~28ma2?}}

9 Note: sin(nx) sin(mx)dx=sin(x(n m))n m+sin(x(n+m))n+m, n6=m sin2(y)ydy=12y 14sin(2y) Quantum MechanicsQEID#43228029 July, 2019 Qualification ExamQEID#432280297 Problem :QM-U-260 The wave function of a particle of massmtrapped in an infinite square well potential,symmetric about the origin and of width 2a,V(x) ={ ,for|x| a0,for a < x < ais found to be: (x) =C[cos( x/2a) + sin(3 x)/a+i4cos(3 x/2a)]inside the well and (x) = 0 Evaluate the If a measurement of the total energy of the particle is made, what are thepossible results of such a measurement and what is the probability of obtainingeach value?3. What is the mass current atx=a/2?Problem :QM-U-286A particle of massmmoves under the influence of an attractive central force~F=k~ the assumptions of the Bohr Model to this system to find an expression forthe allowed, Quantum mechanical very briefly any significant difference between the lowest energy stateof this system in this model and that which would result from a solution of theappropriate Schr odinger equation for this :QM-U-2963.}

10 Consider the 1 Dsymmetric potentialV(x) given by:V(x) = V0,for|x|< a0,fora <|x|< b ,forb <|x|,whereV0anda < bare Sketch the approximate character of the two lowest energy solutions to the time -independent Schr odinger equation for this potential. (Call them 1and 2andthe corresponding energiesE1andE2and assumeV0is greater thanE1andE2.)2. A particular solution of the time - dependent Schr odinger equation for this po-tential can be constructed by superimposing 1eiE1t/~and 2eiE2t/~.Construct a wave packet which at timet= 0 is (almost) entirely in theleft-hand well. Describe the motion of this wave packet as a function of MechanicsQEID#43228029 July, 2019 Qualification ExamQEID#432280298 Problem :QM-U-322 The wave for the two lowest lying states of the one-dimensional harmonic oscillator are 0(x) =A0e x2/2a2and 1(x) =A1xe x2/2a2, whereais the corresponding Determine the constantsA0andA1by normalizing the wave Calculate the ground state expectation values Calculate the ground state expectation values Assume that the uncertainty in the position for a harmonic oscillator is x= x2 x 2and that the uncertainty in momentum is p= p2 p that these uncertainties are consistent with the Heisenberg :QM-U-3381.