Peliminay Exam: Quantum Physics /4/ 9:-: Answe a total of SIX questions of which at least TWO ae fom section A and at least THREE ae fom section B Fo you answes you can use eithe the blue books o individual sheets of pape If you use the blue books put the solution to each poblem in a sepaate book If you use the sheets of pape use diffeent sets of sheets fo each poblem and sequentially numbe each page of each set Be sue to put you name on each book and on each sheet of pape that you submit Some possibly useful infomation: = x + y + z = + dx x n e ax = n! a n+ Hemite polynomial = H n (x = ( n e x + θ + dx e a x = π/ a dn dx n e x cos θ sin θ θ + sin θ φ dx xe a x = a H (x = H (x = x H (x = 4x Laguee = L n ( = e dn d n ( n e associated Laguee = L q n+q( = ( q dq d q L n+q( d l Legende polynomial = P l (x = l l! dx l (x l P (x = P (x = x P (x = (x + dwp l (wp l (w = (l + δ ll associated Legende polynomial = Pl m (x = ( x m / d m dx P l(x m [ (l + (l m! spheical hamonic = Yl m (θ φ = ( m 4π(l + m! ( / ( Y = Y = 4π 4π ( / 5 Y = ( cos θ Y ± = 6π spheical Bessels : ( 5 8π / cos θ Y ± = / sin θ cos θe ±iφ Y ± = ( l ( j l ( = ( l l d sin d ] / P m l (cos θe imφ ( / sin θe ±iφ 8π ( / 5 sin θe ±iφ π ( l n l ( = ( (l+ l d (cos d with asymptotic behavio j l ( cos( lπ/ π/ n l ( sin( lπ/ π/ j ( = sin j ( = sin n ( = cos sin j ( = sin cos n ( = cos cos n ( = cos + cos sin sin
Section A: Statistical Mechanics A An ideal gas of N atoms at tempeatue T is confined by an isotopic thee-dimensional hamonic tap with potential enegy U( = mω whee m is the atom mass and ω is the chaacteistic tapping fequency (a Calculate the system patition function (b Show that the entopy of the tapped ideal gas is equal to [ ( ] ( hω S = Nk 4 + ln N(kT (c If the fequency ω is doubled adiabatically calculate by how much the tempeatue changes (d Show that density of the ideal gas in a tap is distibuted as: ( mω / n( = N exp ( mω πkt kt A The elementay excitations of a weakly inteacting Bose-Einstein condensate in an ultalow-tempeatue dilute atomic gas ae analogous to sound waves with dispesion elation whee c is the analog of the speed of sound (a Calculate the aveage enegy of the excitations ɛ(k = hc(k x + k y + k z / (b How does the heat capacity of the excitations depend on tempeatue? A In an ideal gas with N electons the aveage numbe of paticles occupying a single paticle quantum state with enegy ɛ is equal to (a n = N/V < n(ɛ >= [ exp ( ɛ µ kt + ] Obtain a fomula that can be used to detemine the chemical potential µ in tems of the paticle density (b What is the value µ that µ takes at T = K? (c Show that the expession fo the chemical potential educes to the Boltzmann distibution in the limit λ n whee λ is the de Boglie wavelength: ( / π λ = h mkt (d Make a sketch of < n(ɛ > vesus ɛ at T = K and at T = µ /k Label significant points on both axes
Section B: Quantum Mechanics B A qubit is a quantum analog of a classical infomation bit a supeposition of two states living in a twodimensional Hilbet space H spanned by two vectos and Duplication of a qubit entails that thee is anothe qubit that is initially in some nomalized efeence state e and that a tansfomation U : H H H H exists so that U ψ e = ψ ψ fo any nomalized ψ H Show that if thee is such a tansfomation in the fist place it cannot be unitay and theefoe cannot esult fom any Hemitian Hamiltonian acting on the two qubits This is the no-cloning theoem of quantum infomation science B A thee-level Λ scheme incopoates thee states (say in an atom and as in the figue below and monochomatic light fields to dive the tansitions and The lase fields ae chaacteized by thei Rabi fequencies Ω and Ω basically poducts of the electic field amplitude and the appopiate dipole moment matix element In this otating fame the esonance conditions ae govened by the intemediate detuning and the two-photon detuning δ whose oles as the amounts by which the enegies of the photons undeshoot the esonance conditions ae also sketched in the figue We conside the system on exact two-photon esonance δ = wheeupon the Hamiltonian eads H h = + Ω + Ω + Ω + Ω (a Show that the Hamiltonian has the eigenvalue E = and find the coesponding nomalized eigenvecto This is the so-called dak state D It has no component along the excited state (b Thee is anothe nomalized supeposition of the states and the bight state B which is othogonal to the dak state Find the eigenvalues of the Hamiltonian in the subspace spanned by the vectos B and and show that they ae nonzeo wheneve Ω o Ω (c Suppose that the system stats in the state The Rabi fequencies (lase field amplitudes ae vaied so that initially Ω Ω and at the end Ω Ω Ague that if the vaiation of the Rabi fequencies is slow enough the system ends up in the state without eve visiting the excited intemediate state This scheme is known as STIRAP Δ Ω Ω δ
B Take what may be simplest possible model fo a symmetic double-well potential in one dimension namely V (x = V fo x < a V (x = fo a x b and V (x = fo x > b with V > and < a < b (a Show that the equations fo the enegies of even and odd bound states with < E < V ae espectively whee tan[(b ak] k = coth aκ κ tan[(b ak] k = ( / ( / me m(v E k = h κ = h tanh aκ κ (b Show that in the limit V the eigenstate enegies ae: and that the states ae doubly degeneate E n = n π h n = m(b a (c Suppose now that V is asymptotically lage much lage than any othe elevant enegy in the poblem but finite Fo the puposes of the pesent agument we theefoe assume that κ is a constant equal to κ = (mv / h / Each degeneate pai of enegy eigenstates splits into a doublet with a small diffeence between the enegies Show that the splitting is appoximately E n 8E n (b aκ e κa (d In case (c which state in the doublet is lowe in enegy even o odd? B4 Fo the tial wave function ψ = (a / ρ l e bρ Yl m (θφ whee ρ = /a a = h /me use the vaiational method to calculate the lowest hydogen atom enegy level coesponding to angula momentum l Fo the hydogen atom the Coulomb potential is V ( = e / 4
B5 (a Fo a point paticle of momentum of magnitude k which is scatteed by a potential V ( deive the fist ode Bon appoximation fo the elastic scatteing amplitude (b Suppose the scatteing potential is adially symmetic: V = V ( Show that the coesponding fist ode Bon appoximation fo the scatteing amplitude is given by f Bon (θ = m h q whee θ is the scatteing angle and q = k sin(θ/ (c Conside scatteing fom a spheical potential of the fom d V ( sin(q V ( = V < a V ( = > a Compute the fist ode Bon appoximation fo both the diffeential and total coss sections Hint: The Helmholtz Geen s function G( x y which satisfies ( + k G( x y = δ ( x y is given by G( x y = eik x y 4π x y B6 Conside the addition of two angula momentum opeatos accoding to L + L = L Eigenstates l m ae associated with the opeatos L and L z eigenstates l m ae associated with the opeatos L and L z and eigenstates L M ae associated with the opeatos L and L z (a In tems of the quantum numbes (l m and (l m detemine (ie deive as well as state the answe the values which ae allowed fo the quantum numbes (L M (b In tems of the basis vectos l m and l m constuct the paticula eigenstates L M which possess the thee highest allowed positive M values (Instead of solving this pat fo geneal l l fo patial cedit you can solve fo l = l = 5