8.05 Quantum Physics II, Fall 2011 FINAL EXAM Thursday December 22, 9:00 am -12:00 You have 3 hours.

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1 8.05 Quantum Physics II, Fall 0 FINAL EXAM Thursday December, 9:00 am -:00 You have 3 hours. Answer all problems in the white books provided. Write YOUR NAME and YOUR SECTION on your white books. There are seven questions, totalling 00 points. None of the problems requires extensive algebra. No books, notes, or calculators allowed.

2 Formula Sheet Conservation of probability ρx, t + Jx, t = 0 t x [ ] ρx, t = ψx, t ; Jx, t = ψ ψ im x ψ x ψ Variational principle: E gs dx ψ xhψx, for all ψx satisfying dxψ xψx = Spin-/ particle: e Stern-Gerlach : H = µ B, µ = g S = γs m e S µ B =, µ e = µ m B, e 0 In the basis z; + = + =, z; = = i 0 S i = σ i σ x = ; σ ; 0 y = σ z = i 0 0 [σ i, σ j ] = iɛ ijk σ k [ ] S i, S j = i ɛijk S k ɛ 3 = + σ i σ j = δ ij I + iɛ ijk σ k σ a σ b = a b I + i σ a b e imθ = cos θ + im sin θ, if M = exp a i a σ = cos a + i σ a sin a, a = a expiθσ 3 σ exp iθσ 3 = σ cosθ σ sinθ expiθσ 3 σ exp iθσ 3 = σ cosθ + σ sinθ. S n = n S = n x S x + n y S y + n z S z = n σ. n x, n y, n z = sin θ cos φ, sin θ sin φ, cos θ, S n n; ± = ± n; ± n; + = cosθ/ + + sinθ/ expiφ n; = sinθ/ exp iφ + + cosθ/

3 Complete orthonormal basis i i j = δ ij, = i i i O ij = i O j O = i,j O ij i j i A j = j A i hermitian operator: O = O, unitary operator: U = U Position and momentum representations: ψx = x ψ ; ψp = p ψ ; xˆ x = x x, x y = δx y, = dx x x, xˆ = xˆ pˆ p = p p, q p = δq p, = dp p p, x p = exp ipx ; ψp = π dx p x x ψ = π pˆ = pˆ ipx dx exp ψ x d nψx x pˆn ψ = ; p ˆx n ψ = i d n ψp ; [ˆp, fˆx] = f i dx dp xˆ i expikxdx = δk π Generalized uncertainty principle A A A = A A A B Ψ [A, B] Ψ i x p x = and p = for a gaussian wavefuntion ψ exp x + dx exp ax = π a H t, t Q d Q dt 3

4 Commutator identities [A, BC] = [A, B]C + B[A, C], e A Be A = B + [A, B] + [A, [A, B]] + [A, [A, [A, B]]] +..., 3! e A Be A = B + [A, B], if [[A, B], A] = 0, [ B, e A ] = [ B, A ]e A, if [[A, B], A] = 0 e A+B = e A e B e [A,B] = e B e A e [A,B], if [A, B] commutes with A and with B Time evolution Ψ, t = Ut, 0 Ψ, 0, U unitary Ut, t =, Ut, t Ut, t 0 = Ut, t 0, Ut, t = U t, t d d i Ψ ˆ dt, t = Ht Ψ, t i Ut, t 0 = HtUt, ˆ t 0 dt [ ˆ i ] Time independent H: Ut, t 0 = exp Ĥt t 0 = e i E nt t 0 n n n [ ˆ ˆ i t ] [ Ht, Ht ] = 0, t, t, Ut, t 0 = exp dt Ĥt t 0 A = Ψ, t A S Ψ, t = Ψ, 0 A H t Ψ, 0 A H t = U t, 0A S Ut, 0 [A S, B S ] = C S [A H t, B H t ] = C H t d i Aˆ H t = [Aˆ H t, Hˆ H t], for A S time-independent dt Two state systems H = γ S B spin vector n precesses with Larmor frequency ω = γb NMR magnetic field B = B 0 e z + B cos ωt ex sin ωt e y i i ψt = exp ωts z exp γb eff S t ψ0 ω B eff = B 0 ez + B e x ω0 4

5 Harmonic Oscillator Ĥ = pˆ + mω xˆ = ω N ˆ +, N ˆ = â â m mω â = ˆx + iˆp mω, â = ˆx iˆp, mω mω mωâ xˆ = mω â + â, ˆp = i â, [x, ˆ pˆ] = i, [a, ˆ â ] =. n = a n 0 n! Ĥ n = E n n = ω n + n, Nˆ n = n n, m n = δ mn Coherent states and squeezed states â n = n + n +, â n = n n. mω /4 ψ 0 x = x 0 = exp mω x. π pˆ x H t = xˆ cos ωt + sin ωt mω p H t = pˆ cos ωt mω xˆ sin ωt T x0 e i pˆx 0, T x0 x = x + x 0 x 0 T x0 0 = e i px ˆ 0 0, x 0 = e x 0 4 d e x 0 a d 0, x x 0 = ψ 0 x x 0, d = mω α Dα 0 = e αa α a xˆ 0, Dα exp αa α pˆ a, α = + i d d α = e αa α a 0 = e α e α a 0, â α = α α, α, t = e iωt/ e iωt α γ 0 γ = Sγ 0, Sγ = exp a a aa, γ R 0 γ = exp tanh γ a a 0 cosh γ S γ asγ = cosh γ a sinh γ a, α, γ DαSγ 0 D α adα = a + α C 5

6 Orbital angular momentum operators ˆL i = ɛ ijk xˆj pˆk Spherical Harmonics = + r r r + + cot θ + r θ θ sin θ φ ˆL = ˆ L z = i φ ; θ + cot θ θ + sin θ ˆL± = e ±iφ Y l,m θ, φ θ, φ l, m φ ± θ + i cot θ φ 3 3 Y 0,0 θ, φ = ; Y,± θ, φ = sin θ exp ±iφ ; Y,0 θ, φ = cos θ 4π 8π 4π 5 5 Y,± θ, φ = 3π sin θ exp±iφ ; Y,± θ, φ = sin θ cos θ exp ±iφ ; 8π 5 Y,0 θ, φ = 3 cos θ 6π Algebra of angular momentum operators J orbital or spin, or sum [J i, J j ] = iɛ ijk J k ; [ J, J i ] = 0 J jm = jj + jm ; J z jm = m jm, m = j,..., j. J ± = Jx ± ij y, J ± = J Jx = J + + J, J y = J + i J [J z, J ] = ± J, [J, J ] = 0 ; [J +, J ] = J ± ± ± z J = J + J + Jz J z = J J+ + Jz + J z J ± jm = jj + mm ± j, m ± 6

7 Radial equation d ll + + V r + u m dr νl r = E νl u νl r bound states mr u νl r r l+, as r 0. E > 0 and lim V r = 0, Hydrogen atom r π lim u l r = sin kr l + δ l E, k = r me e E n = a0 n, ψ n,l,m x = u nlr Yl,m θ, φ r n =,,..., l = 0,,..., n, m = l,..., l e a 0 =, α = me c, c 00 MeV fm 37 r u,0 r = exp r/a 3/ 0 a 0 r r u,0 r = exp r/a a 0 3/ 0 a 0 r u, r = exp r/a 3 a 0 3/ 0 a 0 Factorization method d d A = + W x, A = + W x dx dx H d = A A = + W dx W H d = AA = + W + W dx x W x dx φ 0 = N exp 7

8 Addition of Angular Momentum J = J + J Uncoupled basis : Coupled basis : j j ; m m CSCO : {J, J, J z, J z } j j; jm CSCO : {J, J, J, J z } j j = j + j j + j... j j j j ; jm = m + m =m J J = J + J + J Combining two spin / : j j ; m m j } j ; m m j j ; jm {{} J Clebsch Gordan coefficient + + J z J z = 0, =,, 0 = +, 0, 0 =, =. 8

9 . True or false questions [0 points] No explanations required. Just indicate T or F for true or false, respectively. The anti-commutator of two hermitian operators is hermitian. The Heisenberg Hamiltonian and the Schrödinger Hamiltonian are equal if the Schrödinger Hamiltonians at different times commute. 3 The length scale a 0 α is the classical electron radius a 0 is the Bohr radius and α the fine structure constant. 4 The length scale a 0 α is the Compton wavelength of the electron. 5 In the factorization method either H or H must have a normalizable zero energy eigenstate. 6 If J and J are sets of operators each satisfying the algebra of angular momentum the combination J J also satisfies the algebra of angular momentum. 7 Let J and J be angular momentum operators. The operator J J commutes with J x + J x. 8 The traces of J x, J y, and J z are all zero for any representation j = /,, 3/, The states on the first excited level of the spherically symmetric harmonic oscillator potential V r = mωr fit into an l = multiplet of angular momentum. 0 The states on the second excited level of the spherically symmetric harmonic oscillator fit into an l = multiplet of angular momentum.. A short problem on Harmonic Oscillator [0 points] Associated with the annihilation and creation Schrödinger operators â and â there are Heisenberg operators ât and â t. Calculate these time-dependent Heisenberg operators in terms of a, ˆ â and other physical constants. How is the Heisenberg counterpart Nˆ H t of the number operator Nˆ related to N? ˆ 9

10 3. Deriving and testing an inequality [5 points] a Let u denote a Hermitian operator whose eigenvalues are all non-negative so that the following expectation value is non-negative: u u u 0. Use the above to prove an inequality relating u and. u b Verify your inequality for the operator u = r and the ground state of the hydrogen atom by direct calculation of r and. The second can be obtained without r doing integrals by using the virial theorem which tells you that the T = V, where T and V are, respectively, the kinetic and potential energies. Possibly useful integral: dx x n e x = n! for n =,, 3, A curious rewriting of the Hydrogen Hamiltonian [5 points] Consider the hydrogen atom Hamiltonian We will write it as H = p e H = m r. 3 xˆi pˆi + iβ ˆp i iβ ˆx i m i= r r + γ, where pˆi and xˆi are, respectively, the Cartesian components of the momentum and position operators, and β and γ are real constants to be adjusted so that the two Hamiltonians are the same. a Calculate the constants β and γ. b Explain carefully why for any state H γ. c Find the wavefunction of the state for which the above energy inequality is saturated. You may assume that this wavefunction just depends on the radial coordinate. 0

11 5. Hamiltonian for three spin- particles [5 points] Consider 3 distinguishable spin- particles, called,, and 3, with spin operators S, S and S 3, respectively. The spins are placed along a circle and the interactions are between nearest neighbors. The hamiltonian takes the form H = S S + S S S S, with > 0 a constant with units of energy. For this problem it is useful to consider the total spin operator S = S + S + S 3. a What is the dimensionality of the state space of the three combined particles. Write the Hamiltonian in terms of squares of spin operators. b Determine the energy eigenvalues for H and the degeneracies of these eigenvalues. c Calculate the ground state, expressing it as a superposition of states of the form m, m, m 3, m, m, m 3, where m i is the eigenvalue of S z i and applying some suitable constraint. [Hint: The general superposition with arbitrary coefficients has 7 candidate states. Show that the coefficient of 0, 0, 0 is zero and determine all others. Write your answer as a normalized state.] 6. Factorization method structure [0 points] In solving a central potential problem we need to find the spectrum of the set of Hamiltonians H l, with l 0. Assume that with some superpotential W l we suceed in constructing Hamiltonians H l and H l such that H l = H l + α l, with some constants α l assumed known and H l = H l+ + β l, with some constants β l also assumed known. Since we know the superpotential, the operators A l and A l are determined as well. Finally, assume that H l has a zero-energy solution φ 0,l, known for each l. a Write expressions for the three lowest energy eigenstates φ n,l of H l n = 0,, and give their energies E n,l. Your expressions for the states should use the zero energy wavefunctions of H and the A or A operators. Your expressions for the energy should use the constants α and β introduced above. b Generalize to write an expression for the energy eigenstate φ n,l of H l with arbitrary n 0 and find its energy E n,l.

12 7. Time dependent uncertainty on a squeezed vacuum [5 points] Recall the definition of the squeezed vacuum state γ 0 γ = Sγ 0, with Sγ = exp â â ââ, γ C. a Calculate the action of the squeezing operator S on the position and momentum operators: S γ xˆ Sγ =... S γ pˆsγ =... Write your answers in terms of xˆ, pˆ, and simple functions of γ. b Let 0 γ, t denote the state that results from time evolution of the squeezed vacuum 0 γ at time zero. Determine the time dependent uncertainty xt on the state. Your answer should be of the form xt = Gγ, ωt, mω where Gγ, ωt, to be determined, is a unit-free function of γ and ωt. [A little help with the algebra: 0 xˆpˆ + pˆxˆ 0 = 0. ]

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