PHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.
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1 PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms of unknown answer define clearly. If you don t know a formula ask, I might be able to help. All parts pts., ma=. Problem required, attempt of remaining 3 problems; circle which ones you want graded. Possibly helpful formulae and constants ˆL + ψ lm = h ll + ) mm + )ψ lm+ ˆL ψ lm = h ll + ) mm )ψ lm E n = m [ ] e h 4πɛ n E n = hω n + ) ˆL = ˆL + ˆL + ˆL z hˆl z = ˆL ˆL+ + ˆL z + hˆl z Hψ = Eψ Hψ = i h ψ t H = h m + V r) d n e = n! d e = π d e = π ψ ) = ψ ) = ψ ) = e π / ) ) ) e π / ) ) 4 8π / e )
2 ψ 3 ) = ) ) π / R = a 3/ e r/a R = a 3/ ) r e r/a a R = a 3/ r e r/a 4 a L ± = he ±iφ ± θ + i cot θ ) φ Y = 4π 3 Y = 4π cos θ 3 Y ± = sin θe±iφ 8π [ ] σ = [ ] i σ y = σ z = i [ ] µ = eg m S h = 34 J s m e = 9 3 kg m e e 4 h = Rydberg = 3.6eV ev =.6 9 J a = h /mze ) e )
3 . Short Answer. Must attempt only) 4 of 6. Circle answers to be graded. a) Identify and discuss: wave packet A wave packet is a wave function describing a propagating particle localized in space. In general, we can write dp ψ, t) = e ip/ h φp, t) π h where φp, t) is an envelope function describing a distribution of momenta present in the wave function ψ. If φ is sharply peaked around one value of p, ψ will look like a plane wave and be spread out over a large region of space. On the other hand, if φ is roughly constant in p, ψ will look like a delta function, and be strongly peaked at a particular for given t). Ehrenfest theorem Ehrenfest s theorem says that the motion of epectation values in quantum mechanics is classical. More specifically, d p dt = ī h [H, p] = ī h i hdv d = d V d, where the right hand side is now just the classical force on a particle, i.e. this is Newton s law on the average. Pauli principle Pauli s principle for particles states that the wave function must be symmetric or antisymmetric under echange of particle labels, depending on whether the particles are integer bosons) or half-integer fermion) spin, respectively. Mathematically, if represents all the labels, coordinates or quantum numbers associated with particle, and is the same for particle, we must have ψ, ) = ±ψ, ), where + is for bosons and - for fermions e.g. electrons). b) Consider two electrons in a potential well with V = ecept for V = for a. What is the ground state wavefunction for the -particle system if the particles have parallel spins? Label your answer in terms of the eigenstates φ i n) of particle i i =, ), and state what these are. For the infinite square well problem as stated, the eigenfunctions are φ n ) = /a sin nπ/a. The ground state for particles with parallel spins will be a spin triplet S=), ψ, ) = φ )φ ) φ )φ ))χ 3
4 since if both particles were in the single-particle ground state φ, such that the energy was E, antisymmetry of the overall wavefunction would force the spin wave function to be antisymmetric, χ. So the best we can do with two parallel spins / is to put one particle in φ and one in φ, then antisymmetrize. The lowest energy with parallel spins is E + E. c) Given a physical system with Hamiltonian H which has an orthonormal set of eigenfunctions ψ n ) at time t=, show that these same eigenfunctions at a later time t, ψ n, t) are still orthonormal. We need to show that ψ n, t), ψ m, t)) = δ mn for all times t. But this is just ψ n, t), ψ m, t)) = e iht/ h ψ n ), e iht/ h ψ m )) = e ie nt/ h ψ n ), e ie mt/ h ψ m )) = e ie m E n )t/ h ψ n ), ψ m )) = e ie m E n )t/ h δ mn = δ mn, where the last step follows because δ mn is only nonzero when m = n. d) Define the uncertainty in the value of an operator ˆQ as Q = ˆQ ˆQ. ) Suppose an electron is known to be in an eigenstate of ˆL and ˆL z, i.e. ψ = lm. Show eplicitly that L =, L z =, but L. Eplain. Calculate L = lm L lm = h ll + ) lm lm = h ll + ) L z = lm L z lm = hm L ) = lm L ) lm = h ll + ) lm L lm = h 4 l l + ) L = lm L + + L lm = lm ).. lm lm = ) lm L L+ + L lm = lm lm = lm L L + + L + L lm = h lm ll + ) mm + )) + ll + ) mm )) lm = h ll + ) m ), so there must be an uncertainty in a measurement of L if the system is prepared in an L, L z eigenstate, since [L, L z ]. e) Compare the wavelengths of the p s transitions in hydrogen one proton, no neutrons) and deuterium one proton, one neutron). Give your answer in terms of the ratio between the two transition wavelengths λ H : 4
5 λ D, which depends only on m e /m p neglect the proton-neutron mass difference, and give only the leading term in powers of ). The energy which must be carried away by a photon in the transition is E E = hc/λ, so λ H = hc/e E ) = 8 h3 3µ ) 4πɛ. where µ = mm p /m + m p ) is equal to m up to an error of order O 3 ). For Deuterium we have µ = mm p /m + m p ), so e λ H = m + m p) + λ D m + m p f) What is the degeneracy of the nd ecited state E = 7/) hω) of the isotropic 3D simple harmonic oscillator? The energies of the eigenstates of the 3D SHO are given by E = E + E y + E z, where E α = hωn α + /), α =, y, z, where n α are positive integers. So altogether E = hωn + n y + n z + 3/). To get 7/) hω, we need two quanta, which can be distributed in any way among, y, z. So we have states labelled n n y n z with possibilities,,,,,, for a degeneracy of 6.. Hydrogen. An electron in a H-atom is in a state described by ψ = 6 [ψ + ψ + ψ ] ) a) Calculate the epectation value of ˆL z in this state. L z = ) ψ + ψ + ψ, h)ψ + h)ψ + h)ψ = 6 b) What is the probability a measurement of the energy will yield the value.5 Rydberg? Wave function has amplitude 6 ψ + ψ ) to be in the n = Bohr orbit with energy -.5 Ryd. Probability is therefore /6+/6 = /3. An additional electron is now added to the H-atom, forming an H ion. 5
6 c) Write down the Schrödinger equation for the system in terms of the electron coordinates r and r, and show that, if you neglect the Coulomb interaction between the electrons, it separates into two decoupled equations, one for each of the two electrons. H = h m e h 4πɛ r m e 4πɛ r + e 4πɛ r r. If we neglect the last term, the Hamiltonian separates into a sum of two independent Hamiltonians, the st of which acts only on the coordinates of the first particle, the second on those of the second. In such a case we know that ψr, r ) = ψ r )ψ r ) and E = E + E, where each obeys H α ψ α = E α ψ α, for α =,. d) Continuing to neglect the Coulomb interaction between the electrons, write down a valid -electron wavefunction assuming that each of the electrons is in a s state be sure to specify the spin state!) Do the same if one electron is in a s state, one is in a s state, and the two spins are parallel. If both electrons are in s states, the orbital part of the -electron wavefunction Ψ, ) is just ψ s r )ψ s r ), which is symmetric under echange. So we have to multiply by an antisymmetric spin state to satisfy Wolfgang: Ψ, ) ss = ψ s r )ψ s r )χ If the two spins are parallel spin echange symmetric), we must put the electrons into an antisymmetric orbital linear combination of s and s in order to preserve overall antisymmetry: Ψ, ) ss = ψ s r )ψ s r ) ψ s r )ψ s r ))χ 3. Spin. A charge +e, spin- particle with gyromagnetic ratio g is initially in an eigenstate ψ of Ŝz corresponding to eigenvalue + h/. a) Evaluate the epectation value µ of the magnetic moment operator ˆ µ = ge/m)s in this state. In which direction does it point? µ = ge/m) S. Of the three components of S, only the z-component has a nonzero epectation values because S and S y can be epressed as S + ± S. Since S z is h/ in the state given, µ = ge h/4m)ẑ. 6
7 b) What is the probability of obtaining a value of h/ if a measurement of Ŝ is made on this state? If we epand z =, ) in terms of the eigenstates of S, =, )/ and =, )/, we see that z = + )/. The probability of obtaining a value of h/ in a measurement of S is therefore / ) = /. c) At t = a homogeneous magnetic field B is applied in the ŷ-direction. Show that the time evolution operator for this system may be epressed in the form ψt) = Ût) ψ), Ût) = cos θ + iσ y sin θ 3) where σ y is a Pauli matri, and find the form of θt). Hint: remember that the eponential of an operator is to be understood as a Taylor series in that operator, and that σ y =.) H = µ B = ge/m)b S y = gµ B σ y / Ut) = e iht/ h = ep[ igµ B σ y /] with µ = e h/m). The eponential can be epanded e ix = + ix + ix) /! +..., and we see that every term will be proportional to an even or odd power of σ y. Even powers give σy n =, while odd powers therefore give σy n = σ y. The sum of all the even terms therefore gives cos gµ B t/ h, and the sum of all the odd terms gives iσ y sin gµ B t/ h. So defining θ = gµ B t/ h, we find the desired result. d) Determine the precession period T define what you mean by period!) and find the form of the state ψt) after a time t = T/4. In which direction does the magnetic moment µ point now? Let s define, as in class, the period as the time when the time evolution operator takes a state into minus itself, because it is then again an eigenstate. Here we start with an eigenstate ψ of S z with eigenvalue h/, and wait until it evolves into ψ, which is again an eigenstate of S z. This occurs when U =, i.e. when θ = π, or T = π h/gµ B ). When t = T/4, we have U = cos π/4 + i sin π/4σ y. So ψt/4) = cos π/4 + sinπ/4)σ y ) ψ) = [ ] [ ]) [ i + i i = [ ] = z z ) which is an eigenstate of S with eigenvalue h/. So ψt/4) S ψt/4) ˆ. ] 7
8 4. Simple harmonic oscillator. a) Write down the Schrödinger equation for a D simple harmonic oscillator, i.e. a mass m particle oscillating with classical angular frequency ω. The ground state ψ is given on the first page of the eam; it has energy eigenvalue hω/. Show eplicitly that ψ is an eigenstate and find the natural length scale in terms of m, ω, and h show your work!). Hψ = h m + mω ) e π / ) = hω ψ Calculate: e ) e = e ) = 4 ) ) e ), so we get a solution if h ) + m 4 mω = hω = h mω b) Calculate the epectation value of the kinetic energy ˆp /m in the ground state ψ n=). p = h, so m p /m = h m ψ ψ = h m = h m = h m d ) ) e π / 4 ) 3 π / 4 π/) π) = hω 4. Note this is consistent with classical virial theorem, where the energy is shared equally between kinetic and potential energy! 8
9 c) Suppose the harmonic oscillator is in its ground state at time t = when the particle absorbs a particle of mass 3m neglect any momentum transfer), thus instantaneously changing its mass to 4m. What is the probability it remains in its ground state? Just like the Helium problem we did in class, the only thing that changes in the eigenfunctions is the, which is proportional to the inverse square root of the mass of the oscillator, after = before /. Probability it remains in ground state is then ψ before ψ after. The inner product is ψ before ψ after = = π before after π before ) so probability is 4/5 to stay in ground state. d e π 5 before =, 5 ) before + ) after ) d) Consider the two eigenstates of the 3D simple harmonic oscillator, ψ a = ψ )ψ y)ψ z) ; ψ b = [ψ )ψ y)ψ z)+iψ )ψ y)ψ z)] 4) Give the energy corresponding to each eigenvector, and specify the total orbital angular momentum quantum number l and its z-component m for each. Use again theorem which says if H is sum of independent H s, E = E + E y + E z. So E a = 3 hω/ ground state) and E b = 5 hω/. ψ a, y, z) ep r, spherically symmetric so l =, m =. ψ b is + iy) ep r, but this is Y, so l =, m =. 9
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