Fall 2008 Homework Assignments
1) Determine the matrix elements corresponding to the spin operators Sx, Sy, and Sz in the basis which makes Sx diagonal. 2) Expand the expressions eA+B and eA eB in series of powers of A and B, where A and B are two operators which do not commute. Keep sufficient number of terms to demonstrate that the two expressions are not equal. 3) Given that for a state ket |a> the projection on the coordinate basis (i.e. the wavefunction) is given by <x|a> = C exp(-(x/x0)2). x0 is a constant length. (a) Normalize the wavefunction (i.e. determine C) so that this wavefunction corresponds to a probability distribution. (b) Determine the "momentum space wavefunction" <p|a>. (c) Comment on the dependence on x0. 4) Repeat problem 3 for the wavefunction <x|a> = C exp(-([x-r]/x0)2 + ikx). What are the physical significances of the constants x0, r, and k?
1) A particle in a one dimensional harmonic oscillator potential is known at time t=0 to be in the state |α,0> = exp(ik0X) |0> , where k0 is a constant and X is the coordinate operator. (a) What are the expectation values (at t=0) for X and the momentum P for this state? (b) Expand |α,0> in terms of the basis formed by |n> . Hint: Write X in terms of the operators a and a+ and break up the exponential (carefully!). The algebra simplifies if you remember that am |0> = 0 for all m > 1. (c) What is the expectation value of the energy? (d) Find the expectation values of the kinetic and potential energies and show that their sum is equal to the previous result. (e) Find the time dependence of the expectation values of position and momentum. 2) A particle has a Hamiltonian operator in the form H = E0 (a+a + βa+a+aa) where E0 is some constant energy and β is a constant. (a) Write down this operator in terms of the X and P operators. (b) Show that the eigenkets |n> of the harmonic oscillator are still eigenkets of this operator. Find the corresponding energies. (c) Find the commutators [X,H] and [P,H] in terms of X and P, and write down the corresponding equations of motion for the X(H) and P(H) operators in the Heisenberg picture. (d) Find the time dependence of the expectation value of X if the initial state of the particle is given by |α,0> = A (|0> + |1>) . (Determine constant A from normalization.)
Consider the potential energy { V0 (1 - x2/x02) for |x| < x0 V(x) = { { 0 otherwise. Using the WKB approximation, find the transmission coefficient for particles incident on this barrier with an energy E < V0.
Problems 2.29, 2.30, 2.31 in the textbook.
Consider the problem of a particle in a one dimensional box, such that the potential energy V(x) = zero if 0 < x < a , (where a is the size of the box) and is infinite elsewhere. (a) Construct the propagator for this system. (b) If the initial wavefunction is equal to { C sin(2πx/a) for 0 < x < a/2 Ψ(x,0) = { { 0 otherwise, use the propagator to calculate Ψ(x,t). (c) Calculate Ψ(x,t) numerically and plot it for t = 0, τ, 3τ, and 9τ where τ = 2ma2/(π2 hbar ) .
The spherical harmonics Ylm are functions of θ and φ, listed for example, in this page. (a) Using the Lz and L2 operators displayed in equations (3.6.9) and (3.6.15) of the textbook, show that the equations L2 Ylm = l(l+1) h2 Ylm and Lz Ylm = m h Ylm are satisfied for Y00, Y11, and Y21. (b) Consider the harmonic oscillator problem in three dimensions, with a potential V(r)=kr2/2. Note that this can be written as V(x,y,z)=k(x2+y2+z2)/2 in cartesian coordinates. Show that the solution in cartesian coordinates is "separable", i.e. the energy eigenstates can be written as ψijk(x,y,z)= ui(x) uj(y) uj(z). (c) For the problem in part (b), write down the ground state eigenfunction ψ000(x,y,z) as a function of r, θ and φ, and verify that it is still an eigenfunction of the energy operator, expressed in spherical coordinates. What are the z-component and total angular momentum values for this state?
Check the Student BAIS system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.