Fall 2014 Homework Assignments
1) Determine the constants A for the following wavefunctions so that the corresponding probability density fuctions are normalized to unity: a) ψ1(x) = A sin( k0 x) exp(- |x| / x0 ) b) ψ2(x) = A cos(π x / x0) for |x| < x0/2, zero otherwise where A and x0 are constants. 2) Find the Fourier transforms of the wavefunctions given in problem 1 above. 3) Find the expectation values <x>, <x2>, <p>, and <p2> for the wavefunctions in problem 1 and comment on the quantity Δx Δp .
Find the bound state energies for the potential V(x) = - w [ δ(x + a/2) + δ(x - a/2) ] where the value of w is positive, and a is the width of the structure. You may express your results in terms of a solution to a transcental equation.
1) Given the relationships <x|ψ> = A exp( -α |x| ) <k|φ> = δ(k) <x|k> = exp(ikx)/√(2π) Find the inner products <x|φ> , <ψ|φ> , <k|ψ> and ∫ dk <φ|k> <k|ψ> 2) The total energy operator of a system has the following matrix elements in a basis formed by the vectors |q1 > and |q2> : H11 = H22 = E0 H12 = H21 = E1 (a) Find the possible values of an energy measurement and the eigenvectors corresponding to these values. (b) If the measurement is to be carried out on a particle in a state ( 3 |q1> + 4 |q2> ) / 5, find the probabilities for the possible measurement values.Homework 4: (Due Thursday, December 4)
1) Consider the second problem of Homework 3. Find the time development of the state given in part (b) of this problem. 2) Given the matrix 0 1 0 A = 1 0 1 0 1 0 Find sin(A).Homework 5: (Due Tuesday, December 16)
In an experiment, the total angular momentum and its z-component of a particle was measured to be l=1 mz=0. Then, a second experiment is carried out to measure the y-component of the angular momentum of this particle. (a) What are the possible outcomes and their probabilities for this second measurement? (b) If the result of this second measurement turned out to be my=1, what are the possibilities and their probabilities corresponding to a third measurement, again in the z-direction? (Hint: Find the matrix representation of L+ and L- operators in the |l,mz> basis, from which you can determine representation of the Ly operator. You will need to expand your input state in terms of the eigenvectors of this operator.)
Check the Stars 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.