Fall 2011 Homework Assignments
1) A two-dimensional ket space is spanned by the orthonormal kets |a1> and |a2> . Consider the two operators A = |a1> <a1| + 2i |a1> <a2| - 2i |a2> <a1| + 2 |a2> <a2| and B = |a1> <a1| + 2i |a1> <a2| + 2i |a2> <a1| + 2 |a2> <a2| (a) Are these operators Hermitian? Why? (b) Find the eigenvalues and eigenkets of both operators. (c) Are operators A, B , BB+, AB, ABA, BAB, BAB+ Hermitian? Why? (d) Find the eigenvalues of AB and BB+ . 2) Suppose that I want to measure the square of the spin angular momentum of an electron in any direction. (a) What result(s) would I get? What does this imply? What does this mean for the eigenvalues of the corresponding operator? (b) How would your answers to part (a) would change if the particle had spin 1 (with possible measurements results of +ħ 0 and -ħ ? )
1) Consider the operator A = |a1> <a1| + |a2> <a2| + |a3> <a3| with <ai|aj> = δij . (a) What are its eigenvalues? (b) Since you have degeneracy, you have the freedom to choose your orthonormal eigenkets. Suppose you choose one of the eigenkets as ( |a1> - |a2> + |a3> ) C . What must be the value of C so that this ket is normalized? (c) Find two other eigenkets so that you have an orthonormal set. 2) An electron is known to be in the spin state |α> = (1/3) |+z> + c |-x> where the kets |+z> and |-x> are eigenkets of the Sz and Sx operators respectively. o Find the constant c, assuming that it is real and positive. o What is the probability that a measurement of the in the +z direction will give the result ħ/2?
1) Use the WKB method to estimate the bound state energies of a particle of mass m in the asymmetric harmonic oscillator potential { k1 x2/2 if x < 0 V(x) = { { k2 x2/2 otherwise. 2) Construct the propagator for a particle inside a harmonic oscillator potential. Using this propagator, find the wavefunction of a particle inside this potential at time t, if it is given that at time t=0 the particle was in the coherent state |λ>.
1) Using the obvious properties for the three σ matrices corresponding to the directions x, y, and z σi2 = 1 and Tr( σi ) = 0, show that these properties are also satisfied for any σ matrix σn corresponding to a general direction n. Show also that this implies these matrices have the eigenvalues +1 and -1. 2) An arbitrary unit vector associated with the angles θ and φ in the spherical coordinate system may be obtained by rotating a unit vector in the z direction first around the y-axis by the angle θ, then by rotating it around the z-axis by the angle φ. Rotate the spin operator in the Sz direction in the manner described above using rotation operators and show that the operator you thus obtain may be written as a sum of the operators Sx, Sy, and Sz with coefficients consistent with a unit vector with angles θ and φ. (You may work in the Pauli 2-component formalism - i.e. with σ matrices - if you like.)
Problem 3.25 in the textbook.
Find the spherical harmonic functions Y2m(θ,φ) by relating them to the matrix elements of the appropriate rotation operators, and then by calculating these matrix elements.
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