1 of 2 PHYS3320 Fall 2009 Phys 3320, HW #10, Due at start of class, Friday November 20th Q1. QUANTUM WAVES VELOCITIES The quantum mechanical wave function that describes a particle of mass m moving in the z direction with momentum p and energy kinetic energy E is (z,t) = Aei( pzEt )/. The motion is non-relativistic, i.e., E = p2/2m. Calculate both the wave (phase) velocity and the group velocity for this wave function and compare them with the classical speed p/m. Q2. DISPERSION IN HYDROGEN GAS When we first studied polarization in matter, we used a very simple atomic model to estimate the atomic polarizability from a static electric field. Now let’s apply it for non-static fields. A. Assume the hydrogen atom is a point nucleus of charge +e and mass mp surrounded by a sphere of uniform charge density of radius rH and total charge –e. In the absence of any external field, the equilibrium position of the nucleus is at the center of sphere. Find the force on the nucleus if it is displaced from the center by a distance d and use this to find an “effective” spring constant k. What is the natural frequency of oscillation for the hydrogen atom in this model? Putting in the actual values for the variables (the radius will be approximate of course), where in the electromagnetic spectrum (see Table 9.1, Griffiths’ p. 377) does this frequency lie? B. For EM waves with frequencies far from the region of anomalous dispersion, we can ignore the damping and use the simplified formula (Eq. 9.173) to predict the coefficients of refraction and dispersion for hydrogen. Calculate these coefficients for hydrogen gas at 0°C and 1 atm pressure, and compare them with the measured values of A = 1.36 104, B = 7.7 1015m2. Q3. POWER FLOW IN RECTANGULAR WAVEGUIDE Consider the TE10 mode in a rectangular waveguide oriented for transporting waves in the ˆ z direction, with side lengths a in the x direction and b in the y direction. Assume a > b. A. Find the time-averaged energy density u (as a function of x and y) and the time-averaged Poynting vector S (as a function of x and y) inside the waveguide. B. Now spatially average these quantities over the cross-sectional area of the waveguide to find u and S. C. In chapter 8 we used the equationS = u vˆ z to “define” the speed v of energy flow; use it here to find the speed of energy flow in the waveguide. Does this speed make sense?2 of 2 PHYS3320 Fall 2009 Q4. TM MODES IN A RECTANGULAR WAVEGUIDE In the text, the theory of TE modes is derived for a rectangular waveguide. Now it’s your turn to derive the theory of TM modes for a rectangular waveguide of side-lengths a and b oriented to propagate waves along the z direction. A. Find the general solution for the longitudinal (Ez) electric field and using the boundary conditions find the allowed modes for TM waves. B. Find the cutoff frequency, the wave (phase) velocity, and the group velocity for the TMmn mode. C. What is the lowest TM mode? Find the all the components of the electric and magnetic fields for this mode as a function of x, y, z, and t. Make a sketch showing the magnetic field as a function of x and y at a z and t such that ei(kzt )=