701
HOMEWORK PCS 2
1: A molecule of interest has a lowest-lying nuclear spin configuration that is a degenerate pair of states, 1 and 2 . Under certain conditions the initial composition of the system is given by the expression n1 /(n1 + n2 ) = 1.00 At a given instant of time a perturbing field of intensity equivalent to 10-27 J was applied for a duration of 10 s. After an interval of 10 s a second pulse of half the intensity of the first was applied for 5 s. Assuming that spontaneous changes in the 10 s between the pulses are negligible, calculate the probable composition of the system (a) at the end of the first pulse, and (b) at the end of the second pulse. Hint: you will find the Rabi formula useful here. 2: A hydrogen atom in its 2s state passes into a region of space where it experiences a z-directed electric field for a time . Calculate the total wavefunction of the atom and its dipole moment during the time of exposure in the field. Hints: The field will induce mixing of 2s and 2pz states; you to need find the coefficients of the combination in order to find an expression for tot. Remember that to calculate the value of an observable you need to find the expectation value of its operator, and that in polar coordinates z = r cos where = co-latitude. 3: The radiation from a ceramic plate heated to 1000 K was used to irradiate a population of hydrogen atoms. Calculate the rates of spontaneous and stimulated emission at the H line of atomic hydrogen. Hints: You are interested in the 3 p 2 s transition. You need to calculate the transition dipole moment (remember that the 3p level is triply degenerate) and the Einstein A and B coefficients for this transition. You also need to calculate the density of states of the radiation field at 1000 K. To simplify the integration you may assume that 3 pq q 2 s = -1.769a0 m where a0 is the Bohr radius. However, just for the practice of it I recommend that you write out the complete form of one of the transition dipole moments (with full wavefunctions).