Problems

3.1 The ground state wave function of the hydrogen atom is ipo — Nq e~r/a° , where ao is the Bohr radius, ao = 1 /ma, and No is a normalization constant. The first excited state is four-fold degenerate. Four linearly independent wave functions which span the space of excited states are

where N\ is another normalization constant, and x, y, and 2 are the three spatial coordinates of the election and r = \Jx2 + y2 + z2.

(a) Derive a formula for the lifetime of the state ipiz. Reduce your answer to an integral over the spatial coordinates x, y, z or r, 9, </> and constants (No, Ni, a0 and other constants). It is not necessary to fully evaluate the integral, but you should reduce the triple integral to a single integral.

(b) Is the photon which is emitted by the decay polarized? If so, what is its polarization (i.e., in which direction does ea point)?

(c) What is the lifetime of the states tp\x and ip\y1 Are the photons emitted by these decays polarized? If so, in which direction?

3.2 A nonrelativistic particle of mass m and charge e is trapped in an infinite one-dimensional square well described by the potential

Calculate the lifetime of the first two excited states. (Suggestion: treat the EM field as one-dimensional.)

3.3 [Taken from Sakurai (1967).] Suppose a photon of energy w is incident on a hydrogen atom in its ground state. The photon may be absorbed, ionizing the atom. This is a simple model for the photoelectric effect.

(a) Using the formalism developed in this chapter, write the matrix element for the lowest order contribution to this process. (Note that the final state is a scattering state of an electron and a proton.)

(b) If the energy of the incident photon is so large that the final electron-proton scattering state can be approximated by plane waves, show that the differential cross section, defined in Eq. (3.52), is where the spherical coordinate variables 6 and 4> are defined so that the incident photon momentum is along the 2-axis, its polarization is along the x-axis, and a0 is the Bohr radius.

PART II

RELATIVISTIC EQUATIONS

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 4

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