1. An electron and a proton each have kinetic energy of 45 keV (and are therefore approximately non- relativistic because E m0c2). What are their de Broglie wavelengths? Note: For electrons and protons, mec2 = 511 103 eV and mpc2 = 938 106 eV, respectively.

2. Define the following terms: (a) wavenumber (as a function of the wavelength ) (b) angular frequency (as a function of the frequency f ) (c) phase velocity (for a continuous wave) (d) group velocity (for a continuous wave) (e) position and momentum uncertainties (x and px, respectively). (f) position-momentum and energy-time Heisenberg uncertainty relations

3. Show that for a relativistic particle with rest mass m and momentum p the product of group velocity vg and phase velocity vp is equal to c2, that is vg vp = c2. a) If the particle velocity is 0.5 c, how much is the phase velocity? Is this in contradiction with the Special Theory of Relativity? Explain your answer. Hint: Use the relativistic equation for the energy E of a massive particle. Then taking into account that E = and p = k find as a function of the wavenumber. Use the definitions of phase and group velocities [see Problem 2 (c) and (d)] to compute vp, vg , and their product.

4. A baseball, with a mass of 0.5 kg, is known to be inside a shoebox 30 cm long. (a) What is the (minimum) uncertainty in its momentum? (b) What is the (minimum) uncertainty in its velocity? (c) At this speed, how long would it take to get from one end of the box to the other? (d) In view of this result, how relevant is the uncertainty principle for macroscopic objects?

5. Although an excited atom can radiate at any time from t = 0 to t , the average time after excitation at which a group of atoms radiates is called the lifetime, , of a particular excited state. (a) If = 1.0 108 s (a typical value), use the uncertainty principle to compute the line width f of light emitted by the decay of this excited state. (b) If the wavelength of the spectral line involved in this process is 500 nm, find the fractional broadening f /f .

6. Find the formula for the radius of an electron orbit in the ground state of hydrogen atom, and the ground state energy of hydrogen atom, using the Heisenberg uncertainty principle.