Physics 325                   Quiz 3                           April 25, 2003

 

Open book and notes, calculator allowed. Show HOW you arrive at your answers. No credit for random scratch work on the page. If you cannot get it to work out, explain in words how you are going about it. M(proton)=1.6 x 10 –27 kg, M(electron)=9.1 x 10 –31 kg, h=6.6 x 10 –34 J-sec

1 eV=1.6 x –19 J,  k= 1.38 x 10 –23 J/K.

 

 

1.  [40] This question concerns the optical spectrum of Li, which has Z=3.

a. What is the ground state configuration?

b. Construct a level diagram, similar to that on p. 326(322) of TL (except that you may ignore spin for now) including the ground state and first four excited states. (This is a schematic of energy on the vertical axis and angular momentum on the horizontal axis. Try to estimate the correct energy scale.)

c. On this diagram, indicate with arrows all optical transitions which are allowed by (electric dipole) selection rules.

d. If spin is included, three of the excited states will have two values of j. Which are these, and give the values of j for each of them.

 

2. [40] A hydrogen atom in the 2p state with m= +1 (ignore spin).

a. Write the correctly normalized wave function for this state in spherical coordinates.

b. Find the probability that a measurement of the distance of the electron from the nucleus will yield a value of r between 0.5 and 1.5 a0, where a0 is the first Bohr radius. (Set up the integral: do not waste time evaluating it unless you get bored later.)

c.  For each of the following physical properties, does this state have a well-defined value of this physical property? If so, state the value. (Just state it: do not grind around calculating something.)

Square of the angular momentum (L2); Z-component of the angular momentum(Lz); Momentum(p); Energy (H).

d. Make a sketch of the radial wave function for a 4s state; for a 4f state. (Of course you cannot do this exactly, but use what you know about hydrogen wave functions to make a good guess.)

 

3. [10] An atomic hydrogen gas is held at a temperature of 6000 degrees Kelvin (by gravity: not in a box on earth!). What is the ratio of the number of atoms in the n=2 state to that in the ground state? (Ignore fine structure.)

 

4. [10] The ionic compound KCl has a dissociation energy of 4.40 eV. The ionization energy of K is 4.34 eV and the electron affinity of Cl is 3.36 eV. From these numbers, calculate the equilibrium separation of the K and Cl in the molecule. Ignore the “exclusion principle” energy.

 

5. [40] A cubical box of edge 10 –14 m has V=0 inside and infinity outside. 14 identical spin-one- half non-interacting particles with the mass of a proton (i.e., neutrons) are put into the box.

a) What is the energy of the lowest single-particle energy state in the box?

b) What is the energy of the highest single-particle energy state which will be occupied in the ground state of the system? Note that the particles are Fermions.

c) Using the relationship between number density and Fermi energy, find the Fermi energy for this system. Compare this to your answer to (b). Should they agree? Why or why not? Discuss.

d) To what temperature would you have to raise the box for this system to act like a classical free particle gas?