Homework problem set 1

Here follow some homework problems that are meant to be illustrative but simple. You are supposed to solve the problems with the help of books that you can easily find. Collaboration with other participants is encouraged.

Each set has five problems, which when solved fully can give 3 points. The 0-15 points that you achieve are stored in the student server under the heading H1. The result in this column and the other 9, 7 or 3 performance columns (for 5p-, 4p-, and 2p-students, respectively) are summed and divided by 10, 8, and 4, respectively. If you are a Chalmers student, who should be given grades; a result of 6 points then gives a good basis for grade 3, 9 ponts for grade 4, and 12 points for grade 5. However, there should be the concluding dialogue, which might change the examination result up or down a little.

We expect to get your solutions to this homework set, when the course is halfway through.

1. Two-state systems are common in physics. Already in the most approximative version the solution to this quantum-mechanical problem has a lot of valuable physical and chemical information.
       a) Solve this problem for the hydrogen molecular ion and give explicit quantum-mechanical expressions that illustrate the concepts bonding and antibonding orbitals.
       b) Generalize the solution to a case, where the two ineracting states have different energy levels, and give explicit quantum-mechanical expressions that illustrate the concept ionic bond by the forms of bonding and antibonding orbitals.
       c) Give at least two more examples of two-state systems.
   
   
2. The infinite quantum well (sv: "partikel i låda") is another useful quantum system for molecular and materials applications.
       a) Calculated the energy eigenvalues in one, two and three dimensions (let the side be L in all cases).
       b) Mention at least two molecular or materials system where it is a useful approximation.
       c) Calculate the density of states (DOS) at an energy E above the bottom of the potential well.
   
   
3. The harmonic oscillator is a third quantum system with lots of applications in the molecular and materials world.
       a) Calculate the energy eigenvalues in one, two and three dimensions for isotropic oscillators.
       b) Calculate the density of states (DOS) at the frequency nu (the Greek letter).
       c) What are the characteristic temperature (T) dependencies of the low-temperature specific heat  c_V(T) for phonons with a dispersion nu = cq, where q is the wavevector (crystal momentum) and for ferromagnetic magnons (spin waves) with a dispersion nu = A q²?
   
   
4. The free-electron model is useful for the description of conduction electrons in metals.
       a) What is the key parameter in that theory?
       b) What determines the band width?
       c) How does the electronic density of states (DOS) depend on E, the energy above the bottom of the band?
       d) How does the electronic specific heat depend on temperature?
       e) For a simple metal, how does the total specific heat depend on temperature at low temperatures T?
       f) Find a metal for which the free-electron model gives an accurate value of the bandwidth.
   
   
5. The tight-binding approximation is very useful in its simplest approximations, e.g. lowest-order overlap, s-band, simple cubic structure.
       a) What are the key parameters in that theory?
       b) What determines the band width for a full band?
       c) Find a material, for which the tight-binding approximation in its simplest approximations gives a good account of the electron structure.
       [c') How does the electronic density of states (DOS) depend on E, the energy above the bottom of the band?]

Textansvarig: Bengt Lundqvist