Chemistry 3730 Spring 1997 Test 1

Please note the useful information at the end of this paper.

Aids permitted: calculator, Maple.

Your solutions should explain exactly what computations need to be performed, even if Maple is used to help with the algebra.

1. State whether each of the following sets of hydrogenic states is possible or impossible. Give the quantum numbers for possible states. For the incorrect states, explain which rule is being violated. [6 marks]

   (a) (b) (c)

2. Calculate the average kinetic energy for a particle in a one-dimensional box in the ground and first excited states. [10 marks]
3. 1. Explain what is meant by ``compatible observables''. [2 marks]
   2. In one dimension, are position and kinetic energy compatible observables? [8 marks]
4. 1. Calculate the approximate energies of the ground state and of the first excited state of a particle in a one-dimensional box of length L in which the particle experiences a potential energy

      

      as a function of position. [10 marks]  

   2. State the condition(s) under which you expect your answers to question 4a to be accurate. [2 marks]
   3. Compare the energy levels of a particle in a plain box to your answers to question 4a. In which case is the difference greatest? Why? [2 marks]

      Hint: Sketch V(x).

5. Calculate the probability that an electron in a hydrogen atom is between 0 and distance units away from the nucleus in the 1s and 2s states. Then calculate the probability that the electron is between and from the nucleus in the same two states. Round your answers to 4 decimal places. The 1s and 2s wavefunctions are

   

   [10 marks]

   Maple hint: is exp(x) in Maple.

Useful information

The number is Pi in Maple.

For a particle in a one-dimensional box of length L, the wavefunctions are

and the corresponding energies are

In spherical polar coordinates, the volume element is . Conventionally, and .