Chemistry 3730 Fall 1997 Test 1

Duration: 90 minutes

Aids permitted: calculator, Maple

All answers must be recorded in full in the exam booklets provided. If you use Maple, you should explain what you used it for, although you need not write down every Maple command which you used to solve a problem.

The value of each question is noted. The aggregate value of all questions is 50.

Note the useful information which follows the questions.

Answer all questions.

1. The ultraviolet spectrum of a molecule has a band centered at a wavelength of 290nm. What is the energy of the photons associated with this transition? [3 marks]
2. Suppose that you are assigned to referee a journal article for the prestigious Physical Review. The author of the article claims to have designed an experiment which measures proton positions to an accuracy of 1Å ( ) while simultaneously measuring velocities between 1 and 1000m/s to an accuracy of approximately 1%. Is this possible? The mass of a proton is . [7 marks]
3. What is the probability that a particle in a box occupies the interval in the ground state? Report your answer to four significant figures. [8 marks]
4. Calculate the expectation value of for a particle in a box in the n=3 state. Report your answer to four significant figures. [10 marks]

   Note: Your answer may also contain some symbols.

5. Using the relations

   

   show that . What is the physical interpretation of this result? [10 marks]

6. In the limit as (the classical limit), the wavefunction for a particle in a one-dimensional box of length L approaches for some constant C. Find the value of C which normalizes this wavefunction. What is the physical significance of normalization? [6 marks]
7. The particle-in-a-box energy levels can be derived using simple arguments. The following steps will guide you through a rederivation of this equation:
   1. Find a relationship between the wavelength of the particle-in-a-box wavefunction and the length of the box. [2 marks]

      Hint: Sketch the wavefunction for a few values of n.

   2. Use de Broglie's relation to obtain the momentum. [2 marks]
   3. Use the relationship between kinetic energy and momentum to obtain an equation for the energy. [2 marks]