Chemistry 3730 Fall 1998 Test 1

Aids allowed: Maple, calculator

Useful information is provided at the back of this exam paper.

You can use Maple to solve the problems, but you must write down your complete solution in the booklet provided. It is not necessary to write down what you typed into Maple, but you should explain what you calculated using normal mathematical notation.

1. The spectrum of mercury has a bright blue line at 436nm. Convert this wavelength to a frequency and to a wavenumber. [2 marks]
2. What is the minimum value of the square of the angular momentum consistent with a z component of ? [2 marks]
3. Write down the Hamiltonian for a system consisting of two electrons in a one-dimensional box of length L. Explicitly state the boundary conditions of this problem. [6 marks]
4. Calculate the probability that a particle in a one-dimensional box occupies the interval (the middle quarter of the box) for n=1 and 2. Report your answers to four decimal places. Explain briefly why one answer is significantly larger than the other. (Hint: What do the wavefunctions look like?) [10 marks]

   Maple hint: is typed Pi in Maple.

5. Calculate the following quantities for a particle in a one-dimensional box in the n=1 energy level: , and . Give exact answers. [10 marks]

   Maple hints: is typed Pi in Maple. Be careful with your parentheses.

6. Suppose that a proton is placed in a narrow box of length 1cm at . The kinetic energy of a particle confined in one dimension is approximately , where k is Boltzmann's constant, or at . Compute the quantum number n of a proton in a box at this kinetic energy. Then calculate the energy necessary for this particle to make a transition from your computed value of n to n+1. Do you expect quantum mechanical effects to be important in this system? The mass of a proton is . [10 marks]

   Hints: To calculate the transition energy, it will be convenient to derive a formula first, and only then to plug in the numbers. This is easy to do by hand, but if you want to use Maple to do this, use simplify() to get the result in the simplest possible form. Quantum mechanical effects are important if the separation of the energy levels is a large fraction of the energy of the particle.

7. We will see later that it is possible to use approximate wavefunctions when we cannot solve a quantum mechanical problem exactly. Suppose that you wanted to use

   

   as an approximate wavefunction for a particle in a one-dimensional box of length L. Derive a value for the normalization constant A. [10 marks]

Useful formulas and data

Electrostatic potential energy:

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

and .