Chemistry 3730 Fall 1998 Test 2

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 and how you calculated it using normal mathematical notation. You need not write down the results of intermediate steps of a calculation if they are complicated.

1. Write down the electronic Hamiltonian of a technetium atom. Use summation notation and clearly indicate the summation limits. [5 marks]
2. Give the ground-state electronic configuration of a technetium atom. Use any convenient notation. [5 marks]
3. The fundamental vibrational frequency of is . Give an approximation to the fundamental vibrational frequency of . [5 marks]
4. Calculate the average potential energy of a harmonic oscillator in the n=2 state. The wavefunction is

   

   with . Give your answer in terms of and . On average, what fraction of the total energy is potential? [10 marks]

   Maple hints: is Pi in Maple and is exp(x). Do the calculation without defining . Maple will need to know that is a positive constant however. It is probably easiest to complete the calculation by hand.

5. The normalized wavefunction is

   

   In spherical polar coordinates, the operator is

   

   Calculate the expectation value of for the state of a hydrogen atom. [10 marks]

   Maple hints: The function is typed exp(x) in Maple and is sqrt(y). The number is Pi and the imaginary unit i is I. Maple will need to know that is a positive constant.

   Bonus: From the method by which real-valued wavefunctions such as the are obtained and the properties of inner products, prove that the result obtained above is generally true.

6. For the normal particle in a box, there is no potential energy. However, we can imagine problems in which a particle is confined to a box and there is a potential energy within the box. Suppose that, in a box of length L whose left edge is at x=0, a particle is subjected to a potential energy

   

   This is a harmonic oscillator with hard walls to either side. Suppose that , and . Compute an approximation to the ground-state energy using the variational wavefunction

   

   Is this model with the given parameters more like a harmonic oscillator or is it more like a particle in a box? Explain briefly. [15 marks]

   Maple hints: The number is Pi in Maple. Type all numerical values needed to solve this problem into your Maple session after after you have calculated the integrals. Use subs() to substitute values of the variational parameter into the variational energy.

   Note: We are imagining here a single mass connected by springs to immovable walls so , the mass of the particle.

Useful formulas and data

;

Particle in a box

where

Harmonic oscillator

,

where and .

The reduced mass .

Spherical polar coordinates

Volume element:

Angular ranges: ,