CHM 3411, Dr. Chatfield, Spring 2009

Problem Set 2

Due Friday, Jan 23

 

Suggested in Chapter 8 but not to turn in: Discussion questions 1,2,3; Exercises [all (b)] 1,4,8,12,14

In this problem set we will explore several principles of quantum mechanics developed in Chapter 8.  Problem 1 applies the Born interpretation, which relates the wavefunction to the probability of locating a particle in a particular region.  Problem 2 applies the concept of normalization, which is required of any wavefunction so that the total probability of locating a particle someplace in space is one.  This problem also explores the concept of orthogonality of functions.  Problem 3 is the calculation of an expectation value, which is the average value of an observable over many individual measurements.  Finally, problem 4 relates the concept of expectation value to the Heisenberg uncertainty principle.  Note that the material for problems 3 and 4 will be covered in class on Wednesday.

 

1. Atkins Problem 8.4

 

2. Atkins Problem 8.14

 

3. The wavefunction for a 1s electron in a hydrogen atom and the potential energy operator are:

     

Where e is the magnitude of the charge of an electron  and eo is the vacuum permittivity (see in text).  Calculate the expectation value <V>, which is the average potential energy of a 1s electron in a hydrogen atom.

 

4. Consider the following wavefunction, which we will later see is the n=3 wavefunction for the one dimensional particle in a box of length L:

      where 0 < x < L

 (a) Calculate the probability that the particle is in the region between 0 and 0.2 L.

(b) Calculate <x> and <x2> (expectation values).  The first is the average position, the second is the average of the square of the position.  Note that the interval for the integrations are 0 to L (the wavefunction is defined as 0 outside of this range).

(c) Calculate <p> and <p2>.

(d) As discussed (or to be discussed) in class, the uncertainty in position and momentum, Dx and Dp, are defined as:             and    

Using your results from (b) and (c), calculate Dx and Dp for the n=3 state of the particle in a box, and show that the Heisenberg uncertainty principle is satisfied.