Chemistry 3730
Fall 2009 Test 1
October 27, 2009
Instructions:
1)
2)
3)
4)
5)
6)
7)
Write all your answers in the booklets provided.
You are permitted to use a “cheat sheet”, that is comprised of one 8.5”x11”
piece of paper containing any information you want.
Hand held calculators are permitted.
The use of Maple is permitted along with its help system. No other aids are
permitted.
If you chose to use Maple to solve a problem, it is your responsibility to make
it clear what you are doing. All calculations have to be reproduced in the
exam book. Show all intermediate steps. Use the mathematical expressions
NOT the Maple commands used to evaluate them.
You have three hours to complete this exam.
The marks should come to a total of 75.
Part I
Answer all of the following questions three questions. (35 marks)
1)
Consider the set wave functions x) = (x-1)n over the interval 0 < x <1, for n > 0.
i) Determine the normalization constant. (3)
ii) Determine whether the normalized wave functions form an orthnormal set. (5)
iii) Compute the expectation value of x and K for n = 1. (4)
iv) Compute x, and p, and determine whether the uncertainty principle is
obeyed for n = 1. (6)
2)
i) Show that the average momentum for a 1D particle in a box is always zero
regardless of the quantum number. (3)
ii) Give at least two reasons why the quantum number of a 1D Particle in a Box
wave function must always be 1 or larger. (2)
iii) Sketch the contour diagram of a 2D particle in a box wave function with n x =
2, and ny = 3. How many nodes are there? What is the energy of the particle? (3)
3)
i) State three requirements acceptable wave function must meet and give the
reasons for these requirements. (3)
ii) Determine whether f(x) = sinm(nx/L) an eigen function for a 1D particle in a
box where the barrier in infinite when x < 0 and x > L. (6)
Part II
Answer only 15 marks of the following five questions.
1) Show that the eigenvalues of a hermetian operator are real.(5)
2) Show that if f(x) is an eigenfunction of A and B, then A and B commute.(5)
3) Show that the propagator commutes with the Hamiltonian.(10)
4) Show that the Hamiltonian must be explicitly time independent for energy to be
conserved (10).
5) Show that any combination of eigenfunctions of a Hailtonian that share the same
eigenvalue is itself an eigenfunction of the Hamiltonian. (5)
Part III
Answer at least 25 marks in the following eight questions. Use equation and diagrams to
aid your explanations. (all excess marks are bonus)
1)
Determine if the position operator, x, is Hermitian. (5)
2)
i) State the operator for linear momentum along the x-axis in classical mechanics
and its corresponding operator in quantum mechanics. (1)
ii) Given that the position operator for both quantum mechanics and classical
mechanics is x = x, determine whether position and momentum commute in
classical mechanics and quantum mechanics. (4)
iii) Comment on the result. (2)
3)
i) State the one-dimensional classical wave equation for a wave moving at 0.1 c
along the x-axis.(2)
ii) State the corresponding wave function for the fundamental mode, assuming
that the wavefunction has nodes at x = 0 and x = 10 m. (2)
iii) Determine the frequency of the wave function in ii). (2)
iv) Determine the amplitude of wave function in ii) at t = 0.001s x = 1 m. (2)
iv) Compute the wavelength of a 100 g baseball moving at 100 km/hr? Comment
on the significance of this result. (2)
4)
With aid of diagrams and equations, describe black body radiation, and explain
how it contradicts classical mechanics. Explain what assumptions have to be
made to reconcile the experimental data with the theory. (5)
5)
Explain what is meant by a stationary state. What conditions must it fulfill? (5)
6)
i) Describe and explain the phenomenon of tunneling and what factors determine
whether and to what extent it occurs.(4)
ii) Why can't a classical particle undergo tunneling. (2)
iii) Give an example of where tunneling occurs or is utilized. (2)
7)
If we want to design a dye that has an absorption wavelength of 100 nm,
determine the length of the conjugated carbon chain required where each carbon
takes up 1.5 *10-10 m. (5)
8)
Describe a quantum well and explain how it can be thought of as a 1D particle in
a box. What factors influence its operational wavelength and would be important
in the design of a laser. (5)
Useful Information
One dimensional Particle in a box wave function
n 
2
 n x 
sin 
L
 L 