CHM 3411, Dr. Chatfield, Spring 2009

Problem Set 4

Due Friday, Feb. 13

Suggested Exercises from Ch. 9[not to be turned in; all (b)] 8, 9, 10, 11, 13.

1. In this question, we explore the Schrödinger equation for the harmonic oscillator and the solutions to it.

(a) We discussed in class that the Schrödinger equation for a one dimensional harmonic oscillator could be rearranged to the yield the following equation, which has known solutions:

where and

and l is a constant. Show how this rearrangement is done, and give an expression for l in terms of other constants such as E, m, k etc.

(b) The first several Hermite polynomials are given on p. 293 of the text. Show by substitution that the harmonic oscillator wavefunctions for v=0, 1 and 2 satisfy the Schrödinger equation and that the corresponding energies have the values predicted by the equation E = (v+½) hn.

2. This question explores the application of the harmonic oscillator to vibrations of diatomic molecules. The vibration of a diatomic molecule is analogous to vibration of a single particle attached by a spring to an immovable wall, which we discussed in class. The only difference for the equations is that in place of the mass, m, we must use the reduced (or effective) mass, m, which is defined as:

where m_A and m _B are the masses of the individual atoms. [We will demonstrate this later when we get to spectroscopy.] Following are vibrational frequencies for selected diatomic molecules [note that frequencies are often given in units of cm^-1 (wavenumber units: ) instead of s^-1]

³⁵Cl₂ (560 cm^-1) ³⁹K³⁵Cl (281 cm^-1) ¹H₂ (4401 cm^-1)

(a) What are the force constants for these molecules, treating them as harmonic oscillators?

(b) Predict the fundamental frequency for ³⁷Cl₂, assuming that the force constant is the same as for ³⁵Cl₂. (This is a good assumption. The force constant is nearly independent of isotope.) Give the fundamental frequency both in terms of radians per second (w) and cycles per second (n).

(c) What is the zero-point energy of ³⁷Cl₂?

(d) Calculate the energies of the first three vibrational levels of ³⁷Cl₂ (v=0,1,2).

3. This question concerns the classical turning points for the v=1 state of a harmonic oscillator (in one dimension). Recall that classically, the turning points are the maximum extension and the maximum compression the oscillator can have (you might want to picture a vibrating diatomic molecule, and let x represent the difference between the bond length at any point in time and the equilibrium bond length). At a classical turning point, the total energy (E_v) equals the potential energy (V).

(a) Calculate the values of the classical turning points (for the v = 1 state).

(b) What are the positions (values of x) of maximum probability density?

(c) What is the probability of finding the oscillator between x = 0 and x = ∞? (If you are thinking in terms of a diatomic molecule, the question is asking for the probability of finding the diatomic molecule stretched beyond the equilibrium bond length.)

(d) Calculate the probability of finding the oscillator beyond its positive classical turning point (i.e. for a diatomic molecule, the probability of finding the molecule stretched beyond the classical turning point). You will need to use a table of erf(z). See Atkins, Table 9.2. You will also need to evaluate the integral below. This can be done with integration by parts, and the result is:

HINT: You will find that the units will cancel when you convert to y. You will need to evaluate the normalization constant N_v for a harmonic oscillator, which is:

4. The potential energy of a three-dimensional harmonic oscillator is given by

(a) Construct the Schrödinger equation for the three-dimensional harmonic oscillator, and use the method of separation of variables to deduce the energies and wave functions. Give expressions for the energies and wave functions in terms of quantum numbers, and specify what values the quantum numbers can have.

(b) What is the zero-point energy?

(c) If k_x = k_y = k_z, the harmonic oscillator is said to be isotropic. What are the energies and degeneracies of the first five energy levels?

5. Atkins Problem 9.16.