CHEM 151: ELEMENTARY QUANTUM CHEMISTRY 2ND SEM – AY 2013-2014

PROBLEM SET 1 Problem Solving Problems with a triangle () will be solved in class. Other problems are to be submitted on MONDAY, JAN 20, 2014 2:30pm.

1. What must be the temperature of an ideal blackbody so that photons of its radiated light having the peak-intensity wavelength can excite the electron in the Bohr-model hydrogen atom from the ground state to the third excited state?

2. Radiation has been detected from space that is characteristic of an ideal radiator at T = 2.728 K, a rel-ic of the Big Bang at the beginning of the universe. For this temperature, at what wavelength does the Planck distribution peak? In what part of the electromagnetic spectrum is this wavelength?

3. A sample of hydrogen atoms is irradiated with light with wavelengths 85.5 nm, and electrons are ob-served leaving the gas. (a) If each hydrogen atom were initially in its ground level, what would be the maximum kinetic energy in electron volts of these photoelectrons? (b) A few electrons are detected with energies as much as 10.2 eV greater than the maximum kinetic energy calculated in part (a). How can this be?

4. Show that the Debye expression for the heat capacity is proportional to T3 as T 0.

5. Using the Bohr theory, calculate the ionization energy (in electron volts) of singly ionized helium.

6. Calculate the de Broglie wavelength for (a) an electron with a kinetic energy of 100 eV (b) a proton with kinetic energy of 100 eV, and (c) an electron in the first Bohr orbit of a hydrogen atom.

7. Through what potential must a proton initially at rest fall so that its de Broglie wavelength is 1.0 x 10-

10 m?

8. One of the most powerful modern techniques for studying structure is neutron diffraction. This tech-nique involves generating a collimated beam of neutrons at a particular temperature from a high-energy neutron source and is accomplished at several accelerator facilities around the world. If the speed of a neutron is given by v = (3kBT/m)1/2, where m is the mass of a neutron, then what tempera-ture is needed so that the neutrons have a de Broglie wavelength of 50 pm?

9. Consider the wavefunction

where k is positive. Is this a valid time-independent wavefuction for a free particle in a stationary state? What is the energy corresponding to this wavefunction?

10. Show that <p> = 0 for all states of a particle-in-a-one-dimensional-box of length l.

11. Solve the Schrödinger equation and an expression for the energy of a particle in a two-dimensional box.

12. A classical particle in a box has an equi-likelihood of being found anywhere within the region 0 ≤ x ≤

l. Consequently, its probability distribution is

Show that <x> = l/2 and <x2> = l2/3 for this system. Now show that <x2> and σx for a quantum-mechanical particle in a box takes on the classical values as n ∞.

13. Construct quantum mechanical operators in the position representation for kinetic energy in (a) one

dimension, (b) three dimensions.

14. Show that if ψ1 and ψ2 are solutions to the Schrodinger equation, then c1ψ1 + c2ψ2 satisfies this equa-tion, where c1 and c2 are constants.

15. The ground-state wavefunction of a hydrogen atom has the form , b being a collection of fundamental constants with the magnitude 1/(53 pm). Normalize this spherically symmetrical function. The volume element is , with 0 0 0

Review Questions Answers to these questions are not required, but it is highly recommended that you go over them to check your understanding of the concepts discussed in class.

1. What is the ultraviolet catastrophe? Explain how classical mechanical assumptions would lead to these predictions for blackbody radiation.

2. Blackbody radiation is the radiation emitted by molecular or atomic oscillators on the surface of the blackbody after complete absorption of incident radiation. Contrast the assertions made by the Ra-leigh-Jeans Law and the more accurate Planck’s Radiation Law.

3. Contrast the results hypothesized from classical mechanical principles and the experimental results

observed for the photoelectric effect experiment. What are the predicted and experimental results on the stopping potential and photocurrent when the following variables are increased:

a. Frequency of incident light b. Intensity of incident light

4. How was the time-independent Schrodinger equation derived from the time-dependent Schrodinger

equation? Can a time-independent wavefunction be derived from any ψ(x, t)?

5. What are the requirements for a well-behaved wavefunction?

6. For a particle in a box, we chose k = nπ/L with n = 1, 2, 3, … to fit the boundary condition φ = 0 at x = L. However, n = 0, -1, -2, -3… also satisfy that boundary condition. Why didn’t we also choose those values of n?

7. What are the properties of linear operators?

8. What is the significance of quantum mechanical operators being Hermitian? Recall the properties of Hermitian operators. Are the kinetic energy and potential energy operators Hermitian? What about the Hamiltonian?

9. Explain why electron tunneling occurs. Under which conditions is tunneling most likely to occur? Consider height and width of the barrier and the size of the tunneling particle.

10. What is the significance of the eigenvalues of a quantum mechanical operator that corresponds to a physical observeable? What happens if the wavefunction is an eigenfunction of ?