s that there are two extrema ... locations. Problem Set 4: Quantum Mechanics: Atoms and Molecules I

Problem Set 4: Quantum Mechanics: Atoms and Molecules I
1. Show by differentiation of the 2s radial wavefunction for a hydrogen-like atom
that there are two extrema (maxima or minima) in its amplitude, and find their
2. Write an expression for the radial distribution function for an electron occupying
a 2p orbital in a hydrogen-like atom (in other words, an expression for the
probability distribution function that the electron be located a distance r from the
nucleus irrespective of direction/angle). By taking derivatives of this expression
with respect to r, determine the radius (distance from the nucleus) at which the
electron is most likely to be found.
3. How many electrons can occupy each of the following subshells: (i) 1s, (ii) 3p,
(iii) 3d, and (iv) 6g ?
4. “Positronium” consists of an electron and a positron (a particle with mass equal to
that of an electron, but opposite charge) orbiting each other around their common
center of mass. The general features of the spectrum are thus expected to be
hydrogen-like, with the major differences arising due to the difference in mass
between a proton and a positron. Predict the binding energy of the ground state of
5. In what is sometimes called the “free electron molecular orbital” theory, the 
electrons in a conjugated organic molecule are treated as being independent
particles in a one-dimensional box of length L. (The term “conjugated”, as used
here, implies an alternating succession of single and double bonds linking carbon
atoms). Each energy level of the corresponding one-dimensional box can then be
occupied by a pair of electrons with opposite spins. Consider the molecule:
CH2=CH-CH=CH-CH=CH-CH=CH2, which has 8 carbon atoms, each of which
contributes one  electron to the molecule. Assume that these electrons can be
treated as though they are confined to a box of length L = 8 b, where b represents
the average bond length between carbon atoms and is approximately 1.4 x 10-10
m. (An extra half-bond-length has been added to each extremity of the box).
Calculate (i) the minimum excitation energy for this molecule (that is, the energy
corresponding to the transition between the highest occupied and lowest
unoccupied/available energy levels), and (ii) suggest what color a sample of this
compound might appear when viewed in white light.