Jobec D. Pugales January 01, 2009
BS Chemistry â" IV PhyChem 75- Quantum Chemistry
Take Home Exam
1. Evaluate the commutators (a) [H,px], and (b) [H,x] where H = p2x/2m + V (x). Choose (i) V (x) = V, a constant, (ii) V (x) = Â½ kx2, (iii) V (x) --> V (r) = e2/4 0r.
2. The ground-state wave function of a hydrogen atom has the form Ï (r) = Ne-ar, a being a collection of fundamental constants with the magnitude 53 pm. Normalize this spherically symmetrical function. Hint. The volume element is d = sinÎ¸dÎ¸dÏr2dr, with 0 Î¸ , 0 and 0 r â. 'Normalize' always means 'normalize to unity' in this case.
3. A particle of mass m is confined to a one-dimensional box of length L. Calculate the probability of finding it in the following regions: (a) 0 x Â½ L, (b) 0 x Â¼ L. Deduce the expressions for general values of n, and then specialize to n = 1.
4. Energy is required to compress the box when a particle is inside: this suggests that the particle exerts a force on the walls, (a) On the basis that when the length of the box changes by dL the energy changes by dE = FdL, find an expression for the force, (b) At what length does F = 1N when an electron is in the state n = 1?
5. Identify the locations of the nodes in the wave function with n = 4 for a particle in a one-dimensional square well.
6. The oscillation of the atoms around their equilibrium positions in the molecule HI can be modeled as a harmonic oscillator of mass m mH (the iodine atom is almost stationary) and force constant k = 313.8Nm-1. Evaluate the separation of the energy levels and predict the wavelength of the light needed to induce a transition between neighboring levels.
7. (a) Confirm that the Spherical Harmonics Y1,+1 and Y2,0 are solutions of the Schrodinger equation for a particle on a sphere, (b) Confirm by explicit integration that Y1,+1 and Y2,0 are normalized and mutually orthogonal. Hint. The volume element for the integration is sinÎ¸dÎ¸dÏ, with...