1. State what physical property is associated with each of the following quantum numbers in a one-electron atom and give the value of this physical property in terms of the quantum number: a) I b) m: c) s d) "'13 2. Using the L; operator, evaluate the z-component of the angular momentum for a particle on a ring that is described by \ = ext. 3 What are the physical basis behind selection rules? A short explanation is ne. 4. Calculate the zero-point energy of an harmonic oscillator consisting of a particle of mass 3.16 x 10'25 kg and a force constant of 347 N m'l. Express your answer in wavenumbers (cm'l). 5. A quantum mechanical (QM) 3D rotor is different from a classical rotator. a) Write down I, the classical moment of inertia for a rotor of mass m and radius r. b) What is the total angular momentum of a QM 3D rotor of fixed radius r as a function of E? c) Draw an accurate energy level diagram for the first 3 energy levels of the rotor. Give the energies in terms of Planck's constant and the moment of inertia, I. d) Draw a properly scaled picture showing the angular momentum vectors for a 3D rotor with quantum number i? : 1. Be sure to indicate and label the allowed components of the angular momentum. Properly label the axes and the vectors with their magnitudes. 6. Estimate the rotational quantum number of a skateboard wheel of diameter 5 cm and mass 0.2 kg when the skateboarder is travelling at 10 km h". 7. Examine the A2 (Legendrian) operator and a few of its eigenfunctions, the Yim, on page 1: a) A'2 is the Hamiltonian for what system (short answer is ne). b) Prove that the Ylt} spherical harmonic is an eigenfunction of this operator by computing its eigenvalue. Circle your answer carefully. 8. The quantum mechanical approach to the hydrogen (and hydrogenlike) atom requires a Hamiltonian describing the specic interactions in the system. As always, it's the potential energy term that has these interactions: a} Write down the potential energy term, Ky, for the H atom treatment and explain each term in Vel b) Draw a graph of how each component of Vegdepends on r. Include on your plot a graph of Vegitself and label where you expect the electron to be most of the time with mif if the electron has non-zero orbital angular momentum (1:0). Constants: Speed of light: c = 2.99 x 108 m/s Planck's Constant: h ~ 6.62 x 10-34 Jos and h = 1.01 x 10-34 Jos Atomic mass unit: 1 amu = 1.68 x 102 kg Mass of an electron: me = 9.11 10-31 kg Bohr radius: ao = 5.29 x 10"m Rydberg Constant RH = 109,677 cm- Other Potentially Useful Information Table 13.1 Hydrogenic radial wavefunctions Orbital 3 1/2 Yo = cos O 3/2 o 4 TT 2 (2 ) e-p / 2 3 1/2 2.s 1 (2\\ 3/2 N O YA = F 2(2) 1/2 \\do (2 - 2p)e-p/4 sin O etio 3/2 20 2 4(6) 1/2 do pe p/4 1/ 2 Yo = 5 3s 3 3/2 (3cos2 0-1) 0 9(3) 1/2 (6- 20 + ;p? )e-p/6 167 3/2 3p w ho - (4-3p)pe-P/6 L = ' 27(6) 1/2 3/2 i ap 3a 3 81(30) 1/2 do pre-p/6 [re ar do = n! The full wavefunction is w = RY, where Y is given in Table 12.3. In the table, p = 2Zr/do. tie = cos O ti sin 0 dV = r dr do sin O de Elm, = 1(1+1)h /21 Energy of a particle on a sphere 1 2 = - 1 02 E,, - -hcZ2Rx Energy of H-like atoms sin 0 202 sin 0 sin 0 20 p = 2Zr/a9. You have discovered a new hydrogen-like element, Professorium (Ps), Exposure causes people to immediately fall into a coma. The black market is very interested in your nd and would like to have a spectroscopic method (1) to identify whether Ps has been released. You measure its line spectrum and nd that its ionization potential (from the ground state) is 105 cm". In addition you observe a strong x-ray absorption line at 88,890 am1 that you presume originates from the ground state as well. a) What is the Rydberg constant, RPS for this new element? b) Show that the observed x-ray line is indeed a transition from the ground state to a higher lying state and calculate the principal quantum number for this state. 10. What is the energy and magnitude of the angular momentum for the 3d eigenfunction of the hydrogen atom? 11. Consider the following possible wavefunctions for helium (He). (3) 18(1)u(1)13(2)B(2) (b) ls(1)o.(1)1 s(2)ot(2) (C) 15(1)25(2)[a(1)(2) - BUMZH (d) [ls(l)25(2) - 25(1)ls(2)] (1(1 )a(2). Classify these functions as symmetric, antisymmetric, or neither with respect to the exchange of electrons l and 2. Which of these wavefunctions are acceptable for describing He? 12. The radial distribution function gives a somewhat different picture of the electronic structure of the H-like atom: a) Write down the radial distribution function of a ground-state hydrogen-like atom with charge Z. b) Sketch this radial distribution function and the ground state wavefunction as well. How do they differ? c) Show that the maximum in this radial distribution function is an/Z . (5 pts.)