Indicate if the statement is True (T) or False (F) by circling the letter (1 pt each):

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1 Indicate if the statement is (T) or False (F) by circling the letter (1 pt each): False 1. In order to ensure that all observables are real valued, the eigenfunctions for an operator must also be real valued.. The ground state of a quantum mechanical rigid rotor has an energy that is equal to zero. False 3. The function y = 5x is an eigenfunction of xˆ with eigenvalue equal to 5. False 4. The emission for the n=3 to n= transition for the hydrogen atom occurs at a shorter wavelength than the n= to n=1 transition. 5. The energy levels of the quantum mechanical harmonic oscillator are all equally spaced. False 6. In quantum mechanics, eigenfunctions must be normalized or else they are not solutions to a given operator equation. 7. The total angular momentum of an electron in a 3p orbital is equal to. 8. It is possible to know both the total and the z-component of the angular momentum of an electron in a p orbital of the hydrogen atom to the same precision. False 9. There are 4 nodes in the n=4 energy eigenfunction for the particle on a line problem.. The energy state for an electron in a p orbital of hydrogen is triply degenerate. Multiple Choice Section ( pts. each). Circle the letter corresponding to the one best choice. 11. The work function for a certain metal is 7.67 x -19 J. If light of wavelength 518 nm hits the surface, what will happen? 518nm= a. electrons will be ejected with their kinetic energy equal to 4.41 x x -19 J J. b. electrons will be ejected with their kinetic energy equal to 3.84 x -0 J. c. electrons will be ejected if the light is of sufficient brightness d. no electrons will be ejected since the light wavelength is insufficient 1. The debroglie wavelength of an object whose mass is m and is traveling a speed of v is: a. mv h b. mv c. mv d. hv m 13. What is the degeneracy of the energy level equal to a cube (all sides equal to L)? 9h 8mL for the three-dimensional particle in a. 0 b. 1 c. 3 d. 6

2 14. The reduced mass of a certain diatomic molecule is 19.4 amu and its force constant (k) is 160 kg/s. What is its vibrational frequency, v? a Hz b..4 x -14 Hz c. 1.1 x 14 Hz d x 13 Hz 15. f(x) = e -6x is an eigenfunction of the x operator. What is its eigenvalue? a. 18 b. 36 c. 7 d. 144 Short Answer/Problems. Please show your work and circle your final answer. (pts each) 16. Consider a system described as a quantum mechanical harmonic oscillator with frequency equal to 1.48 x 14 Hz. (a) Calculate the zero point energy in Joules (b) Calculate the first vibrational transition (0 to 1) in cm -1. (a) ( )( ) = ν = = E 0 0 pt h J s s J (b) hc ε = hν = = hc ν λ 14 1 ν 1.48 s ν = = = 1 c.998 cm s 4940 cm An experiment repeated many times on identical systems will yield an average value that can be predicted at high precision even though the outcome of any individual measurement only has a finite probability of occurring. Provide a formula for how to compute this average value. BE AS GENERAL AS POSSIBLE. a = Ψ * ÂΨdτ * ΨΨ dτ

3 18. A particle confined to a line of length L is in an n= energy state. What is the probability that the particle exists within % of the center of the line (from 4 L 6L to )? b a 0.5L ψ dx = 0.4L ψ dx 0 a b L.5L πx x sin sin 4 1 π P = dx = ( ax) a = L L L 4a L.5L.4 L.4L 4 5L 4L L 4π 5L 4π 4L P = sin sin L 0 0 8π L L 1 1 = 4 [ sin( π) sin(1.6 π) ] 0 8π 1 = [ 0 ( ) ] = or 4.86% 8π

4 19. The frequency of light observed for the lowest pure rotational transition ( =0 to =1) of the H 7 Cl molecule is 3.8 x 11 Hz. Given the masses of H and 7 Cl are.014 and a.m.u. respectively, calculate the bond length of this diatomic molecule in meters. (.014)( ) 1 µ = = kg I 34 h 6.66 J s πν 4 π (3.8 s ) = = = kg m 47 r I = = 1.7 µ m 0. Write an expression (that you could type into Mathcad to get a number) for the probability Hatom that the electron in a pz ( Ψ ) orbital lies between 1 and Bohr [ao] from the nucleus and also lies within 45 degrees of the positive z axis. π /4 4 r 1 r e dr cos θ sinθ dθ

5 1. Working with just the radial part of the hydrogen atom problem, what is the average value of <r 3 > for an electron in a p orbital in units of Bohr [a0]? r r 1 = r e dr 7! 4 = = = 64 bohr. A certain diatomic molecule behaves as a quantum mechanical oscillator with force constant equal to 185 N/m and a reduced mass of x -6 kg. Assuming that it is in the lowest energy vibrational state, write a one line expression which when entered into Mathcad would give a number representing its average value of position squared <x > in picometers: 6 π (185) (6.854 ) kµ α = = = m = pm π 0 x e x dx

6 3. The following figures show, from left to right, the radial probability distribution and an orbital isosurface for a one-electron orbital. Identify each of these hydrogen orbitals. Write your answer in the third column (only n and need to be specified). Probability Isosurface Orbital s 3s 4d 5f