Last Name: First Name: PSU ID #: Chem 45 Mega Practice Exam 1 Cover Sheet Closed Book, Notes, and NO Calculator The exam will consist of approximately 5 similar questions worth 4 points each. This mega-exam has approximately points. Useful Equations: ( x) = L 1/ nπx sin( ), L n h E n = 8mL Ψn h p = i d dx Please tell me which other equations you think will be useful.
Point total Problem Possible Score Your score 1 4 16 3 1 4 8 5 4 6 1 7 8 Multiple Choice 56 Total 140 The actual exam will only have 100 points!
Problems: (4 points) 1. Below are energy levels and sketches of wave functions for a particle in an infinite box. Assume L=1 and h /8mL =1 V(X) X a. Write down the Hamiltonian for this system. Give the explicit functional forms for the operators. 3
b. Give the expression for the normalized ground state (n=1) wave function and the value of energy. c. Show that the ground state wave function is an eigenfunction of the Hamiltonian. 4
d. Evaluate the integral given below. If you only give a number, please also state your logic at arriving at the answer S L = Ψ 1( x) ( x) dx 1 Ψ 0 e. There are mistakes in the figure for the n=3 state. State what is wrong. f. For the ground state, give the integral expression for the probability of finding the particle between x=0 and x=0.5. Give the probability or evaluate the integral. 5
(16 points). Below is a sketch of the harmonic oscillator potential. The grid lines are a guide for graphing. The units of the potential energy are hv = ħω = 1. The force constant k=1. V(X) a. Draw the ground state (v=0) and the first excited state (v=1) energy levels. X b. Identify the zero point energy on the graph and give its value. 6
c. Set up the expression for the finding the particle outside the classically allowed region for the ground state. Use Ψ 0 to designate the wave function. Be sure to give the limits of integration. You do not have to evaluate the integral(s). d. Set up the expression for the average value of the square of the momentum <p > for the ground state. Include the explicit functional form of the operator. You can assume that the wave function is normalized. 7
(1 points) 3. For the particle on a sphere (or rotating diatomic molecule) with quantum numbers l=3, ml =-. a. What is the length of the total angular momentum? b. What is the z- component of angular momentum (or projection of the angular momentum along the z-axis) or l z? c. What is the value of l x + l y for this state? 8
(8 points) 4. Einstein discovered the photo electric effect. The metal sodium has a work function of.30 ev. a. If the incident light has hν = ħω = 7 ev, what is the kinetic energy of the emitted electron? b. If the incident light has hν = ħω = 1.5 ev, what is the kinetic energy of the emitted electron? 9
(4 points) 5. The particle in a d ring has the wave function Ψ m ( ϕ) = 1 (π ) 1/ e imϕ The operator. l z h = i d dϕ Evaluate the average value of l z for the m = 4 state. 10
(1 points) 6. For the traveling wave or free particle, the unnormalized wavefunction is ikx Ψ ( x ) = e h where k = ħ/p = π/λ. The operator for p is given by p = i d dx a) Evaluate xpψ b) Evaluate pxψ c) Evaluate (xp-px)ψ, that is subtract the answer in part b) from the answer in part a). This expression shows that x and p do not commute and thus cannot be simultaneously known (the uncertainty principle). 11
(8 points) 7. The energy expression for a particle in a d square box is given by h E nx ny = ( n x + n 8mL, y ) a) What are the values of n x and n y for the ground state? What is the degeneracy? b) What are the values of n x and n y for the first excited state? What is the degeneracy? 1
(4 points each) MULTIPLE CHOICE (circle your answer) 1. A classical harmonic oscillator obeys the equation x(t) = 3.1cm cos(0.16s -1 t) This oscillator a. is at x = 0 when t = 0 b. makes one complete oscillation in 0.16 seconds. c. oscillates between extreme values of +1 and -1 on the x axis. d. is at a turning point when t = 0 e. has its maximum kinetic energy when t = 0. An important experimental confirmation of the de Broglie relation was provided by a. Measurements of the low temperature heat capacities of solids. b. Measurements of the energy of black-body radiation as a function wavelength. c. Measurements of the energies of photoelectrons emitted by metals versus wavelength of incident light. d. Measurements of the intensity patterns of electrons scattered from a solid surface. e. All the above. 13
d 3. When one operates with dx on the function 6 sin(4x), one finds that a. The function is an eigenfunction with eigenvalue -96. b. The function is an eigenfunction with eigenvalue 16. c. The function is an eigenfunction with eigenvalue -16. d. The function is not an eigenfunction. e. None of the above is a true statement. 4. For the operator d 3 dx a. sin(x) is an eigenfunction having eigenvalue 6. b. sin(x) is an eigenfunction having eigenvalue -. c. exp(x) is an eigenfunction having eigenvalue -6. d. exp(x) is an eigenfunction having eigenvalue. e. None of the above is a true statement. 5. Which statement is true? Planck s assumption that the oscillators in the walls of a black-body radiator have quantized energies a. explained the line spectra in black-body emissions. b. overcame the ultraviolet catastrophe. c. predicted a frequency of maximum power output that is independent of temperature. d. required the assumption that different metals have different work functions. e. None of the above is a true statement. d 6. Which one of the following functions is not an eigenfunction of dx? a. cos(3x) b. exp(4. i x) c. sin(-56.x) + cos(3x) d. exp(4. i x) sin(4.x) e. cos(3x) + sin(3x) 14
7. It is found that a particle in a one-dimensional box of length L can be excited from the n = 1 to the n = state by light of frequency ν. It the box length is doubled, the frequency needed to produce the n = 1 n = transition becomes a. ν / 4 b. ν / c. ν d. 4 ν e. None of the above is correct. 8. Which one of the following is a true statement for the particle in a onedimensional box with infinite walls at x = 0 and L? a. When n =, the probability for finding the particle on one side of the box is positive, and the probability for finding it on the other side is negative. b. When n = 1, the probability distribution for the particle is uniform (i.e., constant from x = 0 to x = L). c. As n increases, the energy levels get closer and closer together. d. n = 1 gives the lowest-energy state this system can have. e. As L increases, the energy levels get farther apart. 9. For a particle in a one-dimensional box with infinite walls at x = 0,L, which one of the following statements is true? a. The probability for finding the particle in the left-hand side of the box depends upon the quantum number n. b. The average value of the position of the particle is L/ for any allowed state. c. The lowest-energy state has zero kinetic energy. d. Ψ for the lowest-energy state gives the closest approach to the classical distribution of mass. e. The separation between energy levels increases as the mass of the particle increases. 15
10. Which one of the following statements conflicts with the quantum mechanical results for a particle undergoing a one-dimensional harmonic oscillation? a. The lighter the mass of the particle, the greater will be its vibrational zeropoint energy. b. The vibrational frequency of a quantum oscillator is the same as that of a classical oscillator. c. Increasing the vibrational force constant increases the spacing between successive energy levels. d. The spacing between successive energy levels increases as the vibrational quantum number is increased. 11. A quantum-mechanical harmonic oscillator a. spends most of its time near its classical turning points in its lowest-energy state. b. has Ψ = 0 at its classical turning points. c. has doubly degenerate energy levels. d. has energy levels which increase with the square of the quantum number. e. None of the above is a correct statement. 16
1. Light of wavelength 4.33 x 10-6 m excites a harmonic oscillator from its ground to its first excited vibrational state. The wavelength of light that would accomplish this same excitation if the force constant for the oscillator were doubled is a. 4.33 x 10-6 m b..16 x 10-6 m c. 3.06 x 10-6 m d. 6.1 x 10-6 m e. 8.66 x 10-6 m 13. Light of wavelength 4.33 x 10-6 m excites a harmonic oscillator from its ground to its first excited vibrational state. The wavelength of light that would accomplish this same excitation if the oscillator mass were doubled but no other change occurred is a. 4.33 x 10-6 m b..16 x 10-6 m c. 3.06 x 10-6 m d. 6.1 x 10-6 m e. 8.66 x 10-6 m 14. Which one of the following statements is true about the particle-in-a-onedimensional box system with infinite walls? (all integrals range over the full length of the box.) a. Ψ 3 dx = 0 because these wave functions are orthogonal. b. Ψ 1 Ψ 3 dx = 1 because both of these wave functions are normalized. c. Ψ 1 Ψ dx = 0 shows that these wave functions are normalized. d. Ψ 1 Ψ 3 dx = 0 because these wave functions are orthogonal. 17