(2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)?
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1 Part I: Quantum Mechanics: Principles & Models 1. General Concepts: (2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)? (4 pts) b. How does separability between time and space help you in solving the time-dependent Schrödinger Equation for a one-dimensional particle? (4 pts) c. Use a physical argument to justify why the zero-point energy of the particle-in-a-box goes up with decreasing box size. 2
2 2. In this problem, we will study a harmonic oscillator of frequency, ω, and mass, m. You are also given the normalized wave function, ψ a (x) = Ne ax2 where a is some arbitrary value, and the normalization constant is given by: π N 2 = 2a (5 pts) a. What is the expectation value, ˆx 2, for the state, ψ a (x), as a function of a and other physical variables? (5 pts) b. What is the expectation value, ˆp 2, for the state, ψ a (x), as a function of a and other physical variables? 3
3 (5 pts) c. What is the expectation value of the energy for the state, ψ a (x), as a function of a and other physical variables? (5 pts) d. Use the variational principle to show how a is related to ω in the optimal variational estimate of the ground state wave function. 4
4 Part II: Atoms & Molecules 3. General Concepts (3 pts) a. How are one-electron atom states used to approximately solve the Schrödinger equation for a many-electron atom? (If there is more than one way, describe each of them.) (3 pts) b. How are one-electron atom states used to approximately solve the Schrödinger equation for a molecule? (If there is more than one way, describe each of them.) (4 pts) c. What is an appropriate model for describing the center-of-mass motion of a molecule, and why? 5
5 4. (5 pts) a. What is the orbital angular momentum of an electron in a 4p state? (5 pts) b. What is the nodal structure of a 4p one-electron state? (5 pts) c. What are the term symbols for the eigenstates of two electrons occupying the 4p state? (5 pts) d. What is the molecular orbital energy level diagram that typically arises from the combination of 4p states in a homonuclear diatomic with each atom having TWO electrons in the 4p state? 6
6 Part III: Atomic & Molecular Spectroscopy 5. General Concepts (3 pts) a. Why are selection rules useful? (3 pts) b. How are selection rules determined? (4 pts) d. Assuming that the harmonic selection rule is exact, how may infrared peaks do you expect to see in the spectrum of a molecule consisting of 27 atoms? Justify your answer. 7
7 6. WARNING: the frequencies provided here are not correct; they are given in order to make the problem statements simpler... (5 pts) a. Suppose that the vibrational frequency of H 35 Cl were determined to be cm 1. What is the force constant for this bond? (5 pts) b. Given the information provided in part (a), what is the vibrational frequency of the isotope D 37 Cl? 8
8 (5 pts) c. Suppose that the energy difference between the J = 1 to J = 3 rotational energy levels of CH 4 were determined to be 20.0 cm 1 using a Raman experiment. What is the CH bond length in CH 4? (5 pts) b. Given the information provided in part (c), what would be the energy difference between the J = 1 and J = 3 rotational energy levels in CD 4? 9
9 Part IV: Statistical Mechanics 7. General Concepts (2 pts) a. How do Legendre transforms help you in constructing the Maxwell relations? (2 pts) b. How are Legendre transforms between free energies related to the corresponding partition functions? (2 pts) c. In what way, is the partition function of an ideal system simpler than the partition function of a non-ideal system? (4 pts) d. Provide two examples of thermodynamic properties of a system that you could calculate if only you knew Q(T, N, V ). 10
10 8. Note that the average bond length of 12 C 16 O is m, and you can assume that it is unchanged in 13 C 16 O. (5 pts) a. What is the ratio of the translational partition function for 12 C 16 O to that of 13 C 16 O at 300K? (Make sure to show your work!) (5 pts) b. What is the ratio of the vibrational partition function for 12 C 16 O to that of 13 C 16 O at 300K? (Make sure to show your work!) 11
11 (5 pts) c. What is the ratio of the rotational partition function for 12 C 16 O to that of 13 C 16 O at 300K? (Make sure to show your work!) (5 pts) d. What is the ratio of the electronic partition function for 12 C 16 O to that of 13 C 16 O at 300K? (Make sure to show your work!) 12
12 BONU Answer concisely, but convincingly! (5 pts) a. Why isn t classical mechanics sufficient to describe molecular properties and reactivity? (5 pts) b. Why isn t thermodynamics sufficient to describe and predict the properties of a protein dissolved in water at concentration of parts per million? 13
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