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Chemistry 354 - Homework Set IV (Due 3/19/13) Name: 1. (10 points) An electron is confined to a one-dimensional box with dimensions Lx = 2 bohr and Ly = 1.5 bohr. Find all energy levels below 2500 eV, and provide a table with energies, quantum numbers and the level degeneracy associated with each energy. 1 2. (10 points, requires some google research) Express the following quantities in both SI and atomic units: the energy content of a “King Size” (108 g) Butterfinger bar; the enthalpy of combustion of a single molecule of sucrose; the distance from sideline to sideline on a football field; the time required for an electron to make one circuit of the nucleus in the Bohr atom; the mass of a hydrogen atom; the mass of Haystacks Calhoun (erstwhile professional wrestler); the distance between the n = 1 and n = 2 energy levels of an electron in a one-dimensional box of length 1 nm; and finally, the speed of light. Make a table of these numbers, and note that the “atomic-scale” quantities are all “reasonable” numbers in atomic units, while those for “regular life” quantities are “unreasonable”, and that the opposite behavior is seen for the numbers in SI units. Finally, come up with names for the atomic units of mass, time and speed. The “best” answers will be recognized. 2 3. (10 points) Show, by direct application of the Hamiltonian, that the state function " πx 2πy Ψ = N sin sin Lx Ly ! 2πx πy − π sin sin Lx Ly !# is an eigenfunction of the Hamiltonian only under one condition. Find this condition, the associated eigenvalue, and also solve for the value of the normalization constant N . 3 4. (10 points, independent thinking rewarded) Make up, and solve, a problem dealing with the quantum mechanical harmonic oscillator. Laura will be on the lookout – and will give a grade of zero – for problems taken directly from internet sources or textbooks, so rely on your own creativity and initiative for this one. 4 5. (10 points, extra credit) Using the solution to the expectation value for the energy of the one-dimensional particle in the box state function Ψ = N x sin( πx ). Lx obtained in the last problem set and the content of the recent handout on an alternative solution to the same problem, prove the identity: ∞ X n4 π 4 + 6π 2 = 2 4 12288 n=1 (4n − 1) If you can show this using just series summation techniques, that would certainly represent an acceptable (and impressive) answer. 5