| The exam contains 7 questions, each worth 25 points. You should answer 4 questions, totaling 100 points. You may answer less than 100 points but not more. Clearly indicate on this cover page those whole questions you wish graded. Write in pen. |
| Closed Book. Closed Notes. Show all work. Point values for subsections are indicated parenthetically. Box answers where appropriate. |
| 1 | Consider a particle with mass m confined to a box with walls at x=0 and x=L, that is V(x) = 0 for 0 < x < L and V(x) =  otherwise. For parts (a) and (b) below be sure your answer covers the entire x-axis - < x < . |
| a(3) | Write down the Schrodinger equation for this system. |
| b(3) | Write down the eigenvalues and eigenfunctions for the particle in the box. It is NOT necessary to solve the Schrodinger equation just write down the solutions. See cover page. |
| c(19) | The expectation value of an operator g is given by The uncertainty in g,  g, is defined by g2 = < (g-< g >)1 >Let g be the linear momentum operator o(ˆ,p)x and let  be n a particle in the box eigenstate. Determine <o(ˆ,p)x> and px as a function of n. Hint: n is normalized. |
| 2a (7) | What is wrong with the following wave function,(1,2) = 1s(1) (1)2s(2)Here 1 and 2 denote the coordinates of particles 1 and 2 respectively, and a is the 'spin up' function. |
| 2b (6) | >List and describe the good quantum numbers for the electron in a hydrogen atom obtained from in the Schrodinger's equation. Which of these quantum numbers are present in Bohr's model? |
| 2c (6) | Identify and describe one aspect of the photoelectric effect that could be explained on the basis of classical mechanics and one aspect that required a new quantum hypothesis. |
| 2d (6) | State the Pauli exclusion principle using the concept of quantum numbers. |
| 3. | The following questions refer to a free particle with mass m constrained to the x-axis. Here V(x)=0 for all x. |
| 3a (9) | Write down the Schrodinger equation, and all its eigenvalues and eigenfunctions. It is NOT necessary to solve the Schrodinger equation just write down the solutions. See cover page. |
| 3b (6) | Are any of the eigenfunctions of the free particle Schrodinger equation, eigenfunctions of o(ˆ,p)x the linear momentum operator? If so what are the eigenvalues? What is their degeneracy? Explain. |
| 3c (3) | Are any of the eigenfunctions of the free particle Schrodinger equation eigenfunctions of o(ˆ,p)x2? If so what are the eigenvalues? |
| 3d (7) | For the free particle determine an expression for finding the particle in the region between x and x+dx. Rationalize this result using the Heisenberg uncertainty principle. |
| 4a(9) | Construct the correlation diagram for B2including the 1s, 2s and 2p orbitals. Justify your ordering of the  2p and the  2p orbitals. |
| 4b(7) | Assume that the ground state of B2 is paramagnetic. Explain how this result can be used to distinquish between the possible orderings of the 2p and the 2p orbitals in part (a). |
| 4c(2) | On the basis of your correlation diagram will the ground state of B2 be bound? Explain. |
| 4d(7) | Assume that NaF is an ionic compound. Explain how molecular orbital theory describes the ionic bond by describing the HOMO (highest occupied molecular orbital), LUMO (lowest unoccupied molecular orbital) and the associated electron configuration. |
| 5 | Explain statements (5a)-(5d) on the basis of the ideas leading periodic trends. |
| 5a(5) | The ionization potential for oxygen is less than that for nitrogen. |
| 5b(5) | The atomic radius of Cl ( 0.99 Å), atomic number 17, is greater than that of Ar (0.97 Å), atomic number 18. |
| 5c(5) | The screening of a 2s electron by a 1s electron is more complete than that of a 2p electron by a 2s electron. |
| 5d(5) | The electron affinity of F is greater than that of neon. |
| 5e(5) | Consider the atomic radii of the following, consecutive, alkaline earth metal ions: Mg2+(12), Ca2+(20), Sr2+(38). The atomic number is given in parenthesis. Is the change in ionic radius expected to be greater going from Mg+ to Ca2+ or from Ca2+ to Sr2+? Explain. |
| 6. | Consider the molecule CH3CH Ð HCCH3 as pictured below The carbon atoms are in the xy plane. Assume the standard hybridization for carbon. |
| 6a (5) | Describe, in terms of hybrid atomic orbitals and atomic orbitals the character of the in plane bonds between atoms 1 - 2 and between atoms 2-3. |
| 6b (5) | Draw pictures to indicate the structure of the molecular orbitals as linear combinations of atomic orbitals. Clearly indicate the phase relationships (signs of coefficients of combination) and any zero coefficients. |
| 6c(5) | On an orbital energy diagram indicate the relative energies of the orbitals in part (b). Indicate the - electron configuration of the ground electronic state and the first excited state |
| 6d(5) | Describe the principal difference in bond length between CH3CH Ð HCCH3 and CH3CHÐHCCH3+. |
| 6e(5) | Shining light on this molecule facilitates trans-cis isomerization. Explain. |
| 7a(20) | For CO construct the molecular orbital correlation diagram and describe the character of all orbitals. Explain in detail your description.
To receive credit you must: (i) use the appropriate hybrization for carbon and oxygen and carefully include its consequences; (ii) justify all splittings in terms of orbital interaction diagrams, and (iii) fully describe the relative atomic orbital character of the molecular orbitals. Citations to the text without the above justification will not receive credit. |
| 7b(5) | Which nonbonding orbital has the largest spatial extent along the internuclear axis? EXPLAIN. Hint: Notions from periodic trends will help here. |