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120A_PS3 - Chemistry 120A Problem Set 3(due 1 Problems 4-17...

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Chemistry 120A Problem Set 3 (due February 12, 2010) 1. Problems 4-17, 4-20 and 4-24 in McQuarrie and Simon (M&S). 2. Two-state quantum mechanics . Many chemical systems are usefully approximated in terms of two quantum states. One class of examples is illustrated in Figure 1 – an electron associated with two sites, A and B ,. The simplest molecule, H + 2 , is one realization, where one site refers to the region near one proton, and the other site refers to the region near the other proton. Electron transfer between donor and acceptor sites in oxidation/reduction chemistry is another realization. If the electron is located at or near site A , its wave function is denoted by φ A ( r ) where r is the position of the electron. Similarly, if it is located at or near site B , its wave function is φ B ( r ). In Dirac’s notation, these wave functions are h r | A i and h r | B i , respectively, and corresponding state vectors are | A i and | B i , respec- tively. Assume these state vectors are normalized and orthogonal, i.e. h B | A i = R d r φ * B ( r ) φ A ( r ) = 0, and h A | A i = h B | B i = 1. Further, assume that these two vectors span all physically pertinent states. Thus, in this case, | A i and | B i form a complete and orthonormal set. The Hamiltonian operator in this representation is the 2 × 2 matrix H = A - K - K * B , where A , B and K are constants. Because the Hamiltonian is Hermitian, A and B are real numbers. For simplicity, when carrying out this exercise, you may also
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