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Unformatted text preview: Chem 3890 Physical Chemistry I Fall 2010 Chemistry 3890 Problem Set 7 Fall 2010 Due: in class, Friday, October 29 Last revised: October 22, 2010 Problem 1 The Hermite polynomials H n [ ] (Griffith, Sec. 2.3, see also http://mathworld.wolfram.com/ HermitePolynomial.html ) satisfy the following relations ( n a non-negative integer) H prime n = 2 nH n- 1 (1a) H n +1 = 2 H n- 2 nH n- 1 (1b) where H prime n d H n / d . 1. Using these relations, show that the Hermite polynomials satisfy Hermites equation: H primeprime n- 2 H prime n + 2 nH n = 0 . (2) Normalized HO eigenstates are (McQ&S, 5.6) n ( x ) = bracketleftbigg 2 n n ! bracketrightbigg 1 2 H n ( x ) e- x 2 / 2 , n = 0 , 1 , 2 , . . . (3) where m/ planckover2pi1 . 2. Use the above information to derive expressions for the matrix elements ( n prime | x | n ) integraldisplay + - d x * n prime ( x ) x n ( x ) (4a) and ( n prime | p | n ) integraldisplay + - d x * n prime ( x ) p n ( x ) (4b) for general n and n prime (non-negative integers). 3. Evaluate the expectation values ( x ) n ( n | x | n ) , ( x 2 ) n , ( p ) n , and ( p 2 ) n . 4. Show that ( T ) n = ( V ) n , where T is the kinetic energy operator and V is the HO potential operator....
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