ps1_1976

ps1_1976 - oscillator: v | x | v = v x v dx = unless v = v...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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± ² ³ ´ µ ³ ´ µ MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 1976 Problem Set #1 1. Given [ x k , p j ] = ( x k p j p j x k ) = i δ jk and x ˆ y ˆ z ˆ L r × p x y z p x p y p z Show that (a) [ L i , L i ] = 0 i = x , y , z (b) [ L i , L j ] = i L k ( i , j , k ) = ( x , y , z ) and cyclic permutations. 2. Consider the Hamiltonian matrix constructed in the { φ } basis. α β 0 φ 1 H = β α β φ = φ 2 0 β 2 α φ 3 α, β are real numbers. (a) Give eigenvalues, their corresponding eigenvectors, and the unitary transformation which brings H to diagonal form ( U HU ) i j = E δ i j . (HINT: Use trigonometric form of solution for cubic equation.) (b) Can the eigenvalues be evenly spaced for clever choice of α, β ? (c) Can two of the eigenvalues ever be degenerate for nonzero β ? 3. Carry out the angular momentum algebra and show explicitly that ±± z | j x | ±± z ± = 0, and that J 2 J x ± 2 0. x 4. (a) Given the matrix elements of the coordinate x for a harmonic
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Unformatted text preview: oscillator: v | x | v = v x v dx = unless v = v 1 and v + 1 | x | v = (2 ) 1 / 2 ( v + 1) 1 / 2 v 1 | x | v = (2 ) 1 / 2 ( v ) 1 / 2 where = 4 2 m / h , and = 2 1 [ k / ] 1 / 2 is the vibrational harmonic frequency. Evaluate the nonzero matrix elements of x 2 , x 3 , and x 4 ; that is, evaluate the integrals v | x r | v = v x r v dx for r = 2 , 3, and 4 (without actually doing the explicit integrals, of course!). (b) From the results of part a, evaluate the average values of x , x 2 , x 3 , and x 4 in the v th vibrational state. Is it true that x 2 = ( x ) 2 , or that x 4 = ( x 2 ) 2 ? What conclusions can you draw about the results of a measurement of x in the v th vibrational state?...
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ps1_1976 - oscillator: v | x | v = v x v dx = unless v = v...

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