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P-Chem II Course HWK 4 solutions

P-Chem II Course HWK 4 solutions - Phys—Chem...

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Unformatted text preview: Phys—Chem. ll [CHM4411—02], Spring 2012 (Distributed/posted on 2/17/11, due on Wed. 2/22/2011) Homework 4 You name: ------ Problem 7.10 M 1 A particle is in a state described by the wavefunction, (/1 = (420% 6"”, where a is constant and x obeys: OSXSW. 1) is w normalized? 2) Determine the expectation value of the commutator of the position and momentum operators: [X, Px]. Exercise 7.1.73 Determine the commutators of the operators: (a) fl— and —1~ dx X \ , .fl Q“, 4 V K 5 mm M Exercise 7.1 7(b) Determine the commutators of the operators A and A: where the5e operators are defined as: 2+m 2—m A=—-—and A+: J5 J5 gr Exercise 7.24 To the movement of a particle in one dimension x (x varies between ~00 and +00) we associate ~0X2 three wavefunctions: (a) 1/1 = 8”“; (b) (/1 : cos(kx); (c) {/2 z e . . . . A h d g; Find out which one of these IS an eigenfunction of the momentum operator, : ~77. I x w. ngQECalculate the average linear momentum of the particle for the wavefunctions (b) and (c) above. M .7 A” W $2.: .23 Exercise 8.2 Write the Schrodinger equation for a particie moving in 1 dimension potential well of width L (La, box of length L) and express the corresponding normalized wavefunction as function of an integer number n. Can n take any value? (1) Calculate the probability that the particle will be found between 0 and 0.01L in a box of length L when it has: (a) n21, (b) n=2, (c) n=4 (2) Calculate the probability that the particle will be found between 0.65L and 0.67L in a box of length L when it has: (3) n=1, (b) n=2, (c) n=4 Hint: Consider the wavefunction to be constant over these narrow windows. ...
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