PHY4221Fall05_Prob2

PHY4221Fall05_Prob2 - PHY4221 Quantum Mechanics I Fall term...

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PHY4221 Quantum Mechanics I Fall term of 2004 Problem Set No.2 (Due on October 16, 2004) 1. A free particle has the initial wave function Ψ( x, 0) = A exp n - ax 2 o , (a) Normalize Ψ ( x, 0). (b) Find Ψ ( x,t ). (c) Sketch | Ψ( x,t ) | 2 (as a function of x ) at t = 0, and again for very large t . Qualitatively, what happens to | Ψ( x,t ) | 2 as time goes on? (d) Find h x i , h p i , h x 2 i , h p 2 i , σ x and σ p . (e) Does the uncertainty principle hold? At what time t does the system come closest to the uncertainty limit? 2. A particle of mass m in the infinite square well (of width a ) is (at t = 0) equally likely to be found at any point in that region. (a) What is its initial wave function, Ψ( x, 0)? (Assume it is a real function.) (b) What is the probability that a measurement of the energy would yield the value π 2 ¯ h 2 2 ma 2 . 3. Use the Heisenberg’s equation of motion to show that d dt h xp i = 2 h T i - h x dV dx i , where T and V are the kinetic energy operator and potential energy operator
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This note was uploaded on 12/10/2011 for the course PHYS 4221 taught by Professor Cflo during the Spring '11 term at CUHK.

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PHY4221Fall05_Prob2 - PHY4221 Quantum Mechanics I Fall term...

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