# HW2 - x,t | 2 as time goes on(d Find h x i h p i h x 2 i h...

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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 {- a | x |} , where A and a are positive real constants. (a) Normalize Ψ ( x, 0). Then ﬁnd the corresponding wave function in momen- tum space φ ( k, 0). (b) Construct Ψ ( x,t ) in the form of an integral. (c) Discuss the limiting cases ( a very large, and a very small). 2. 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(
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Unformatted text preview: 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? 3. 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 . —— End ——...
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