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3890ps4_10

# 3890ps4_10 - Chem 3890 Physical Chemistry I Fall 2010...

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Chem 3890 Physical Chemistry I Fall 2010 Chemistry 3890 Problem Set 4 Fall 2010 Due: in class, Friday, October 1 Last revised: September 24, 2010 Problem 1 A PIB is prepared in the state Ψ( x,t = 0) = ψ ( x ) = bracketleftBig 2 φ 1 ( x ) - 2 ( x ) + e iπ/ 4 2 φ 3 ( x ) bracketrightBig , (1) at t = 0, where the particle has mass m , 0 x a and the φ are the usual PIB eigenfunctions. 1. Normalize the state ψ ( x ). Use the normalized ψ ( x ) in the rest of the problem. 2. The particle energy (Hamiltonian ˆ H ) is measured at t = 0. What is the probability of finding E = E 1 , E = E 2 , E = E 3 and E = E 4 , respectively, in an individual measurement of ˆ H ? 3. If a large number of such measurements are made, calculate the expectation values ( H ) and ( H 2 ) , and the associated uncertainty σ H . 4. Write down an expression for the probability P ( ξ ) that a PIB in state ψ ( x ) will be found in the range 0 x ξ when the particle position is measured; leave your expression in the form of an integral involving ψ and ψ * . Use Mathematica to evaluate your expression for P ( ξ ) for the state (1). What is the limiting value of P ( ξ ) as ξ 0? ξ a ? Do these values make sense?

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3890ps4_10 - Chem 3890 Physical Chemistry I Fall 2010...

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