120A_PS2 - Chemistry 120A Problem Set 2 (due February 5,...

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Chemistry 120A Problem Set 2 (due February 5, 2010) 2. Consider the particle constrained to move on a circle for which you have found the stationary states in Problem 1. In this problem, suppose this particle is prepared in a non-stationary state where its time-dependent wave function Ψ( θ,t ) is initially Ψ( θ, 0) sin( θ ) - sin(2 θ ) , (a) Show that ψ n ( θ ) = (1 / 2 π ) 1 / 2 exp( inθ ) for n = 0 , ± 1 , ± 2 ,... is an orthonormal set, i.e., that Z 2 π 0 dθ ψ * n ( θ ) ψ m ( θ ) = δ n,m . (b) Show that the general solution to the time-dependent Schr¨odinger equation is Ψ( θ,t ) = X n = -∞ c n exp ± inθ - i ( n 2 ~ / 2 I ) t ² , where the c n ’s are constants. (c) For the given initial wave-function, Ψ( θ, 0), determine the values of the c n ’s. (d) Determine the expectation values of θ and θ 2 as functions of time, t , and describe the time dependence of the probability density of θ . (e) Determine the expectation value of the energy. Does it depend upon time?
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120A_PS2 - Chemistry 120A Problem Set 2 (due February 5,...

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