Consider a wavefunction Ψ0 prepared at t = 0 in an infinite square well of width L, such that Ψ0 = N in the interval −L/8 < x < L/8 and is zero otherwise, as illustrated below. found in image Figure 3: A wavefunction Ψ0, prepared in an infinite square well at time t = 0. a) Derive an expression for the normalization constant N, in terms of L. b) The wavefunction of this system at time t can be written as Ψ(x,t) = Pn cnψn(x)e−iEnt/~, where {cn} are a set of constants and {ψn(x)} are the set of energy eigenstates of the well. Use this equation to derive an expression for the cn in terms of Ψ0 and ψn(x). c) The energy eigenstates in the infinite square well are given by found in image Use these equations to derive expressions for the first five of the cn defined in part b) (i.e. derive expressions for c1, c2, c3, c4 and c5).
Consider a wavefunction Ψ0 prepared at t = 0 in an infinite square well of width L, such that Ψ0 = N in the interval −L/8 < x < L/8 and is zero otherwise, as illustrated below.
found in image
Figure 3: A wavefunction Ψ0, prepared in an infinite square well at time t = 0.
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a) Derive an expression for the normalization constant N, in terms of L.
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b) The wavefunction of this system at time t can be written as Ψ(x,t) = Pn cnψn(x)e−iEnt/~, where {cn} are a set of constants and {ψn(x)} are the set of energy eigenstates of the well. Use this equation to derive an expression for the cn in terms of Ψ0 and ψn(x).
c) The energy eigenstates in the infinite square well are given by
found in image
Use these equations to derive expressions for the first five of the cn defined in part b) (i.e. derive expressions for c1, c2, c3, c4 and c5).
d) Calculate new expressions for the cn if the constant initial wavefunction Ψ0 were changed so that it was restricted to have non-zero values only in the range 0 < x < L/8 (i.e. such that Ψ0 = constant in the region 0 < x < L/8, and is zero everywhere else).
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