Consider a classical particle moving in a one-dimensional potential well u(x), as shown in Figure 6.10 (attached). The particle is in thermal equilibrium with a reservoir at temperature so the probabilities of its various states are determined by Boltzmann statistics. (a) Show that the average position of the particle is given byfxe-Bu(2) drfe-Bu(x) d.u(x) 4where each integral is over the entire r axis.Figure 6.10. A one-dimensional po-tential well. The higher the temper-ature, the farther the particle willstray from the equilibrium point.

Solution: (A model of thermal expansion.) (a) Let the "system” be the particle's position and the…

Answered: (a) Show that the average position of…

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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). 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…
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There is an electron, in a 1-d, infinitely deep square potential well with a width of d. If it is in ground state, 1. Draw the electron's wavefunction. Show the position of the walls of the potential well. 2. Explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.
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(a) Show that the average position of the particle is given by fxe-Bu(2) dr fe-Bu(x) d. u(x) 4 where each integral is over the entire r axis. Figure 6.10. A one-dimensional po- tential well. The higher the temper- ature, the farther the particle will stray from the equilibrium point.