Please answer all parts!! 5. Consider a particle in a box of length L = 1 in a state defined by the wave function,4(x) = Ax²(1 - x)(a) Apply the normalization condition to determine A in op(x). 2(1)(b) Compute the expectation value of the energy directly. Express it in the form E =(const)h2/8m. What is the value of const?(c) Sketch (x) inside the box. Based on visual inspection, what two particle-in-a-boxeigenfunctions probably contribute most to (x)?(d) Expand (x) in terms of the particle in a box eigenfunctions. Write the first four terms inthe wave function with numerical coefficients. If you work this problem by hand, thefollowing integrals will be useful-Ju² sinu du = -u² cosu + 2cosu + 2usinuJu³ sinu du = -u³ cosu + 3u² sinu - 6sinu + 6ucosu(e) Evaluate the average energy using your approximate wave function from part (d), andcompare it to the result you got from part (b).

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Answered: 5. Consider a particle in a box of…

Introductory Chemistry: A Foundation

9th Edition

ISBN:9781337399425

Author:Steven S. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Donald J. DeCoste

Chapter10: Energy

Section: Chapter Questions

Problem 3QAP

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What is the physical explanation of the difference between a particle having the 3-D rotational wavefunction 3,2 and an identical particle having the wavefunction 3,2?
Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction ψn. (a) Without evaluating any integrals, explain why ⟨x⟩ = L/2. (b) Without evaluating any integrals, explain why ⟨px⟩ = 0. (c) Derive an expression for ⟨x2⟩ (the necessary integrals will be found in the Resource section). (d) For a particle in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why <p2x> = n2h2/4L2.
E7B.3(b) Which of the following functions can be normalized (in all cases the range for x is from x = -∞ to ∞, and a is a positive constant): (i) sin(ax); (ii) cos(ax) ex²? Which of these functions are acceptable as wavefunctions?
Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinx
(a) For a particle in the stationary state n of a one dimensional box of length a, find the probability that the particle is in the region 0xa/4.(b) Calculate this probability for n=1,2, and 3.
E7D.5(b) Calculate the probability that a particle will be found between 0.65L and 0.67L in a box of length L for the case where the wavefunction is (i) ₁, (ii) V₂. You may make the same approximation as in Exercise E7D.5(a). pr E7 po pr
[p* L₂ da 0 HO H + →
The given wave function for the hydrogen atom is y =w,00 +210 + 3y2 · Here, ypim has n, 1, and m as principal, orbital, and magnetic quantum numbers respectively. Also, yim an eigen function which is normalized. The expectation value of L in the state wis, is 9h? (a) 11 (b) 11h? 20 (c) 11 (d) 21ħ?
P7B.1 Imagine a particle confined to move on the circumference of a circle ('a particle on a ring'), such that its position can be described by an angle & in the range 0 to 2π. Find the normalizing factor for the wavefunctions: (a) e" and (b) eim, where m, is an integer.
Consider a fictitious one-dimensional system with one electron.The wave function for the electron, drawn below, isψ (x)= sin x from x = 0 to x = 2π. (a) Sketch the probabilitydensity, ψ2(x), from x = 0 to x = 2π. (b) At what value orvalues of x will there be the greatest probability of finding theelectron? (c) What is the probability that the electron willbe found at x = π? What is such a point in a wave functioncalled?
Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In eachcase, give your reasons for accepting or rejecting each function. (i) Ψ(x)=x2; (ii) Ψ(x)=1/x; (iii) Ψ(x)=e-x^2.
The ground-state wavefunction for a particle confined to a one dimensional box of length L is Ψ =(2/L)½ sin (πx/L) Suppose the box 10.0 nm long. Calculate the probability that the particle is: (a) between x = 4.95 nm and 5.05 nm (b) between 1.95 nm and 2.05 nm, (c) between x = 9.90 and 10.00 nm, (d) in the right half of the box and (e) in the central third of the box.

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