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-box eigenfunctions probably contribute most to (x)? (d) Expand (x) in terms of the particle in a box eigenfunctions. Write the first four terms in the wave function with numerical coefficients. If you work this problem by hand, the following integrals will be useful - Ju² sinu du = -u² cosu + 2cosu + 2usinu Ju³ sinu du = -u³ cosu + 3u² sinu - 6sinu + 6ucosu (e) Evaluate the average energy using your approximate wave function from part (d), and compare it to the result you got from part (b).

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-box eigenfunctions probably contribute most to (x)? (d) Expand (x) in terms of the particle in a box eigenfunctions. Write the first four terms in the wave function with numerical coefficients. If you work this problem by hand, the following integrals will be useful - Ju² sinu du = -u² cosu + 2cosu + 2usinu Ju³ sinu du = -u³ cosu + 3u² sinu - 6sinu + 6ucosu (e) Evaluate the average energy using your approximate wave function from part (d), and compare it to the result you got from part (b).

Physical Chemistry

2nd Edition

ISBN:9781133958437

Author:Ball, David W. (david Warren), BAER, Tomas

Publisher:Ball, David W. (david Warren), BAER, Tomas

Chapter11: Quantum Mechanics: Model Systems And The Hydrogen Atom

Section: Chapter Questions

Problem 11.61E: What is the physical explanation of the difference between a particle having the 3-D rotational...

See similar textbooks

Similar questions

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) If Â = 3x? and B = , then show that Â and ß donot commute with respect to the function f(x) = sin x. Show, if the wave function, w) = A cos(kx) + iA sin(kx) is an Eigen-function of the linear momentum operator, P and if so, what is the Eigen value. (Note: A and k are constants). (b)
(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
5. Consider a particle constrained to move in one dimension described by the wavefunction v (x) = Ne2** (a) Determine the normalization constant (b) Is the wavefunction an eigenfunction of d? +16x? dx? (c) Calculate the probability of finding the particle anywhere along the negative x-axis
[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.
2. Which of the following wavefunctions are eigenfunctions d? of the operator dx- For those that are eigenfunctions, what is the eigenvalue (a) Y = ex (b) Y = x? (c) Y = sin x (d) Y = 3 cos x (e) Y = sin x + cos x