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...

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