5. (McQuarrie 3-27) Calculate (p) and (p2) for the n = 2 state of a particle in a 1-D box of length a. Show that: ор = h a What happens to your uncertainty in the momentum as the box size shrinks? Relate this to the Heisenberg uncertainty principle. 7. (McQuarrie 3-30) Show that the normalized wave function for a particle in a 3-D box with sides of length a, b, and c is: P(x, y, z) = √ √ sin ("") sin ("") sin ("-2) 8 abc a пупу b Hint: You can argue the form of the solution based on separation of variables and the known 1-D solutions. Then just show that the resulting 3-D wave function is normalized.

5. (McQuarrie 3-27) Calculate (p) and (p2) for the n = 2 state of a particle in a 1-D box of length a. Show that: ор = h a What happens to your uncertainty in the momentum as the box size shrinks? Relate this to the Heisenberg uncertainty principle. 7. (McQuarrie 3-30) Show that the normalized wave function for a particle in a 3-D box with sides of length a, b, and c is: P(x, y, z) = √ √ sin ("") sin ("") sin ("-2) 8 abc a пупу b Hint: You can argue the form of the solution based on separation of variables and the known 1-D solutions. Then just show that the resulting 3-D wave function is normalized.

Principles of Modern Chemistry

8th Edition

ISBN:9781305079113

Author:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Publisher:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Chapter4: Introduction To Quantum Mechanics

Section: Chapter Questions

Problem 37P: (a) Using Equation 4.36, make a graph of the n=3 wave function and the square of this wave function...

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(a) Using Equation 4.36, make a graph of the n=3 wave function and the square of this wave function for a particle in a box with the edges at x=0 and x=7 . Estimate the probability of finding the particle between x1=3 and x2=4 . You may either do an integral or find the area under the curve by counting the number of rectangular boxes (roughly estimating the fractional boxes), and then multiply by the area of each box, watching your units. Show your calculation with units for credit. (b) Verify that the n=3 wave function in (a) is an allowed quantum state (sometimes called an eigenfunction) of by substituting it for in the left side of the equation =E immediately following Eq. 4.32. Calculate the derivatives to see if your result E is a constant multiplied by the original n=3 wavefunction. What does the constant you get tell you about the energy of thiseigenstate? Is it in agreement with the energy for the n=3 level you would calculate from Equation 4.37?
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