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...

See similar textbooks

Similar questions

(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?
A particle of mass m is placed in a three-dimensional rectangular box with edge lengths 2L, L, and L. Inside the box the potential energy is zero, and outside it is infinite; therefore, the wave function goes smoothly to zero at the sides of the box. Calculate the energies and give the quantum numbers of the ground state and the first five excited states (or sets of states of equal energy) for the particle in the box.
Using Table 5.2, write down the mathematical expression for the 2px wave function for an electronically excited H atom. Estimate the probability of finding the 2px electron if you look in a cubical box of volume of 0.8(pm)3 centered at a distance of 0.5001010m in the =/2 , =0 direction. Does this probability change as you change ? At what angles is the probability of finding the electron smallest and at what angles is the probability the largest? (Note that =2 is the same location as =0 , so don’t double count.)
Estimate the probability of finding an electron which is excited into the 2s orbital of the H atom, looking in a cubical box of volume 0.751036m3 centered at the nucleus. Then estimate the probability of finding the electron if you move the volume searched to a distance of 105.8 pm from the nucleus in the positive z direction. (Note that since these volumes are small, it does not matter whether the volume searched is cubical or spherical.)
Consider a one-dimensional particle-in-a-box and a three-dimensional particle-in-a-box that have the same dimensions. a What is the ratio of the energies of a particle having the lowest possible quantum numbers in both boxes? b Does this ratio stay the same if the quantum numbers are not the lowest possible values?
Indicate which of these expressions yield an eigenvalue equation, and if so indicate the eigenvalue. a ddxcos4xb d2dx2cos4x c px(sin2x3)d x(2asin2xa) e 3(4lnx2), where 3=3f ddsincos g d2d2sincosh ddtan
Draw, label, and explain the functions of the parts of a spectroscope.
Indicate which of these expressions yield eigenvalue equations, and if so indicate the eigenvalue. a ddxsinx2b d2dx2sinx2 c iddxsinx2d iddxeimx, where m is a constant e ddx(ex)f (22md2dx2+0.5)sin2x3 g ddy(ey2)
The maximum in the blackbody radiation intensity curve moves to shorter wavelength as temperature increases. The German physicist Wilhelm Wien demonstrated the relation to be max1/T . Later, Planck’s equation showed the maximum to be max=0.20hc/kt . In 1965, scientists researching problems in telecommunication discovered “background radiation” with maximum wavelength 1.05 mm (microwave region of the EM spectrum) throughout space. Estimate the temperature of space.
What are the values of E, L, and Lz for an F8+ atom whose electron has the following wavefunctions, listed as n,l,ml? a 1,0,0 b 3,2,2c 2,1,1d 9,6,3.
State how many radial, angular, and total nodes are in each of the following hydrogen-like wavefunctions. a 2s b 3s c 3p d 4f e 6g f 7s
List some unexplainable phenomena from the classical science and describe what could not be explained about them at the time.