Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Textbook Question
Chapter 10, Problem 10.57E
Is the uncertainty principle consistent with our description of the wavefunctions of the
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The wave functions p and d are linear combinations of the spherical harmonic functions, which are eigenfunctions of the operators ÂÎ, î², and Î ₂ for rotation in three dimensions. The combinations have been chosen to yield real
functions.
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Part A
Are these functions still eigenfunctions of 1₂? Answer this question by applying the operator to the functions.
Match the items in the left column to the appropriate blanks in the sentences and the equations on the right. Make certain each sentence and equation is complete before submitting your answer.
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0
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an eigenfunction of 1 z.
Help
Consider a 1D particle in a box confined between a = 0 and x = 3. The Hamiltonian for the particle inside the
box is simply given by Ĥ
. Consider the following normalized wavefunction
2m dz²
¥(2) =
35
(x³ – 9x). Find the expectation value for the energy of the particle inside the box. Give your
5832
final answer for the expectation value in units of (NOTE: h, not hbar!). In your work, compare the expectation
value to the lowest energy state of the 1D particle in a box and comment on how the expectation value you
calculated for the wavefunction ¥(x) is an example of the variational principle.
For the model of a particle in a one-dimensional box, sketch the wavefunctions
from n = 1 to n = 4 and the corresponding probability density functions, in order
of increasing energy. Indicate the position of any nodes in both sketches.
Chapter 10 Solutions
Physical Chemistry
Ch. 10 - State the postulates of quantum mechanics...Ch. 10 - Prob. 10.2ECh. 10 - State whether the following functions are...Ch. 10 - State whether the following functions are...Ch. 10 - Prob. 10.5ECh. 10 - Prob. 10.6ECh. 10 - Evaluate the operations in parts a, b, and f in...Ch. 10 - The following operators and functions are defined:...Ch. 10 - Prob. 10.9ECh. 10 - Indicate which of these expressions yield...
Ch. 10 - Indicate which of these expressions yield an...Ch. 10 - Why is multiplying a function by a constant...Ch. 10 - Prob. 10.13ECh. 10 - Using the original definition of the momentum...Ch. 10 - Under what conditions would the operator described...Ch. 10 - A particle on a ring has a wavefunction =12eim...Ch. 10 - Calculate the uncertainty in position, x, of a...Ch. 10 - For an atom of mercury, an electron in the 1s...Ch. 10 - Classically, a hydrogen atom behaves as if it were...Ch. 10 - The largest known atom, francium, has an atomic...Ch. 10 - How is the Bohr theory of the hydrogen atom...Ch. 10 - Though not strictly equivalent, there is a similar...Ch. 10 - The uncertainty principle is related to the order...Ch. 10 - Prob. 10.24ECh. 10 - Prob. 10.25ECh. 10 - For a particle in a state having the wavefunction...Ch. 10 - Prob. 10.27ECh. 10 - A particle on a ring has a wavefunction =eim,...Ch. 10 - Prob. 10.29ECh. 10 - Prob. 10.30ECh. 10 - Prob. 10.31ECh. 10 - Normalize the following wavefunctions over the...Ch. 10 - Prob. 10.33ECh. 10 - Prob. 10.34ECh. 10 - For an unbound or free particle having mass m in...Ch. 10 - Prob. 10.36ECh. 10 - Prob. 10.37ECh. 10 - Prob. 10.38ECh. 10 - Evaluate the expression for the total energies for...Ch. 10 - Prob. 10.40ECh. 10 - Verify that the following wavefunctions are indeed...Ch. 10 - In exercise 10.41a, the wavefunction is not...Ch. 10 - Prob. 10.43ECh. 10 - Prob. 10.44ECh. 10 - Explain why n=0 is not allowed for a...Ch. 10 - Prob. 10.46ECh. 10 - Prob. 10.47ECh. 10 - Prob. 10.48ECh. 10 - Carotenes are molecules with alternating CC and...Ch. 10 - The electronic spectrum of the molecule butadiene,...Ch. 10 - Prob. 10.51ECh. 10 - Prob. 10.52ECh. 10 - Show that the normalization constants for the...Ch. 10 - Prob. 10.54ECh. 10 - Prob. 10.55ECh. 10 - An official baseball has a mass of 145g. a...Ch. 10 - Is the uncertainty principle consistent with our...Ch. 10 - Prob. 10.58ECh. 10 - Prob. 10.59ECh. 10 - Instead of x=0 to a, assume that the limits on the...Ch. 10 - In a plot of ||2, the maximum maxima in the plot...Ch. 10 - Prob. 10.62ECh. 10 - Prob. 10.63ECh. 10 - The average value of radius in a circular system,...Ch. 10 - Prob. 10.65ECh. 10 - Prob. 10.66ECh. 10 - Prob. 10.67ECh. 10 - Prob. 10.68ECh. 10 - Prob. 10.69ECh. 10 - Assume that for a particle on a ring the operator...Ch. 10 - Mathematically, the uncertainty A in some...Ch. 10 - Prob. 10.72ECh. 10 - Prob. 10.73ECh. 10 - Verify that the wavefunctions in equation 10.20...Ch. 10 - An electron is confined to a box of dimensions...Ch. 10 - a What is the ratio of energy levels having the...Ch. 10 - Consider a one-dimensional particle-in-a-box and a...Ch. 10 - Prob. 10.78ECh. 10 - Prob. 10.79ECh. 10 - Prob. 10.80ECh. 10 - Prob. 10.81ECh. 10 - What are x,y, and z for 111 of a 3-D...Ch. 10 - Prob. 10.83ECh. 10 - Prob. 10.84ECh. 10 - Prob. 10.85ECh. 10 - Prob. 10.86ECh. 10 - Prob. 10.87ECh. 10 - Prob. 10.88ECh. 10 - Substitute (x,t)=eiEt/(x) into the time-dependent...Ch. 10 - Write (x,t)=eiEt/(x) in terms of sine and cosine,...Ch. 10 - Prob. 10.91ECh. 10 - Prob. 10.92ECh. 10 - Prob. 10.93ECh. 10 - Prob. 10.95E
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- 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 ddtanarrow_forwardConsider 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?arrow_forwardHow is the Bohr theory of the hydrogen atom inconsistent with the uncertainty principle? In fact, it was this inconsistency, along with the theorys limited application to non-hydrogen-like systems, that limited Bohrs theory.arrow_forward
- Verify that the following wavefunctions are indeed eigenfunctions of the Schrdinger equation, and determine their energy eigenvalues. a =eiKx where V=0 and K is a constant b =eiKx where V=k, k is some constant potential energy, and K is a constant c =2asinxa where V=0.arrow_forwardA particle on a ring has a wavefunction =eim, where =0to2 and m is a constant. a Normalize the wavefunction, where d is d. How does the normalization constant depend on the constant m? b What is the probability that the particle is in the ring indicated by the angular range =0to2/3? Does this answer make sense? How does the probability depend on constant m?arrow_forwardFor a particle in a state having the wavefunction =2asinxa in the range x=0toa, what is the probability that the particle exists in the following intervals? a x=0to0.02ab x=0.24ato0.26a c x=0.49ato0.51ad x=0.74ato0.76a e x=0.98ato1.00a Plot the probabilities versus x. What does your plot illustrate about the probability?arrow_forward
- Show that the normalization constants for the general form of the wavefunction =sin(nx/a) are the same and do not depend on the quantum number n.arrow_forwardIndicate 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)arrow_forwardThe uncertainty principle is related to the order of the two operators operating on a wavefunction. Evaluate the expressions x (pxsinx) and px( x sinx) and demonstrate that you get different results.arrow_forward
- In exercise 10.41a, the wavefunction is not normalized. Normalize the wavefunction and verify that it still satisfies the Schrdinger equation. The limits on x are 0 and 2. How does the expression for the energy eigenvalue differ?arrow_forwardThe wavefunction for the motion of a particle on a ring is of the form ψ = Neimlϕ. Evaluate the normalization constant, N.arrow_forwardGiven that a particle in a one dimensional box has the wavefunctionn 4 = (2)* sin(). n= 1,2,3... for 0arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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