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Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 11, Problem 11.25E
Interpretation Introduction
Interpretation:
The value of
Concept introduction:
In
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4.
Given these operators A=d/dx and B=x², can you measure the expectation
values of the corresponding observables to infinite precision simultaneously?
Without evaluating any integrals, state the value of the expectation value of x for a particle in a box of length L for the case where the wavefunction has n = 2. Explain how you arrived at your answer.
Part A
In normalizing wave functions, the integration is over all space in which the wave function is defined. Normalize the wave function x(a − x)y(b − y) over the range 0 ≤ x ≤ a,
0 ≤ y ≤ b. The element of area in two-dimensional Cartesian coordinates is dx dy; a and b are constants.
Match the items in the left column to the appropriate blanks in the equations on the right. Make certain each equation is complete before submitting your answer.
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30√√
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a5f5
[y(b - y)]
[x(a − x)]
[x(a − x)]²
a
30
a³f³
30
[y(by)]²
N²
0
b
0
N
||
a
dx
dx
||
dy =
0
b
dy = 1
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Chapter 11 Solutions
Physical Chemistry
Ch. 11 - Convert 3.558mdyn/A into units of N/m.Ch. 11 - Prob. 11.2ECh. 11 - Prob. 11.3ECh. 11 - Prob. 11.4ECh. 11 - Prob. 11.5ECh. 11 - Prob. 11.6ECh. 11 - Prob. 11.7ECh. 11 - Prob. 11.8ECh. 11 - Prob. 11.9ECh. 11 - Prob. 11.10E
Ch. 11 - Prob. 11.11ECh. 11 - Prob. 11.12ECh. 11 - a For a pendulum having classical frequency of...Ch. 11 - Prob. 11.14ECh. 11 - The OH bond in water vibrates at a frequency of...Ch. 11 - Show that 2 and 3 for the harmonic oscillator are...Ch. 11 - Prob. 11.17ECh. 11 - Prob. 11.18ECh. 11 - Prob. 11.19ECh. 11 - Use the expression for 1 in equations 11.17 and...Ch. 11 - Prob. 11.21ECh. 11 - Prob. 11.22ECh. 11 - Consider Figure 11.4 and choose the correct...Ch. 11 - Based on the trend shown in Figure 11.5, draw the...Ch. 11 - Prob. 11.25ECh. 11 - Prob. 11.26ECh. 11 - Prob. 11.27ECh. 11 - Prob. 11.28ECh. 11 - Prob. 11.29ECh. 11 - Prob. 11.30ECh. 11 - Compare the mass of the electron, me, with a the...Ch. 11 - Reduced mass is not reserved only for atomic...Ch. 11 - Prob. 11.33ECh. 11 - An OH bond has a frequency of 3650cm1. Using...Ch. 11 - Prob. 11.35ECh. 11 - Prob. 11.36ECh. 11 - Prob. 11.37ECh. 11 - Prob. 11.38ECh. 11 - Prob. 11.39ECh. 11 - What are the energies and angular momenta of the...Ch. 11 - Prob. 11.41ECh. 11 - A 25-kg child is on a merry-go-round/calliope,...Ch. 11 - Prob. 11.43ECh. 11 - a Using the expression for the energy of a 2-D...Ch. 11 - Prob. 11.45ECh. 11 - Prob. 11.46ECh. 11 - Prob. 11.47ECh. 11 - The quantized angular momentum is choose one:...Ch. 11 - Prob. 11.49ECh. 11 - Prob. 11.50ECh. 11 - Prob. 11.51ECh. 11 - Can you evaluate r for the spherical harmonic Y22?...Ch. 11 - Show that 1,0 and 1,1 for 3-D rotational motion...Ch. 11 - Prob. 11.54ECh. 11 - Prob. 11.55ECh. 11 - a Using the he expression for the energy of a 3-D...Ch. 11 - Prob. 11.57ECh. 11 - In exercise 11.57 regarding C60, what are the...Ch. 11 - Draw the graphical representations see Figure...Ch. 11 - Prob. 11.60ECh. 11 - What is the physical explanation of the difference...Ch. 11 - List the charges on hydrogen-like atoms whose...Ch. 11 - Prob. 11.63ECh. 11 - Prob. 11.64ECh. 11 - Prob. 11.65ECh. 11 - Calculate the difference between the Bohr radius...Ch. 11 - To four significant figures, the first four lines...Ch. 11 - What would the wavelengths of the Balmer series...Ch. 11 - Construct an energy level diagram showing all...Ch. 11 - Prob. 11.70ECh. 11 - What is the degeneracy of an h subshell? An n...Ch. 11 - What is the numerical value of the total angular...Ch. 11 - What are the values of E, L, and Lz for an F8+...Ch. 11 - Prob. 11.74ECh. 11 - Why does the wavefunction 4,4,0 not exist?...Ch. 11 - Prob. 11.76ECh. 11 - What is the probability of finding an electron in...Ch. 11 - What is the probability of finding an electron in...Ch. 11 - Prob. 11.79ECh. 11 - Prob. 11.80ECh. 11 - State how many radial, angular, and total nodes...Ch. 11 - Prob. 11.82ECh. 11 - Prob. 11.83ECh. 11 - Verify the specific value of a, the Bohr radius,...Ch. 11 - Prob. 11.85ECh. 11 - Prob. 11.86ECh. 11 - Evaluate Lz for 3px, Compare it to the answer in...Ch. 11 - Calculate V for 1s of the H atom and compare it to...Ch. 11 - Prob. 11.89ECh. 11 - Prob. 11.90ECh. 11 - Prob. 11.91ECh. 11 - Prob. 11.92ECh. 11 - Graph the first five wavefunctions for the...Ch. 11 - Prob. 11.94ECh. 11 - Set up and evaluate numerically the integral that...Ch. 11 - Prob. 11.96E
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- Locate the nodes of a harmonic oscillator wavefunction with v = 2. (Express your answers in terms of the coordinate y.)arrow_forwardA normalized wavefunction for a particle confined between 0 and L in the x direction is ψ = (2/L)1/2 sin(πx/L). Suppose that L = 10.0 nm. Calculate the probability that the particle is (a) between x = 4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm, (c) between x = 9.90 nm and 10.00 nm, (d) between x = 5.00 nm and 10.00 nm.arrow_forwardImagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In eachcase, give your reasons for accepting or rejecting each function. (i) Ψ(x)=x2; (ii) Ψ(x)=1/x; (iii) Ψ(x)=e-x^2.arrow_forward
- What is the kinetic energy of a particle described by the wavefunction cos(kx)? ħ² d² 2m dx² KEarrow_forwardThe wavefunction for the motion of a particle on a ring is of the form ψ = Neimlϕ. Evaluate the normalization constant, N.arrow_forwardConsider the three spherical harmonics (a) Y0,0, (b) Y2,–1, and (c) Y3,+3. (a) For each spherical harmonic, substitute the explicit form of the function taken from Table 7F.1 into the left-hand side of eqn 7F.8 (the Schrödinger equation for a particle on a sphere) and confirm that the function is a solution of the equation; give the corresponding eigenvalue (the energy) and show that it agrees with eqn 7F.10. (b) Likewise, show that each spherical harmonic is an eigenfunction of lˆz = (ℏ/i)(d/dϕ) and give the eigenvalue in each case.arrow_forward
- The rotation of a molecule can be represented by the motion of a particle moving over the surface of a sphere with angular momentum quantum number l = 2. Calculate the magnitude of its angular momentum and the possible components of the angular momentum along the z-axis. Express your results as multiples of ℏ.arrow_forwardIn the octatetraene (C8H10) molecule, what should be the length of the electromagnetic wave required to excite a π electron from the top filled level to the bottom empty level? The length of the molecule can be taken as 9.5 Å. Compare your value with the experimental value λ4.5 = 3020 Å and discuss whether the particle-in-a-box problem is a good approximation.arrow_forward. One way in which a water molecule can vibrate has a measured angular frequency of @= 3.0 × 10¹4 s 1. What is the value of ħo in eV for this system if we model it as a simple harmonic oscillator? what are its lowest three possible energy values?arrow_forward
- For a He atom in the excited state with atomic electronic configuration 1s1 2s1: write out the symmetric and antisymmetric spatial wavefunctions. These two are linear combination of ψ1s and ψ2s each with an electron in them. combine each spatial wavefunction with the appropriate spin wavefunctions for two electrons (there are four of them) such that the total wavefunction is antisymmetric.arrow_forwardIn the octatetraene (C8H10) molecule, one π electron is from the top filled level. What should be the length of the electromagnetic wave required to excite it to the lowest empty level? The length of the molecule can be taken as 9.5 Å. The experimental value you found is λ4.5 =3020 Å Discuss whether the particle-in-the-box problem is a good approximation by comparing it witharrow_forward6. Calculate the expectation value of the radius (r) at which you would find the electron if the H atom wavefunction| is P1,0,0(r).arrow_forward
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