(a)
Interpretation:
The energy of the given 3-D rotational wavefunction is to be calculated.
Concept introduction:
The energy for the 3-D rotational motion is given by,
The energy of the particle depends on the moment of inertia, quantum number and Planck’s constant. The total energy is quantized.
(b)
Interpretation:
Total
Concept introduction:
The total angular momentum for the 3-Dimensional system is given by,
The total angular momentum does not depend on the mass of the particle, radius of the rotation and also the magnetic quantum number.
(c)
Interpretation:
The z-component of the total angular momentum of the given 3-D rotational wavefunction is to be calculated.
Concept introduction:
The z-component of the three dimensional angular momentum that has components in x, y and z direction is quantized. Its value depends on the magnetic quantum number and it is given by,
Want to see the full answer?
Check out a sample textbook solutionChapter 11 Solutions
Student Solutions Manual for Ball's Physical Chemistry, 2nd
- The moment of inertia of an SF6 molecule is 3.07 × 10−45 kg m2. What is the minimum energy needed to start it rotating?arrow_forward8C.4 (a) the moment of inertia of a CH4 molecule is 5.27 x 10^-47 kg m^2. What is the minimum energy needed to start it rotating? 8C.5 (a) use the data in 8C.4 (a) to calculate the energy needed excite a CH4 molecule from a state with l=1 to a state with l=2arrow_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
- (a) If  = 3x? and B = , then show that  and ß donot commute with respect to the function f(x) = sin x. Show, if the wave function, w) = A cos(kx) + iA sin(kx) is an Eigen-function of the linear momentum operator, P and if so, what is the Eigen value. (Note: A and k are constants). (b)arrow_forwardIf two wavefunctions, Wa and Wb, are orthonormal and degenerate, then what is true about the linear combinations 1 1 w. +v.) a a and (a) y+ and y- are orthonormal. (b) y+ and y- are no longer eigenfunctions of the Schrödinger equation. (c) V+ and y- have the same energy. (d) V+ and Y- have the same probability density distribution.arrow_forwardWhich of the following functions can be normalized (in all cases the range for x is from x = −∞ to ∞, and a is a positive constant): (i) e-ax^2; (ii) e–ax. Which of these functions are acceptable as wavefunctions?arrow_forward
- Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinxarrow_forwardThe ground-state wavefunction for a particle confined to a one dimensional box of length L is Ψ =(2/L)½ sin (πx/L) Suppose the box 10.0 nm long. Calculate the probability that the particle is: (a) between x = 4.95 nm and 5.05 nm (b) between 1.95 nm and 2.05 nm, (c) between x = 9.90 and 10.00 nm, (d) in the right half of the box and (e) in the central third of the box.arrow_forward(a) For a particle in the stationary state n of a one dimensional box of length a, find the probability that the particle is in the region 0xa/4.(b) Calculate this probability for n=1,2, and 3.arrow_forward
- Calculate the momentum of an X-ray photon with a wavelength of 0.17nm. How does this value compare with the momentum of a free electron that has been accelerated through a potential difference of 5000 volts? (Hint: electron mass, m, = 9.10938 x 10" kg; electron charge e = 1.602 x 10"C; speed of light e = 3.0 x 10° m.s'; 1.00 J= 1.00 VC; h = 6.626 x 10"J.s. The various energy units are: 1 J=1 kg.m's", 1.00 cV =1VC, leV = 1.602 x 10"J, 1J=6.242 x 10" eV, etc.). %3D %3Darrow_forwardWhich of the following functions can be normalized (in all cases the range for x is from x = −∞ to ∞, and a is a positive constant): (i) sin(ax);(ii) cos(ax) e-x^2? Which of these functions are acceptable as wavefunctions?arrow_forwardFor the system described in Exercise E7C.8(a), evaluate the expectation value of the angular momentum represented by the operator(ħ/i)d/dϕ for the case ml = +1, and then for the general case of integer ml.arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,Principles of Modern ChemistryChemistryISBN:9781305079113Author:David W. Oxtoby, H. Pat Gillis, Laurie J. ButlerPublisher:Cengage LearningChemistry: Principles and PracticeChemistryISBN:9780534420123Author:Daniel L. Reger, Scott R. Goode, David W. Ball, Edward MercerPublisher:Cengage Learning
- Introductory Chemistry: A FoundationChemistryISBN:9781337399425Author:Steven S. Zumdahl, Donald J. DeCostePublisher:Cengage Learning