Z 4. The spherical harmonics are the eigenfunctions of 1² and Î₂ for the rigid rotor and the hydrogen atom (and other spherically symmetric problems). In this problem, we will examine the nature of the angular nodes for these systems. Since the spherical harmonics include a factor of eim, which never has magnitude zero, for this exercise we will construct some linear combinations of the spherical harmonics so we are working with real-valued functions. Two of the real-valued spherical harmonics are: 1/2 = · (Y₁¯¯¹ + Y₁¹): 1 3 2 2π sin cos p 1 2i (Y3² - Y3²) = 1 4 105 2π -sin² 0 cos sin 20 (a) Determine the angles at which nodal surfaces will occur for each of these functions, and describe the nature of the nodal surfaces that they represent. In other words, identify the locations of nodal planes and other surfaces in the Cartesian axis system. (b) What atomic orbitals (e.g. 1s, 2p, etc.) are represented by these functions and what is the total number of distinct angular nodal surfaces?
Z 4. The spherical harmonics are the eigenfunctions of 1² and Î₂ for the rigid rotor and the hydrogen atom (and other spherically symmetric problems). In this problem, we will examine the nature of the angular nodes for these systems. Since the spherical harmonics include a factor of eim, which never has magnitude zero, for this exercise we will construct some linear combinations of the spherical harmonics so we are working with real-valued functions. Two of the real-valued spherical harmonics are: 1/2 = · (Y₁¯¯¹ + Y₁¹): 1 3 2 2π sin cos p 1 2i (Y3² - Y3²) = 1 4 105 2π -sin² 0 cos sin 20 (a) Determine the angles at which nodal surfaces will occur for each of these functions, and describe the nature of the nodal surfaces that they represent. In other words, identify the locations of nodal planes and other surfaces in the Cartesian axis system. (b) What atomic orbitals (e.g. 1s, 2p, etc.) are represented by these functions and what is the total number of distinct angular nodal surfaces?
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
Related questions
Question
The spherical harmonics are the eigenfunctions of ?̂2 and ?̂ ? for the rigid rotor and the
hydrogen atom (and other spherically symmetric problems). In this problem, we will
examine the nature of the angular nodes for these systems.
Since the spherical harmonics include a factor of eim
, which never has magnitude zero, for
this exercise we will construct some linear combinations of the spherical harmonics so we
are working with real-valued functions. Two of the real-valued spherical harmonics are:
1
2 (?1
−1 + ?1
1) = 1
2 √ 3
2? sin ? cos ? 1
2? (?3
2 − ?3
−2) = 1
4 √105
2? sin2 ? cos ? sin 2?
(a) Determine the angles at which nodal surfaces will occur for each of these functions, and
describe the nature of the nodal surfaces that they represent. In other words, identify
the locations of nodal planes and other surfaces in the Cartesian axis system.
(b) What atomic orbitals (e.g. 1s, 2p, etc.) are represented by these functions and what is
the total number of distinct angular nodal surfaces?
hydrogen atom (and other spherically symmetric problems). In this problem, we will
examine the nature of the angular nodes for these systems.
Since the spherical harmonics include a factor of eim
, which never has magnitude zero, for
this exercise we will construct some linear combinations of the spherical harmonics so we
are working with real-valued functions. Two of the real-valued spherical harmonics are:
1
2 (?1
−1 + ?1
1) = 1
2 √ 3
2? sin ? cos ? 1
2? (?3
2 − ?3
−2) = 1
4 √105
2? sin2 ? cos ? sin 2?
(a) Determine the angles at which nodal surfaces will occur for each of these functions, and
describe the nature of the nodal surfaces that they represent. In other words, identify
the locations of nodal planes and other surfaces in the Cartesian axis system.
(b) What atomic orbitals (e.g. 1s, 2p, etc.) are represented by these functions and what is
the total number of distinct angular nodal surfaces?
Transcribed Image Text:Z
4. The spherical harmonics are the eigenfunctions of 1² and Î₂ for the rigid rotor and the
hydrogen atom (and other spherically symmetric problems). In this problem, we will
examine the nature of the angular nodes for these systems.
Since the spherical harmonics include a factor of eim, which never has magnitude zero, for
this exercise we will construct some linear combinations of the spherical harmonics so we
are working with real-valued functions. Two of the real-valued spherical harmonics are:
1/2 =
· (Y₁¯¯¹ + Y₁¹):
1 3
2 2π
sin cos p
1
2i
(Y3² - Y3²)
=
1
4
105
2π
-sin² 0 cos sin 20
(a) Determine the angles at which nodal surfaces will occur for each of these functions, and
describe the nature of the nodal surfaces that they represent. In other words, identify
the locations of nodal planes and other surfaces in the Cartesian axis system.
(b) What atomic orbitals (e.g. 1s, 2p, etc.) are represented by these functions and what is
the total number of distinct angular nodal surfaces?
AI-Generated Solution
AI-generated content may present inaccurate or offensive content that does not represent bartleby’s views.
Unlock instant AI solutions
Tap the button
to generate a solution
Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON