The spherical harmonics are the eigenfunctions of ?̂2 and ?̂ ? for the rigid rotor and thehydrogen atom (and other spherically symmetric problems). In this problem, we willexamine the nature of the angular nodes for these systems.Since the spherical harmonics include a factor of eim, which never has magnitude zero, forthis exercise we will construct some linear combinations of the spherical harmonics so weare working with real-valued functions. Two of the real-valued spherical harmonics are:12 (?1−1 + ?11) = 12 √ 32? sin ? cos ? 12? (?32 − ?3−2) = 14 √1052? sin2 ? cos ? sin 2?(a) Determine the angles at which nodal surfaces will occur for each of these functions, anddescribe the nature of the nodal surfaces that they represent. In other words, identifythe 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 isthe total number of distinct angular nodal surfaces? Z4. The spherical harmonics are the eigenfunctions of 1² and Î₂ for the rigid rotor and thehydrogen atom (and other spherically symmetric problems). In this problem, we willexamine the nature of the angular nodes for these systems.Since the spherical harmonics include a factor of eim, which never has magnitude zero, forthis exercise we will construct some linear combinations of the spherical harmonics so weare working with real-valued functions. Two of the real-valued spherical harmonics are:1/2 =· (Y₁¯¯¹ + Y₁¹):1 32 2πsin cos p12i(Y3² - Y3²)=141052π-sin² 0 cos sin 20(a) Determine the angles at which nodal surfaces will occur for each of these functions, anddescribe the nature of the nodal surfaces that they represent. In other words, identifythe 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 isthe total number of distinct angular nodal surfaces?

The objective of this question is to determine the angles at which nodal surfaces will occur for the…

Answered: Z 4. The spherical harmonics are the…

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)...

See similar textbooks

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?