Answered: Table 7F.1 The spherical harmonics Ym…

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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Consider the three spherical harmonics (a) Y_0,0, (b) Y_2,–1, and (c) Y_3,+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 lˆ_z = (ℏ/i)(d/dϕ) and give the eigenvalue in each case.

Table 7F.1 The spherical harmonics
Ym (0,0)
1/2
4
(유)
1/2
3
1
cose
1/2
3
±1
sinde*i
1/2
(3 cos'0-1)
16T
1/2
15
±1
cose sinde*i
tip
1/2
15
+2
32n
1/2
7
(5 cos'e-3cose)
3
16T
1/2
21
±1
(5 cos'e-1)sin Oe**
64T
1/2
105
±2
sin'e coseei0
32n
1/2
35
13
sin'de*
64T
2.

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