The radial eigenfunction for the quantum state of lowest energy for an electron with angular momentum L=√√e(+1) in a Coulomb potential is 10, (P) = N₁.C+1 (C+Da where N is a constant. a) Show that the eigenfunction satisfies the normalization condition ¹₁. (r^)³ dr = 1 0 20+3 2 1 N² (+1)ao. (2l+2)!
The radial eigenfunction for the quantum state of lowest energy for an electron with angular momentum L=√√e(+1) in a Coulomb potential is 10, (P) = N₁.C+1 (C+Da where N is a constant. a) Show that the eigenfunction satisfies the normalization condition ¹₁. (r^)³ dr = 1 0 20+3 2 1 N² (+1)ao. (2l+2)!
Chemistry
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ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
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Chapter1: Chemical Foundations
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![Question 4
The radial eigenfunction for the quantum state of lowest energy for an electron with angular
momentum L=√√e(+1) in a Coulomb potential is
U₁₂ (1) = Nr. ²+¹ (C+Dav
where N is a constant.
a) Show that the eigenfunction satisfies the normalization condition [µ¼0, (r)]³ dr =1 _if
20+3
2
1
N²
-
(l+1)ao.
(2l+2)!
b) Show that the most probable radius of the state is
1'mp = (l+ 1)² ap
c) Show that the average radius of the state is
(r =
_ (2€ + 3) (€ + 1),
2
d) Show that the average of the square radius is
(x-²) = (2l + 4)(2l +3)(l + 1)² ao
4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d23c93b-870e-4ad1-8eae-60e3d2381ea8%2F58117ed1-cf8f-4128-b1e9-96c5792d4d48%2F10o9g4_processed.png&w=3840&q=75)
Transcribed Image Text:Question 4
The radial eigenfunction for the quantum state of lowest energy for an electron with angular
momentum L=√√e(+1) in a Coulomb potential is
U₁₂ (1) = Nr. ²+¹ (C+Dav
where N is a constant.
a) Show that the eigenfunction satisfies the normalization condition [µ¼0, (r)]³ dr =1 _if
20+3
2
1
N²
-
(l+1)ao.
(2l+2)!
b) Show that the most probable radius of the state is
1'mp = (l+ 1)² ap
c) Show that the average radius of the state is
(r =
_ (2€ + 3) (€ + 1),
2
d) Show that the average of the square radius is
(x-²) = (2l + 4)(2l +3)(l + 1)² ao
4
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