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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 ao 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 (1₂²) = (20 + 4)(2l + 3)(l + 1)² aº

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 ao 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 (1₂²) = (20 + 4)(2l + 3)(l + 1)² aº

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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
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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