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)!

Principles of Instrumental Analysis
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Chapter12: Atomic X-ray Spectrometry
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Problem 12.4QAP: The L lines for Ca, Zn, Zr, and Sn have wavelengths of 36.3, 11.9,6.07, and 3.60 Å, respectively....
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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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