The functions of interest in hydrogen atom theory are: 1+1 -2 21+1, f₁(x) = x¹+¹ e "L (+) n-1-1 In Eqn 2.1, n and I are integers and 0 ≤ 1 ≤ n - 1. For l = 1, show that: ²e + Note: L n-1-1 1 2- f,(x) = gx e 21+1 k is the Associated Laguerre Polynomial, L
The functions of interest in hydrogen atom theory are: 1+1 -2 21+1, f₁(x) = x¹+¹ e "L (+) n-1-1 In Eqn 2.1, n and I are integers and 0 ≤ 1 ≤ n - 1. For l = 1, show that: ²e + Note: L n-1-1 1 2- f,(x) = gx e 21+1 k is the Associated Laguerre Polynomial, L
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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![The functions of interest in hydrogen atom theory are:
\[
f_n(x) = x^{l+1} e^{-\frac{x}{2n}} L_{n-l-1}^{2l+1} \left( \frac{x}{n} \right) \quad (2.1)
\]
In Eqn 2.1, \(n\) and \(l\) are integers and \(0 \leq l \leq n - 1\). For \(l = 1\), show that:
\[
f_2(x) = \frac{1}{8} x^2 e^{-\frac{x}{4}} \quad (2.2)
\]
*Note:* \(L_{n-l-1}^{2l+1}\) is the Associated Laguerre Polynomial, \(L_p^k\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefc0e927-4815-4e2f-82d9-9b6b5f419d74%2F4438240c-60ca-4a9e-9d72-02423df46ded%2Fdvmtt29_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The functions of interest in hydrogen atom theory are:
\[
f_n(x) = x^{l+1} e^{-\frac{x}{2n}} L_{n-l-1}^{2l+1} \left( \frac{x}{n} \right) \quad (2.1)
\]
In Eqn 2.1, \(n\) and \(l\) are integers and \(0 \leq l \leq n - 1\). For \(l = 1\), show that:
\[
f_2(x) = \frac{1}{8} x^2 e^{-\frac{x}{4}} \quad (2.2)
\]
*Note:* \(L_{n-l-1}^{2l+1}\) is the Associated Laguerre Polynomial, \(L_p^k\).
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