6-30. Show that the total probability density of the 2p orbitals is spherically symmetric by evaluating EV- (Use the wave functions in Table 6.6.)
6-30. Show that the total probability density of the 2p orbitals is spherically symmetric by evaluating EV- (Use the wave functions in Table 6.6.)
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![## Transcription of Educational Content on Spherically Symmetric Probability Density and Associated Legendre Functions
### 6–30. Spherically Symmetric Probability Density of 2p Orbitals
Show that the total probability density of the 2p orbitals is spherically symmetric by evaluating:
\[
\sum_{m=-1}^{1} \psi_{21m}^{2}.
\]
(Use the wave functions in Table 6.6.)
\[
\sum_{m=-1}^{1} \psi_{21m}^{2} = \frac{1}{32\pi} \left( \frac{Z}{a_{0}} \right)^{3} e^{-r} \left( \cos^{2}\theta + \sin^{2}\theta \cos^{2}\phi + \sin^{2}\theta \sin^{2}\phi \right)
\]
\[
= \frac{Z^3}{32\pi a_0^3} e^{-r} \left[ \cos^2\theta + \sin^2 \theta (\cos^2 \phi + \sin^2 \phi) \right]
\]
\[
= \frac{Z^3}{32\pi a_0^3} e^{-r} (\cos^2 \theta + \sin^2 \theta)
\]
\[
= \frac{Z^3}{32\pi a_0^3} e^{-r}
\]
The sum depends only on the variable \( r \) (through \( \sigma \)), so the total probability density of the 2p orbitals is spherically symmetric.
### 6–6. Generation of Associated Legendre Functions
Use Equation 6.26 to generate the associated Legendre functions in Table 6.2.
\[
P_{l}^{m}(x) = (1-x^{2})^{m/2} \frac{d^{m}}{dx^{m}} P_{l}(x) \quad (6.26)
\]
#### List of Legendre Functions
- \( P_{0}^{0}(x) = (1-x^{2})^{0/2} \frac{d^{0}}{dx^{0}} P_{0}(x) = P_{0}(x) = 1 \)
- \( P_{1}^{0}(x) = (1-x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2f6cfa1-15bf-4f18-84f8-c308fdc99132%2F0d7dfc01-4d5b-43d0-a39c-3ee49649e6ed%2Fj5nvdn_processed.png&w=3840&q=75)
Transcribed Image Text:## Transcription of Educational Content on Spherically Symmetric Probability Density and Associated Legendre Functions
### 6–30. Spherically Symmetric Probability Density of 2p Orbitals
Show that the total probability density of the 2p orbitals is spherically symmetric by evaluating:
\[
\sum_{m=-1}^{1} \psi_{21m}^{2}.
\]
(Use the wave functions in Table 6.6.)
\[
\sum_{m=-1}^{1} \psi_{21m}^{2} = \frac{1}{32\pi} \left( \frac{Z}{a_{0}} \right)^{3} e^{-r} \left( \cos^{2}\theta + \sin^{2}\theta \cos^{2}\phi + \sin^{2}\theta \sin^{2}\phi \right)
\]
\[
= \frac{Z^3}{32\pi a_0^3} e^{-r} \left[ \cos^2\theta + \sin^2 \theta (\cos^2 \phi + \sin^2 \phi) \right]
\]
\[
= \frac{Z^3}{32\pi a_0^3} e^{-r} (\cos^2 \theta + \sin^2 \theta)
\]
\[
= \frac{Z^3}{32\pi a_0^3} e^{-r}
\]
The sum depends only on the variable \( r \) (through \( \sigma \)), so the total probability density of the 2p orbitals is spherically symmetric.
### 6–6. Generation of Associated Legendre Functions
Use Equation 6.26 to generate the associated Legendre functions in Table 6.2.
\[
P_{l}^{m}(x) = (1-x^{2})^{m/2} \frac{d^{m}}{dx^{m}} P_{l}(x) \quad (6.26)
\]
#### List of Legendre Functions
- \( P_{0}^{0}(x) = (1-x^{2})^{0/2} \frac{d^{0}}{dx^{0}} P_{0}(x) = P_{0}(x) = 1 \)
- \( P_{1}^{0}(x) = (1-x
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