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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
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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