Problem 5. Earlier in class, we learned that if Y1 and ¥2 are solutions of the Schrödinger equation that have the same energy En, then a linear combination cYı +c2¥2 is also a solution. Let Y1 = ¥210 and Y2 = ¥211 from Table 6.5 of the McQuarrie and Simon text- book. Evaluate the energy of the hydrogen atom for Y=c¡¥1+c2¥2 assuming c +cz = 1. What does this tell you about the uniqueness of the three p orbitals, px, py, and p2?
Problem 5. Earlier in class, we learned that if Y1 and ¥2 are solutions of the Schrödinger equation that have the same energy En, then a linear combination cYı +c2¥2 is also a solution. Let Y1 = ¥210 and Y2 = ¥211 from Table 6.5 of the McQuarrie and Simon text- book. Evaluate the energy of the hydrogen atom for Y=c¡¥1+c2¥2 assuming c +cz = 1. What does this tell you about the uniqueness of the three p orbitals, px, py, and p2?
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![**Problem 5:**
Earlier in class, we learned that if Ψ₁ and Ψ₂ are solutions of the Schrödinger equation that have the same energy Eₙ, then a linear combination c₁Ψ₁ + c₂Ψ₂ is also a solution. Let Ψ₁ = Ψ₂₁₀ and Ψ₂ = Ψ₂₁₁ from **Table 6.5** of the McQuarrie and Simon textbook. Evaluate the energy of the hydrogen atom for Ψ = c₁Ψ₁ + c₂Ψ₂ assuming c₁² + c₂² = 1. What does this tell you about the uniqueness of the three p orbitals, pₓ, pᵧ, and p꜀?
---
**Table 6.5:**
The table below shows the complete hydrogenlike atomic wave functions for n = 1, 2, and 3. The quantity Z is the atomic number of the nucleus, and σ = Zr/a₀, where a₀ is the Bohr radius.
---
- **n = 1, l = 0, m = 0:**
\[
\Psi_{100} = \frac{1}{\sqrt{\pi}} \left(\frac{Z}{a_0}\right)^{3/2} e^{-\sigma}
\]
- **n = 2, l = 0, m = 0:**
\[
\Psi_{200} = \frac{1}{\sqrt{32\pi}} \left(\frac{Z}{a_0}\right)^{3/2} (2 - \sigma) e^{-\sigma/2}
\]
- **n = 2, l = 1, m = 0:**
\[
\Psi_{210} = \frac{1}{\sqrt{32\pi}} \left(\frac{Z}{a_0}\right)^{3/2} \sigma e^{-\sigma/2} \cos \theta
\]
- **n = 2, l = 1, m = ±1:**
\[
\Psi_{21\pm1} = \frac{1}{\sqrt{64\pi}} \left(\frac{Z}{a_0}\right)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ac106f3-a177-400f-8063-f0df96f817c6%2F2f3d6f58-279c-488c-b06d-f87f1cee6ad4%2Fet093oe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5:**
Earlier in class, we learned that if Ψ₁ and Ψ₂ are solutions of the Schrödinger equation that have the same energy Eₙ, then a linear combination c₁Ψ₁ + c₂Ψ₂ is also a solution. Let Ψ₁ = Ψ₂₁₀ and Ψ₂ = Ψ₂₁₁ from **Table 6.5** of the McQuarrie and Simon textbook. Evaluate the energy of the hydrogen atom for Ψ = c₁Ψ₁ + c₂Ψ₂ assuming c₁² + c₂² = 1. What does this tell you about the uniqueness of the three p orbitals, pₓ, pᵧ, and p꜀?
---
**Table 6.5:**
The table below shows the complete hydrogenlike atomic wave functions for n = 1, 2, and 3. The quantity Z is the atomic number of the nucleus, and σ = Zr/a₀, where a₀ is the Bohr radius.
---
- **n = 1, l = 0, m = 0:**
\[
\Psi_{100} = \frac{1}{\sqrt{\pi}} \left(\frac{Z}{a_0}\right)^{3/2} e^{-\sigma}
\]
- **n = 2, l = 0, m = 0:**
\[
\Psi_{200} = \frac{1}{\sqrt{32\pi}} \left(\frac{Z}{a_0}\right)^{3/2} (2 - \sigma) e^{-\sigma/2}
\]
- **n = 2, l = 1, m = 0:**
\[
\Psi_{210} = \frac{1}{\sqrt{32\pi}} \left(\frac{Z}{a_0}\right)^{3/2} \sigma e^{-\sigma/2} \cos \theta
\]
- **n = 2, l = 1, m = ±1:**
\[
\Psi_{21\pm1} = \frac{1}{\sqrt{64\pi}} \left(\frac{Z}{a_0}\right)
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