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Questions and Answers
What does the polar diagram in Figure 7-7 represent?
What does the polar diagram in Figure 7-7 represent?
For values of r which are small compared to $a_0/Z$, what can be said about the exponential term in the eigenfunctions?
For values of r which are small compared to $a_0/Z$, what can be said about the exponential term in the eigenfunctions?
How does the probability density depend on the coordinate $\phi$ for small values of r?
How does the probability density depend on the coordinate $\phi$ for small values of r?
What is the expression for the three-dimensional behavior of the eigenfunctions $\psi_{n,l,m}$?
What is the expression for the three-dimensional behavior of the eigenfunctions $\psi_{n,l,m}$?
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What is the relationship between the polar diagram in Figure 7-7 and the three-dimensional surface of the eigenfunctions?
What is the relationship between the polar diagram in Figure 7-7 and the three-dimensional surface of the eigenfunctions?
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For small values of r, what can be said about the behavior of the electron near the nucleus?
For small values of r, what can be said about the behavior of the electron near the nucleus?
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What is the form of the factor $\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$ presented in the text?
What is the form of the factor $\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$ presented in the text?
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How does the three-dimensional surface obtained by rotating the polar diagram represent the eigenfunctions?
How does the three-dimensional surface obtained by rotating the polar diagram represent the eigenfunctions?
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What is the relationship between the radial factor $R_{n,l}(r)$ and the directional factor $\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$ in the expression for the eigenfunctions?
What is the relationship between the radial factor $R_{n,l}(r)$ and the directional factor $\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$ in the expression for the eigenfunctions?
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Study Notes
One-Electron Atom Probability Density
- Directional dependence of one-electron atom probability density is illustrated through polar diagrams.
- Radial dependence of eigenfunctions behaves as ( \psi \propto r^l ) when ( r \rightarrow 0 ).
- For small ( r ), probability densities are proportional to ( r^{2l} ).
Probability Density and Angular Momentum
- Probability of finding an electron near the nucleus is significant for ( l = 0 ), but rapidly decreases for higher ( l ).
- Angular momentum ( l ) influences the probability distribution of electrons, affecting atomic structure.
Potential Energy Implications
- High potential energy for electrons near the nucleus is critical, especially in multi-electron atoms.
- Eigenfunctions' behavior for small ( r ) is crucial in understanding multi-electron atom structures.
Normalization of Eigenfunctions
- Eigenfunctions are adjusted to ensure normalization; total probability across all space equals one.
- Verification of quantum states assures confidence in the accuracy of eigenfunctions listed in reference tables.
Verification of Eigenfunction
- The eigenfunction ( \psi_{211} ) and its normalizing eigenvalue ( E_2 ) confirm satisfaction of the time-independent Schrödinger equation for a one-electron atom with atomic number ( Z = 1 ).
- The linear nature of the differential equation allows for simplification by ignoring constant multipliers during verification.
Introduction of Quantum Numbers
- Notation ( \psi_i = f(r, \phi) \sin \theta ) simplifies the representation of wave functions in three-dimensional space.
- The specified dependence on all three coordinates serves as a basis for further exploration of atomic behavior.
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Description
This quiz explores the polar diagram showing the factor determining the directional dependence of one-electron atom probability density. It discusses the behavior of the eigenfunctions and probability densities with varying radial distances.
