## Questions and Answers

What does the polar diagram in Figure 7-7 represent?

The directional dependence of 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?

The exponential term is slowly varying

How does the probability density depend on the coordinate $\phi$ for small values of r?

The probability density does not depend on $\phi$

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?

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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?

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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?

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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.