Polar Diagram for the Directional Dependence of One-Electron Atom Probability Density

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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 $\psi_{n,l,m}$?

$R_{n,l}(r)\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$

What is the relationship between the polar diagram in Figure 7-7 and the three-dimensional surface of the eigenfunctions?

The polar diagram and the three-dimensional surface both represent the directional dependence

For small values of r, what can be said about the behavior of the electron near the nucleus?

The electron is more likely to be found near the nucleus

What is the form of the factor $\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$ presented in the text?

As a polar diagram

How does the three-dimensional surface obtained by rotating the polar diagram represent the eigenfunctions?

The distance from the origin to the surface represents the directional dependence $\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$

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?

They are independent factors that multiply to give the complete eigenfunction

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.

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