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

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@SofterPun

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Questions and Answers

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

  • The probability density for small values of r
  • The directional dependence of the eigenfunctions (correct)
  • The behavior of the electron near the nucleus
  • The radial 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 rapidly varying
  • The exponential term is slowly varying (correct)
  • The exponential term is zero
  • The exponential term is constant

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

  • The probability density is zero for small values of r
  • The probability density is constant with respect to $\phi$
  • The probability density depends on $\phi$
  • The probability density does not depend on $\phi$ (correct)

    • What is the expression for the three-dimensional behavior of the eigenfunctions $\psi_{n,l,m}$?

    <p>$R_{n,l}(r)\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$</p> Signup and view all the answers

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

    <p>The polar diagram and the three-dimensional surface both represent the directional dependence</p> Signup and view all the answers

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

    <p>The electron is more likely to be found near the nucleus</p> Signup and view all the answers

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

    <p>As a polar diagram</p> Signup and view all the answers

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

    <p>The distance from the origin to the surface represents the directional dependence $\Theta_{l,m}(\theta)\Phi_{l,m}(\phi)$</p> Signup and view all the answers

    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?

    <p>They are independent factors that multiply to give the complete eigenfunction</p> Signup and view all the answers

    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.

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