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3.5: Some Useful Expressions

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    Listed below are a number of equations which give the dependence of Image107.gif, Image106.gif and Image105.gif on the quantum numbers n, l and m. They refer not only to the hydrogen atom but also to any one-electron ion in general with a nuclear charge of Z. Thus He+ is a one-electron ion with Z = 2, Li+2 another example with Z = 3.

    The average distance between the electron and the nucleus expressed in atomic units of length is:


    Note that Image232.gif is proportional to n2 for l = 0 orbitals, and deviates only slightly from this for l ¹ 0. The value of Image232_13n7.gifdecreases as Z increases because the nuclear attractive force is greater. Thus Image234.gif for He+ would be only one half as large as Image234_h5q0.gif for H.

    Image110.gif Image111.gif
    Image112.gif Image113.gif

    This page titled 3.5: Some Useful Expressions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard F. W. Bader via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.