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8.3: Appendix 2 - The Hydrogen Atom

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    The box model of the hydrogen atom

    Hydrogen is the simplest element. There are just two components: an electron, and a positively charged nucleus comprised of a single proton.

    Perhaps the simplest model of the hydrogen atom employs our now familiar square potential wells. This approximation cannot be taken very far, but it does illustrate the origin of the shapes of some of the orbitals. If we compare the smooth, spherically symmetric Coulomb potential to our box model of an atom, it is clear that the box approximation will give up a lot of accuracy in the calculation of atomic orbitals and energy. The box, however, does yield insights into the shape of the various possible atomic orbitals.

    The box is a separable potential. Thus, the atomic orbitals can be described by a product:

    If the wall have infinite potential, the possible energies of the electron are given by

    \[ E_{n_{x},n_{y},n_{z}} = \frac{\hbar^{2} \pi^{2}}{2m} \left( \frac{n_{x}^{2}}{L_{x}^{2}} + \frac{n_{y}^{2}}{L_{y}^{2}} + \frac{n_{z}^{2}}{L_{z}^{2}} \right) \label{8.3.1} \]

    where the dimensions of the box are \(L_{x} \times L_{y} \times L_{z}\) and \(n_{x}\), \(n_{y}\) and \(n_{z}\) are integers that correspond to the state of the electron within the box.

    For example, consider the ground state of a box with infinite potential walls. \((n_{x},n_{y},n_{z})=(1,1,1)\)

    Screenshot 2021-06-06 at 15.55.13.png
    Figure \(\PageIndex{1}\): The ground state of a 3 dimensional box. \((n_{x},n_{y},n_{z})=(1,1,1)\)

    Now, consider the orbital's shape if either \(\psi_{x}(x)\) or \(\psi_{y}(y)\) or \(\psi_{z}(z)\) is in the first excited state: \((n_{x},n_{y},n_{z})=(2,1,1)\), \((n_{x},n_{y},n_{z})=(1,2,1)\) or \((n_{x},n_{y},n_{z})=(1,1,2)\)

    The 1s orbital is similar to \((n_{x},n_{y},n_{z})=(1,1,1)\). The p orbitals are similar to the first excited state of the box, i.e. \((n_{x},n_{y},n_{z})=(2,1,1)\) is similar to a \(p_{x}\) orbital, \((n_{x},n_{y},n_{z})=(1,2,1)\) is similar to a \(p_{y}\) orbital and \((n_{x},n_{y},n_{z})=(1,1,2)\) is similar to a \(p_{z}\) orbital.

    Screenshot 2021-06-06 at 15.59.39.png
    Figure \(\PageIndex{2}\): The first excited states of a 3 dimensional box. (a) \((n_{x},n_{y},n_{z})=(2,1,1)\), (b) \((n_{x},n_{y},n_{z})=(1,2,1)\), (c) \((n_{x},n_{y},n_{z})=(1,1,2)\).

    The approximation soon breaks down, however. The 2s orbital, which has the same energy as the 2p orbitals is most similar to the box orbital \((n_{x},n_{y},n_{z})=(3,3,3)\), which has significantly higher energy. Nevertheless, the box does illustrate the alignment of the three p orbitals with the x, y and z axes.

    8.3: Appendix 2 - The Hydrogen Atom is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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