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Homework 18 (Due 5/23/2016)

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    Write the Schrödinger Equation for the following system and identify if they can or cannot be solved analytically and why. The potential energy operator must be expands, but it is ok to leave the kinetic energy operator as a generic \(\hat{T}\) term (i.e., you do not need to explicitly expand it into either Cartesian or Spherical coordinates).

    1. The \(H\) atom
    2. The \(He^+\) ion
    3. The \(He\) atom
    4. The \(H_2^+\) molecule
    5. The \(H_2\) molecule


    What is the difference between covalent bond and ionic bond wave functions in Valence Bond theory?


    Which of the following molecules have the shortest bond: F2, N2, O2? Why?


    Calculate the dipole moment (\(\mu\)) of a two ions of +2e and -2e that are separated by 100 pm. Express dipole moment in both units of \(C\,m\) and \(D\).


    Assuming \(\psi\) is the generic wavefunction (solution) to a general Schrödinger equation

    \[(\hat{T} + \hat{V} ) \psi = E \psi\]

    Show that any new function \(\phi\) that is related to \(\psi\) via

    \[ \phi = N \psi\]

    is also a valid wavefunction to the Schrödinger equation. \(N\) is a scalar (i.e. number). If so many wavefunctions (\(\phi\)) can be generated that are valid solutions to the Schrödinger equation? How do you determine a specific value for \(N\) and why?


    Given that a new function \(\phi\) can be constructed from both \(\psi_1\) and \(\psi_2\), which are valid solutions to the Schrödinger equation,

    \[ \phi = N_1 \psi_1 + N_2 \psi_2\]

    Demonstate that \(\phi\) is also a solution to the Schrödinger equation. How do you determine the \(N_1\) and \(N_2\) numbers? Can \(\phi\) be expanded to three terms of \(\psi\) wavefunctions, instead of two used above? Can \(\phi\) be expanded to an infinite number of \(\psi\) wavefunctions

    \[ \phi= \sum_i^{\infty} N_i \psi_i\]

    and still be a valid wavefunction (solution to Schrödinger equation)?

    Homework 18 (Due 5/23/2016) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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