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Homework 6A

  • Page ID
    143074
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    Name: ______________________________

    Section: _____________________________

    Student ID#:__________________________

    Q1

    The energy of a single electron around a proton is

    \[ E_n = -\dfrac {m_e e^4}{8\epsilon_0^2 h^2 n^2}\]

    with \(n=1,2,3 ...\infty\). The corresponding eigenstates can be expressed as the product of a angular and radial functions

    \[ |\psi(r,\theta,\phi) \rangle = | R(r) \rangle |Y_{\ell}^{m_l}(\theta,\phi) \rangle \]

    which have three quantum numbers to describe an unique wavefunction, \(n\), \(l\) and \(m_l\) each with unique range permissible. The fact that energy depends only on \(n\) indicates that this system has degenerate wavefunction. List all the eigenstates (explicitly pointing out the appropriate quantum numbers) that have the following energies:

    1. \( E_n = -\dfrac {m_e e^4}{32\epsilon_0^2 h^2}\)
    2. \( E_n = -\dfrac {m_e e^4}{72\epsilon_0^2 h^2}\)
    3. \( E_n = -\dfrac {m_e e^4}{128\epsilon_0^2 h^2}\)

    Q2

    How many radial nodes (i.e., nodes in \(R(r)\)) are there in the following hydrogen orbitals?

    1. \(| \psi_{1s} \rangle\)
    2. \(| \psi_{2s} \rangle\)
    3. \(| \psi_{2p_x} \rangle\)
    4. \(| \psi_{2p_z} \rangle\)
    5. \(| \psi_{100} \rangle\) (the subscript refers to the \(n\),\(l\),\(m_l\) triad of quantum numbers)
    6. \(| \psi_{300} \rangle\)

    Q3

    1. Explain why the radial distribution function \(r^2R(r)^*R(r)\) should be used to discussed the probability of finding an electron from a nucleus rather than the square of the radial component of the wavefunction \(R^2\).
    2. Draw a plot of both the radial distribution function and the square of the radial component of the wavefunction.

    Q4

    Show that the Hamiltonian operator \(\hat{H}=(p^2/2m)\) of a rigid rotor commutes with all three components of \(\vec{L}\). Thus \(H\), \(L^2\), and \(L_z\) are mutually compatible observables.

    Q5

    What is \( \langle I \rangle \), the expectation value of the moment of inertia, of the hydrogen atom for the 1s, 2s, and 2p\(_z\) states.

    Q6

    Consider a hydrogen atom. Is an electron in the 1s orbital (on average) further, closer, or the same distance from the nucleus in the 2s orbital? Confirm this intuition quantum mechanically by calculating the expectation value.

    Q7

    What is the orbital angular momentum of an electron in the following orbitals

    1. 1s
    2. 2s
    3. 2p
    4. 3d

    How many angular and radial nodes exist for the wavefunctions described by the above states?

    Q8

    Given:

    \[ \hat{L}_z = -i \hbar \frac{\partial}{\partial \varphi} \]

    \[\hat{L^2} = - \hbar^2\left( \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \theta} \sin \theta \dfrac{\partial}{\partial \theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2} {\partial\varphi^2}\right) \]

    Show that \( Y_0^0 \) is an eigenfunction of \( \hat{L^2} \) and\( \hat{L}_z \).


    Homework 6A is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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