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4.P: Exercises

  • Page ID
    1203
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    1. Demonstrate directly from the fundamental commutation relations for angular momentum, \ref{300}, that \( [L^\pm, L_z] = \mp \,\hbar\,L^\pm\), and \( L_x\)
    2. \( L_y\) \( L_z\) \( \theta\), \( \psi(\theta,\varphi)=Y_{l\,m}(\theta,\varphi)\). Evaluate \( \langle L_y\rangle\), \( \langle L_y^{\,2}\rangle\).
    3. Derive Equations \ref{385} and \ref{386} from Equation \ref{384}.
    4. Find the eigenvalues and eigenfunctions (in terms of the angles \( \varphi\) ) of \( l=1\). A measurement of \( \hbar\). What values will be obtained by a subsequent measurement of \( L_x\) yields the results 0 and \( -\hbar\) .
    5. The Hamiltonian for an axially symmetric rotator is given by \( H = \frac{L_x^{\,2}+L_y^{\,2}}{2\,I_1} + \frac{L_z^{\,2}}{2\,I_2}.\) What are the eigenvalues of \( H\) ?
    6. The expectation value of \( f({\bf x},{\bf p})\) in any stationary state is a constant. Calculate \( 0= \frac{d}{dt}\,(\langle{\bf x}\cdot{\bf p}\rangle) = \frac{\rm i}{\hbar}\,\langle[H, {\bf x}\cdot{\bf p}] \rangle\) for a Hamiltonian of the form \( H = \frac{p^2}{2\,m} + V(r).\) Hence, show that \( \left\langle\frac{p^2}{2\,m}\right\rangle = \frac{1}{2}\left\langle r\,\frac{dV}{dr}\right\rangle\) in a stationary state. This is another form of the Virial theorem. (See Exercise 8.)
    7. Use the Virial theorem of the previous exercise to prove that \( \left\langle \frac{1}{r}\right\rangle = \frac{1}{n^2\,a_0}\) for an energy eigenstate of the hydrogen atom.
    8. Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:
      1. \( R_{1\,0}(r) = \frac{2}{a_0^{\,3/2}}\,\exp\left(-\frac{r}{a_0}\right).\)
      2. \( R_{2\,0}(r)= \frac{2}{(2\,a_0)^{3/2}}\left(1-\frac{r}{2\,a_0}\right)\exp\left(-\frac{r}{2\,a_0}\right).\)
      3. \( R_{2\,1}(r)= \frac{1}{\sqrt{3}\,(2\,a_0)^{3/2}}\,\frac{r}{a_0}\,\exp\left(-\frac{r}{2\,a_0}\right).\)

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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    This page titled 4.P: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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