4.P: Exercises
- Page ID
- 1203
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Demonstrate directly from the fundamental commutation relations for angular momentum, \ref{300}, that \( [L^\pm, L_z] = \mp \,\hbar\,L^\pm\), and \( L_x\)
- \( L_y\) \( L_z\) \( \theta\), \( \psi(\theta,\varphi)=Y_{l\,m}(\theta,\varphi)\). Evaluate \( \langle L_y\rangle\), \( \langle L_y^{\,2}\rangle\).
- Derive Equations \ref{385} and \ref{386} from Equation \ref{384}.
- 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 \( -\hbar\) .
- 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 \( H\) ?
- 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.)
- 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.
- Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:
- \( R_{1\,0}(r) = \frac{2}{a_0^{\,3/2}}\,\exp\left(-\frac{r}{a_0}\right).\)
- \( R_{2\,0}(r)= \frac{2}{(2\,a_0)^{3/2}}\left(1-\frac{r}{2\,a_0}\right)\exp\left(-\frac{r}{2\,a_0}\right).\)
- \( R_{2\,1}(r)= \frac{1}{\sqrt{3}\,(2\,a_0)^{3/2}}\,\frac{r}{a_0}\,\exp\left(-\frac{r}{2\,a_0}\right).\)
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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