6.6: Problems
- Page ID
- 420517
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Calculate the finite-mass Rydberg constant (\(R_ {M}\) ) for
- H
- D
- \(_{7} N\)
- \(_{11} Na\)
- The 1s orbital wavefunction for hydrogen is given by
\[{\psi }_{1s}=\dfrac{1}{\sqrt{\pi }}{\left(\dfrac{1}{a_0}\right)}^{3/2}e^{-\frac{r}{a_0}}\nonumber \]
- Show that this wavefunction is normalized.
- Find the expectation value of \(r\) in units of \(a_{0}\) (the Bohr Radius.)
- Show that the 2s wavefunction for hydrogen is
- Normalized
- An eigenfunction of the Hamiltonian. (What is the eigenvalue?)
- The Laguerre Polynomial \(L_{1}(x)\) is given by
\[L_1\left(x\right)=-x+1\nonumber \]
The Associated Laguerre polynomials are generated from the relationship
\[L^{\alpha }_n\left(x\right)=\dfrac{d^{\alpha }}{dx^{\alpha }}L_n(x)\nonumber \]
- Show that the Associated Laguerre polynomials \(L^0_1\left(x\right)=-x+1\), \(L^1_1\left(x\right)=-1\), and \(L^2_1\left(x\right)=0\). (In fact, \(L^{\alpha }_1\left(x\right)=0\) for any choice of \(\alpha > 1\).)
- Given that the Associated Laguerre polynomials used in the radial wavefunctions of the Hydrogen atom problem are \(L^{2l+1}_{n+l}(x)\), derive a relationship between \(n\) and \(l\) that ensure that \(L^{2l+1}_{n+l}(x)\neq 0\).
- Using the Laguerre polynomials \(L_2\left(x\right)=\dfrac{1}{2}\left(x^2-4x+2\right)\) and \(L_1\left(x\right)=-x+1\), show that
\[\dfrac{d}{dx}L_n\left(x\right)=\dfrac{d}{dx}L_{n-1}\left(x\right)-L_{n-1}(x)\nonumber \]
- Sketch the radial wavefunctions for the 1s, 2s, 2p, 3s, 3p, and 3d orbital wavefunctions of Hydrogen.
- Determine the number of nodes in each of the following hydrogen atom orbital wavefunctions:
| wavefunction | Total nodes | Angular nodes | Radial nodes |
|---|---|---|---|
| 2s | |||
| 3p | |||
| 5d | |||
| 6f |
- Determine the ionization potential for \(_{3} He ^{+}\).
- Find \(R_{He}\) for the He-3 isotope.
- Use the relationship
\[IP=Z^2R_M\left(\dfrac{1}{{\left(1\right)}^2}-\dfrac{1}{{\left(\infty \right)}^2}\right)\nonumber \]
- Based on the following data, find the ionization energy of Rb, using the fact that at high excitation, the quantum defect (\(\delta\)) becomes constant.
| n (for the \(np \leftarrow 5s\) transition) | Wavenumber (\(cm ^{-1}\) ) |
|---|---|
| 5 | 12578.950 |
| 6 | 23715.081 |
| 7 | 27835.02 |
| 8 | 29834.94 |
| 9 | 30958.91 |
| 10 | 31653.85 |
| 11 | 32113.55 |
| 12 | 32433.50 |
| 13 | 32665.03 |
| 14 | 32838.02 |
| 15 | 32970.66 |
| 16 | 33074.59 |
| 17 | 33157.54 |
| 18 | 33224.83 |
| 19 | 33280.13 |
| 20 | 33326.13 |