2.6: The de Broglie-Bohr Model for a Hydrogen Atom Held Together by a Gravitational Interaction
- Page ID
- 154851
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[ \lambda = \frac{h}{mv} \nonumber \]
de Broglie's hypothesis that matter has wave-like properties.
\[ n \lambda = 2 \pi r \nonumber \]
The consequence of de Broglie's hypothesis; an integral number of wavelengths must fit within the circumference of the orbit. This introduces the quantum number, n, which can have values 1,2,3,...
\[ mv = \frac{nh}{2 \pi r} \nonumber \]
Substitution of the first equation into the second equation reveals that linear momentum is quantized.
\[ T = \frac{1}{2} m v^2 = \frac{n^2 h^2}{8 \pi^2 m_e r^2} \nonumber \]
If momentum is quantized, so is kinetic energy.
\[ E = T + V = \frac{n^2 h^2}{8 \pi^2 m_e r^2} - \frac{G m_p m_e}{r} \nonumber \]
Which means that total energy is quantized, where \( - \frac{G m_p m_e}{r}\) is the gravitational potential energy interaction between a proton and an electron.
\[ \frac{d}{dr} \left( \frac{n^2h^2}{8 \pi^2 m_e r^2} - \frac{G m_p m_e}{r} \right) = 0 ~ \text{solve, r} \rightarrow \frac{h^2 n^2}{4 \pi^2 G m_e^2 m_p} \nonumber \]
Minimization of the energy with respect to orbit radius yields the optimum values of r. This expression is subtituted back in the energy expression below to find the allowed energies.
\[ E = \frac{n^2 h^2}{8 \pi^2 m_e r^2} - \frac{G m_p m_e}{r} \text{substitute, r} = \frac{h^2 n^2}{4 \pi^2 G m_e^2 m_p} \rightarrow E = \frac{2 \pi^2 G^2 m_e^3 m_p^2}{h^2 n^2} \nonumber \]
\[ \begin{matrix} \text{Fundamental constants:} & m_p = 1.67262 (10)^{-27} \text{kg} & m_e = 9.10939 (10)^{-31} \text{kg} \\ ~ & h = 6.62608 (10)^{-34} \text{joule sec} & G = 6.67259 (10)^{-11} \frac{m^3}{ \text{kg s}^2} \end{matrix} \nonumber \]
Energy:
\[ E(n) = - \frac{2 \pi^2 G^2 m_e^3 m_p^2}{h^2 n^2} \nonumber \]
Orbit radius:
\[ r(n) = \frac{h^2 n^2}{4 \pi^2 G m_e^2 m_p^2} \nonumber \]
Calculate the first four energy levels and orbit radii.
\[ \begin{matrix} n = 1 .. 4 & \frac{E(n)}{J} = \begin{pmatrix} -4.233 \times 10^{-97} \\ -1.058 \times 10^{-97} \\ -4.704 \times 10^{-98} \\ -2.646 \times 10^{-98} \end{pmatrix} & \frac{r(n)}{m} = \begin{pmatrix} 1.201 \times 10^{29} \\ 4.803 \times 10^{29} \\ 1.081 \times 10^{30} \\ 1.921 \times 10^{30} \end{pmatrix} \end{matrix} \nonumber \]
Prepared by Frank Rioux.