91影视

A potential function is a function of the position of an object. It can be defined only for conservative forces. A forces is conservative if the work it does on n object only on the initial and final position of the object and net on the path. The gravitational force is a conservative force.

The potential is given by:

$V\left(r\right)=-\frac{e^2}{4蟺{系}_0}\frac{1}{r}$

Consdier the Earth-sun system as a gravitational analog to the hydrogen atom. The potential is:

$V\left(r\right)=-G\frac{Mm}{r}m/r.$

Where G is the gravitational constant, $G=6.673脳{10}^{-11}{m}^{3}k{g}^{-1}{s}^{-2}$

translates hydrogen results to the gravitational analogs.

The Bohr radius for the Earth can be found by replacing$e^2$ by$mM$ and$1/4{\mathrm{蟺蔚}}_0$ by G ,so we get:

$a_g=\frac{{鈩�}^2}{GM{m}^2}$

Substitute with the numerical values to get:

$$\begin{gathered} a_g=\frac{{\left(1.0546脳{10}^{-34}Js\right)}^2}{\left(6.6726脳{10}^{-11}m/kg.s^2\right)\left(1.9892脳{10}^{30}kg\right){\left(5.98脳{10}^{24}kg\right)}^2} \\ {a}_g=2.34脳{10}^{-138}m \end{gathered}$$

Equation 4.70 (the allowed energies) is given by:

$鈬�E_n=-\left[\frac{m}{2{魔}^2}(GMm{)}^2\right]\frac{1}{n^2}$

There are many items that could be altered to test the recreation of another. These changing quantities are called variables. A variables is any factor, trait, or condition that can exist in differing amounts or types. An experiment usually amounts or types, An experiment usually has three kinds of variables: independent, dependent, and controlled.

$$\begin{gathered} E_c=\frac{1}{2}m{v}^2-G\frac{Mm}{r_o} \\ G\frac{Mm}{r_o^2}=\frac{m{v}^2}{r_o} \\ 鈬�\frac{1}{2}m{v}^2 \\ =\frac{GMm}{2r_o} \end{gathered}$$

So,

$c{E}_c=-\frac{GMm}{2r_o}$

$$\begin{gathered} =-\left[\frac{m}{2{鈩�}^2}(GMm{)}^2\right]\frac{1}{n^2} \\ 鈬�n^2=\frac{GM{m}^2}{{鈩�}^2}{r}_o \\ =\frac{r_o}{a_g} \end{gathered}$$

$鈬�n=\sqrt{\frac{r_o}{a_g}}.$

$$\begin{gathered} r_o=\mathrm{earth}-\mathrm{sundistance}=1.496脳{10}^{11}m \\ 鈬�n=\sqrt{\frac{1.496脳{10}^{11}}{2.34脳{10}^{-138}}} \\ =2.53脳{10}^{74}{r}_0 \\ =\mathrm{earth}-\mathrm{sun}\mathrm{distance} \\ =1\mathrm{m}. \end{gathered}$$

The wavelength of the emitted photon

$鈭�E=-\left[\frac{G^2{M}^2{m}^3}{2{魔}^2}\right]\left[\frac{1}{{\left(n+1\right)}^2}-\frac{1}{n^2}\right].\frac{1}{{\left(n+1\right)}^2}=\frac{1}{n^2\left(1+1/n^2\right)}鈮�\frac{1}{n^2}\left(1-\frac{2}{n}\right)$

So,

role="math" localid="1658389122151" $$\begin{gathered} \left[\frac{1}{(n+1{)}^2}-\frac{1}{n^2}\right]鈮�\frac{1}{n^2}\left(1-\frac{2}{n}-1\right)=-\frac{2}{n^3}; \\ 螖E=\frac{G^2{M}^2{m}^3}{{鈩�}^2{n}^3} \\ 鈭�E=\frac{{\left(6.67脳{10}^{-11}\right)}^2{\left(1.99脳{10}^{30}\right)}^2{\left(5.98脳{10}^{24}\right)}^3}{{\left(1.055脳{10}^{-34}\right)}^2{\left(2.53{脳}^{74}\right)}^3} \\ =2.09脳{10}^{-41}J \end{gathered}$$

$$\begin{gathered} E_p=鈭�E=hv=\frac{hc}{位} \\ 位=\left(3脳{10}^8\right)\left(6.63脳{10}^{-34}\right)/\left(2.09脳{10}^{-41}\right) \\ =9.52脳{10}^{15}m \end{gathered}$$

$$\begin{gathered} \mathrm{But}1ly=9.46脳{10}^{15}m.that位鈮�1ly,n^2=GM{m}^2{r}_o/{鈩�}^2 \\ 位=\frac{ch}{鈭�E} \\ =c2\mathrm{蟺魔}\frac{{\mathrm{魔}}^2{\mathrm{n}}^3}{{\mathrm{G}}^2{\mathrm{M}}^2{\mathrm{m}}^3} \\ =\mathrm{c}\frac{2{\mathrm{蟺魔}}^2}{{\mathrm{G}}^2{\mathrm{M}}^2{\mathrm{m}}^3}{\left(\frac{{\mathrm{GMm}}^2{\mathrm{r}}_0}{{\mathrm{魔}}^2}\right)}^{3/2} \\ =\mathrm{c}\left(2\mathrm{蟺}\sqrt{\frac{{\mathrm{r}}_0^3}{\mathrm{GM}}}\right) \end{gathered}$$

$\mathrm{But}(\mathrm{from}(\mathrm{c}\left)\right)$

$v=\sqrt{GM/r_o}=2蟺r_o$, where T is the period of the orbit (in this case one year),

so $T=2蟺\sqrt{r_o^3/GM}$and hence ${}^{位=cT}$(one light year).