91影视

From Section 7.8, you know that, all formulas for hydrogen apply to hydrogen like atoms if you simply replace ${\mathbf{e}}^{\mathbf{2}}$to${\mathbf{Ze}}^{\mathbf{2}}$.

Hence,

Energy levels of hydrogen like atoms, ${\mathbf{E}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{m}{\left({\mathrm{Ze}}^2\right)}^{\mathbf{2}}}{\mathbf{2}{\left(4{\mathrm{蟺蔚}}_0\right)}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}{\mathbf{n}}^{\mathbf{2}}}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{2}}}$ 鈥︹赌︹赌︹赌�(1)

Where, z is the atomic mass, h is Plank鈥檚 constant, n = principal quantum number,${\mathbf{蔚}}_{\mathbf{0}}$is the permittivity of free space.

And the Bohr鈥檚 Radius is,

${\mathbf{r}}_{\mathbf{n}}\mathbf{=}{\mathbf{n}}^{\mathbf{2}}\frac{\left(4{\mathrm{蟺蔚}}_0\right){\mathbf{h}}^{\mathbf{2}}}{\mathbf{m}\left({\mathrm{ze}}^2\right)}$ 鈥�.. (2)

Where, m is the mass.

Now, using eq. (1) and replacing mass (m) with Effective mass $\left(\mathrm{渭}\right)$, you get,

$$\begin{gathered} {\mathrm{E}}_{\mathrm{n}}=\frac{\mathrm{m}{\left({\mathrm{Ze}}^2\right)}^2}{2{\left(4{\mathrm{蟺蔚}}_0\right)}^2{\mathrm{h}}^2}\frac{1}{{\mathrm{n}}^2} \\ =\frac{{\mathrm{Z}}^2\mathrm{渭}}{\mathrm{m}}\frac{-{\mathrm{me}}^4}{2{\left(4{\mathrm{蟺蔚}}_0\right)}^2{\mathrm{h}}^2}\frac{1}{{\mathrm{n}}^2} \\ =\frac{{\mathrm{Z}}^2\mathrm{渭}}{\mathrm{m}}\frac{{\mathrm{E}}_1}{{\mathrm{n}}^2} \end{gathered}$$

Hence, the allowed energies are $\frac{{\mathrm{Z}}^2\mathrm{渭}}{\mathrm{m}}\frac{{\mathrm{E}}_1}{{\mathrm{n}}^2}$.

Where, ${\mathrm{E}}_1$is the Energy of hydrogen ground state.

Again, using eq. (2) and replacing mass (m) with Effective mass $\left(\mathrm{渭}\right)$, you get,

$$\begin{gathered} {\mathrm{r}}_{\mathrm{n}}=1^2\frac{\left(4{\mathrm{蟺蔚}}_0\right){\mathrm{h}}^2}{\mathrm{渭}\left({\mathrm{Ze}}^2\right)} \\ =\frac{\mathrm{m}\left(4{\mathrm{蟺蔚}}_0\right){\mathrm{h}}^2}{\mathrm{窜渭}\left({\mathrm{me}}^2\right)} \\ =\frac{\mathrm{m}}{\mathrm{窜渭}}{\mathrm{a}}_0 \end{gathered}$$

Hence, the Bohr鈥檚 Radius is $\frac{\mathrm{m}}{\mathrm{窜渭}}{\mathrm{a}}_0$.

Where, $a_0$is the hydrogen Bohr radius.