$\frac{1}{\lambda}=R y\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

${ Ry }=1.0973731568508(65) \times 10^7 m^{-1}$

$n_1 n_2 \text { integers }$

Spectral series

Lyman: $n_1=1, n_2=2,3, \cdots$

Balmer: $n_1=2, n_2=3,4 \cdots$

Paschen: $n_1=2, n_2=3,4, \cdots$

Brackett, Pfund, Humphreys, Strong and Hansen

Only discrete orbits are allowed

Quantization of Angular Momentum

$m v_n r_n=n \hbar \equiv \frac{n h}{2 \pi}$

Modes of vibration

Analogy with vibrating strings revise.

$m v_n r_n=n \hbar \quad \hbar =\frac{h}{2 \pi}$

$\frac{m v_n^2}{r_n}=\frac{k}{r^2} ; \quad k=\frac{z e^2}{4 \pi \epsilon_0}$

$z=1$

$\frac{n^2 \hbar^2}{m r_n}=k$

$r_n=\frac{n^2 \hbar^2}{m k}$

$\frac{n^2}{r_n}=$constant

$r_n=\frac{n^2 \hbar^2}{m k} \quad$

$V_n=-\frac{k}{r_n}=\frac{-m k^2}{n^2 \hbar^2}$

$T_n=\frac{1}{2} m v_n^2$

$m v_n r_n=n \hbar$

$v_n^2=\frac{n^2 \hbar^2}{m^2 r_n^2} = \frac{1}{2} m \frac{n^2 \hbar^2}{m^2 r_n^2}$

$T_n=\frac{1}{2} \frac{m n^2 \hbar^2}{m^2} \frac{m^2 k^2}{n^4(\hbar)^4}$

$T_n=\frac{1}{2} \frac{m k^2}{n^2 \hbar^2}$

$V_n=-\frac{m k^2}{n^2 \hbar^2}$

$E_n=-\frac{1}{2} \frac{m k^2}{n^2 \hbar^2}$

All the result

Scaling laws

$r_n \sim n^2$

$v_n \sim \frac{1}{n}$

$T_n, V_n \sim \frac{1}{n^2}$

$E_n \sim \frac{1}{n^2}$

$T_n=-\frac{1}{2} V_n$

$E_n < 0$ for all n

Method: Test the model in a different setting

Two examples:

Franck-hertz experiment

Vibrational states of molecules

Mercury(Vapour): Hg

In discharge tubes, Hg emitted radiation of about 250nm. (sharpline)

Franck hertz experiment

Additional evidence

Scattering of electrons by mercury

Franck hertz experiment

Take a small quantity of mercury (in the vacuum tube) and evaporate

Low density Hg vapour in the Vacum tube distributed all over.

They injected accelerated electrons in the tube.

Energy range was a fraction of eV to about $80eV$

$1 \mathrm{eV}-80 \mathrm{eV}$

The cathode ray particles (electron) interact with the electrons in $\mathrm{Hg}$ Atom.

$\Delta E=E_1-E_g$

Additional evidence

Scattering of electrons by mercury

Energy of scattered $e^{-}$ in zero

$7-4.9 = 2.1 eV$

Cathorde $e^{-}$ energy in $7eV$

$E_{in} \quad 10.8 \mathrm{eV}$

Final energy is $10.8 - 9.8 = 1.0 eV$

$\frac{4 \cdot 9 \times 3}{14.7 \mathrm{eV}}$

Waited: Excited vapour emits light of energy $4.9 eV$

Additional evidence

Scattering of electrons by mercury

Modern day Experiment

Neon $\rightarrow$ already a gas atoms

Emitted radiation is in the visible range

Franck hertz experiment

Additional evidence

Scattering of electrons by neon

Example:- Van-der Waals

$\frac{A}{r^6} - \frac{B}{r^7}$

$V(r)$ has a Vanishing derivative at $r=R$;

Which is a position for minima:

$\frac{\partial^2 V}{\partial r^2} |_{r=R}$ >0

$\frac{\partial V}{\partial r} |_{r=R}=0$

Make a Taylor expansion of $V(r)$ around $r=R$

$\left.\frac{\partial^2 V}{\partial r^2}\right|_{r=R}=K>0$

$V(r)=V(R)+\frac{1}{2} K(r-R)^2+$ higher order

$\rho=|r-R| .$

${V=V_0+\frac{1}{2} K P^2}$

Harmonic oscillator potential

$K \rightarrow$ Spring constant

Apply the Bohr model hypothesis to SHO

$E =\frac{1}{2} m v^2+\frac{1}{2} k x^2 = \frac{1}{2} m v^2+\frac{1}{2} m \omega^2 x^2$

$F =-k$

$E =\frac{1}{2} m \vec{v}^2+\frac{1}{2} k \vec{r}^2 =\frac{1}{2} m \vec{v}^2+\frac{1}{2} m \omega^2 \vec{r}^2$

$\vec{F} =-k \vec{r}$ Hooke's law in three dimension

Circular orbits + Bohr quantization.

$\frac{m v_n^2}{r_n}=+k r_n=-m \omega^2 r$

$m v_n r_n=n \hbar$

$m\left(\frac{v_n^2}{r_n^2}\right)=k=\text { constant }$

$v_n=\frac{n \hbar}{m r_n}$

$v_n^2=\frac{n^2 \hbar^2}{m^2 r_n^2}$

$\therefore m \frac{v_n^2}{r_n^2} = m \frac{n^2 \hbar^2}{m^2 r_n^2} \frac{1}{r_n^2}$

Constant = {k} = $r_n^4=\frac{n^2 \hbar^2}{m k}=\frac{n^2 \hbar^2}{m^2 \omega^2} .$

$r_n^4=\frac{n^2 \hbar^2}{m^2 \omega^2}$

$r_n^2={n\left(\frac{\hbar}{m \omega}\right)} \Rightarrow V_n \propto n$

Quantum mechanics Bohr model in giving a new length scale