Phy5645/stability of matter hw1

(Submitted by Team 4-Yuhui Zhang)

As for the motion of electron in hydrogen atom. Assume that the angular momentum $L=n\hbar$, which is the result of Bohr-Sommerfield Condition, try to get $r_{n}$, $E_{n}$ and Rydberg constant.

$L=rmv=n\hbar$

Assume the motion of this electron is circular,

${\frac {e^{2}}{r^{2}}}=m{\frac {v^{2}}{r}}$

so, $mv={\sqrt {\frac {me^{2}}{r}}}$

$mvr=r{\sqrt {\frac {me^{2}}{r}}}=n\hbar$

we get: $r=r_{n}={\frac {n^{2}\hbar ^{2}}{me^{2}}}$

we can see that: $a=r_{1}={\frac {\hbar ^{2}}{me^{2}}}$, which is Bohr radius.

$r_{n}=n^{2}a$

the energy:

$E=E_{n}={\frac {1}{2}}mv^{2}-{\frac {e^{2}}{r_{n}}}=-{\frac {1}{2}}{\frac {me^{2}}{2n^{2}h^{2}}}$

the light spectrum:

$h\upsilon =E_{ni}-E_{nf}=RC({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}})$

so, $R={\frac {me^{4}}{4\pi ^{2}\hbar ^{3}c}}$.