# Bohr-Sommerfeld Rule

1. Oct 22, 2014

### knowLittle

1. The problem statement, all variables and given/known data
Imagine that force for is atom was $F= - \frac{\beta}{r^4}$, rather than $F=- \frac{ke^2}{r^2}$, and consider only circular orbits, it would remain true that $L_n= n \hbar$

a.) From Netwon's law find the relationship between $T$(Kinetic Energy) and $V$,
b.) Find $E$ as a function of $r$
c.) Find quantized values of $r_n$
d.) "" quantized values of $\omega_n$
e.) "" quantized values of $E_n$
f.) Does it remain true that for high $n, \Delta E= E_{n+1}- E_n \approx \hbar \omega_n$

Note: The definition of $\beta$ is not given. It bothers me. They are not saying that from both F's given we could solve for $\beta$, how do I overcome this ambiguity.

3. The attempt at a solution
a.)
Comparing force in a spring
$F_{net}=F_{spring} =-kx= ma$
The description of SHM is closely related to uniform circular motion.
$E= K_E +V$

$E=\frac{1 m v^2}{2} + \frac{kx^2}{2}$

Is this correct?

b.)
$E =K_E + V$
We are given $F=- \frac{\beta}{r^4}$
We know that centripetal force $F_c= \frac{mv^2}{r}$
$r F = mv^2$
According to the problem this F and the F involving $\beta$ are equivalent.
So, $mv^2= r \frac{\beta}{r^4} = \frac{\beta}{r^3}$
Then, $K_E = \frac{\beta}{2r^3}$

Now, note that $V=- \int F dr$
$V= + \int \frac{\beta}{r^4} dr= \beta \frac{1}{-3 r^3}$

Finally, $E= \frac{\beta}{2r^3} - \frac{\beta}{3r^3}$

Last edited: Oct 22, 2014
2. Oct 23, 2014

### Staff: Mentor

The numerical value of $\beta$ is not important to solve the problem, just like the numerical values of $k$ and $e$ wouldn't be needed in the case with the Coulomb interaction.

This is not relevant to the problem, which has nothing to do with harmonic motion. Part of the solution for a) you actually have answered in b).