Bohr-Sommerfeld Rule: Solving for the Quantized Values of E, r, and ω

[knowLittle]

Homework Statement

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.

The Attempt at a Solution

a.)[/B] 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, $E= \frac{\beta}{2r^3} - \frac{\beta}{3r^3}$

 

[DrClaude]

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.

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.

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?

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).