# Quantization using the bohr model

## Homework Statement

A new (fifth) force has been proposed that binds an object to a central body through a potential
energy function given by:
$$U(r) = -Dr^{\frac{-3}{2}}$$
$$2 r > 0$$ and $$D > 0$$

(a) What is the (central) force F(r) associated with this potential energy function?
The object (with mass m) is in an orbit around a central body. The central body is electrically
neutral so we can ignore the Coulomb force. Further, we can ignore the gravitational force
between the object and the central body as it is insignificant compared to the “fifth force.”

(b) What is the total energy of this body in orbit? Is it bound to the central body? Explain

(c) Using the Bohr model that says that angular momentum is quantized, $$L = mvr = n\hbar$$, determine the permitted values of the object’s radius $$r_{n}$$.

(d) The total energy of the object associated with $$r_{n}$$ is also quantized. The general form of this energy expression is $$En = an^{b}$$ where a and b are constants. Determine these constants.

## Homework Equations

$$E=U+K$$
$$F=ma$$ (I think..)

## The Attempt at a Solution

a:
Got this one:
Differentiate the first formula to $$dU(r) = {\frac{3}{2}}Dr^{\frac{-5}{2}}$$

b:
this one too:
$$E=U+K$$
with $$K=\frac{1}{2}mv^{2}$$ and $$U=-Dr^{\frac{-3}{2}}$$

c:
I tried $${\frac{3}{2}}Dr^{\frac{-5}{2}}=ma$$ and $$a=\frac{v^{2}}{r_{n}}$$ with gives $$r_{n}=\frac{9}{4}(\frac{D}{nv\hbar})^{2}$$ but I don't know if it is correct.

d. I don't know how tackle this.

Last edited:

turin
Homework Helper

A new (fifth) force has been proposed that binds an object to a central body through a potential
energy function given by:
$$U(r) = -Dr^{\frac{-3}{2}}$$
$$2 r > 0$$ and $$D > 0$$
(b) What is the total energy of this body in orbit? Is it bound to the central body? Explain

b:
$$E=U+K$$
with $$K=\frac{1}{2}mv^{2}$$ and $$U=-Dr^{\frac{-3}{2}}$$
What is v? Did you answer the rest of the question?

(c) Using the Bohr model that says that angular momentum is quantized, $$L = mvr = n\hbar$$, determine the permitted values of the object’s radius $$r_{n}$$.

c:
I tried $${\frac{3}{2}}Dr^{\frac{-5}{2}}=ma$$ and $$a=\frac{v^{2}}{r_{n}}$$ with gives $$r_{n}=\frac{9}{4}(\frac{D}{nv\hbar})^{2}$$ but I don't know if it is correct.
Again, what is v? In this case, you definitely shouldn't have a v-dependent r, unless v is also quantized. Is it? Ultimately, it is probably best to remove v from all considerations, since this problem is getting at the QM, and v is much more CM than QM.

I got it. Thanks for the help :)