Quantum Physics. Gasiorowicz ch1 problem num15

[rar0308]

Homework Statement

Use the Bohr quantinization rules to calculate the energy states for a potential given by

$V(r)=V_0 (\frac{r}{a})^k$
with k very large. Sketch the form of the potential and show that the energy values approach $E_n \cong Cn^2$

Homework Equations

The Attempt at a Solution

I read textbook of ch1. but I can't understand the problem.
What is Bohr quantinization rules?
What is the energy states?
The potential for what particle?
Such things are not explained in the text. At least what such words indicate is ambiguous. The exact words don't appear in the text.
Problem don't describe a situation enough.

 

[Clever-Name]

You're right, it's very ambiguous the way it's worded, but if you think about what this potential might describe it becomes clear.

I'll give you a hint, this potential describes a particle undergoing circular motion!

You are also going to need to use Bohr's angular momentum quantization rule (page 17)
$$mvr = n\hbar \\ n = 1,2,3,...$$

Using this, and the hint I have given you, can you think of a way to set up an expression that might give you quantized (discrete!) energy levels?

Remember: Energy = Kinetic + Potential!

 

[rar0308]

I can see the potential is of central force.
But how can I know that the particle is undergone uniform ciruclar motion?
The given potential always implies UCM?

 

[Clever-Name]

A central potential will always yield motion defined by
$$F = -\nabla V(r)$$
This might not result in uniform circular motion, that depends on the particle's velocity. However, this permits the force to be written as:
$$F = \frac{mv^{2}}{r}$$

 

[rar0308]

Very helpful.
I thought $F = \frac{mv^2}{r}$ is only for UCM

 

[Clever-Name]

Right, of course it is, I'm not sure what I was thinking when I posted above. It is true however that any central potential will allow uniform circular motion, and I can't seem to think of any other way to go about this problem without first assuming the particle is undergoing uniform circular motion. The solution is perfect when you assume UCM.

Sorry for the late reply.