- #1

steely

- 1

- 0

## Homework Statement

a) Express the total energy of an electron in the Coulomb potential of proton through the electron's angular momentum

**and the shortest distance**

*L***between the proton and the electron's orbit. Hint: The electron's velocity is perpendicular to it's position vector whenever it is distance**

*a***away from the proton.**

*a*b) For a fixed

**, minimize the expression found in (a) with respect to**

*L***. Show that the minimum corresponds to the case of a circular orbit. State the minimum value of the total energy for fixed**

*a***.**

*L*## Homework Equations

L=sqrt(l

^{2}+l)ћ

L

_{z}=m

_{l}ћ

L=r x mv=Iω

E

_{n}=-m

_{e}e4/[2(4πε

_{0})

^{2}ћ

^{2}n

^{2}]

n > l

__>__|m

_{l}|

__>__0

KE=.5Iω

^{2}

## The Attempt at a Solution

E=U+KE => E= .5L*v/a - e^2/[4πε

_{0}a]

I have little confidence in that as a solution. Either I've stopped short or I'm going in the wrong direction, I'm not sure. I really only want help with part (a)... Once I get that far I should be handle (b) somewhat easily, I just wanted to provide context for the problem.