# Derivation of orbital period - Hydrogen

• AwesomeTrains

#### AwesomeTrains

Hey everyone!
1. Homework Statement

I've been giving the equation for a gaussian wave packet and from that I have to derive this formula:
$T_{Kepler}=2\pi \bar n ^3$ by doing a first order taylor series approximation at $\bar n$ of the phase:
$f(x)=f(\bar n)+\frac{df}{dx}|_{\bar n}(x-\bar n)+\sigma(x^2)$

The given wave packet:
$\psi(t)=\frac{1}{(2\pi\sigma_n^2)^{1/4}}\sum_{n=1}^\infty exp\left(-\frac{(n-\bar n)^2}{4\sigma_n^2}\right)r\psi_{n,l,m}exp\left(i\frac{t}{2n^2}\right)$

## Homework Equations

I assume I've posted all the relevant ones in 1.

## The Attempt at a Solution

Why is the dimension of $T_{Kepler}$ not seconds?
Anyways I tried doing the taylor approximation with t as the differentiation variable and got this (assuming $i\frac{t}{2n^2}$ is the phase):
$i\frac{t}{2n^2}=i\frac{\bar n}{2n^2}+i\frac{t-\bar n}{2n^2}+\sigma(t^2)=i\frac{it}{2n^2}$ (Strange)

Am I supposed to do it for the variable n? Or how do I proceed from here?
Any hints are very appreciated.
Kind regards
Alex

Last edited: