Derivation of orbital period - Hydrogen

[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

 

[Asim Riaz]

2. The equation you are trying to approximate is for the phase of the wave packet, so you should differentiate with respect to the variable n. You can use the chain rule to break up the derivative into terms that are easier to evaluate.Start by writing the phase as a function of n:f(n)=f(\bar n)+\frac{df}{dn}|_{\bar n}(n-\bar n)+\sigma(n^2)Now take the derivative with respect to n: \frac{df}{dn}=\frac{df}{dn}|_{\bar n}+2\sigma nEvaluate this at \bar n and substitute it back into your equation:f(n)=f(\bar n)+\frac{df}{dn}|_{\bar n}(n-\bar n)+\sigma(n^2)=f(\bar n)+2\sigma \bar n (n-\bar n)+\sigma(n^2)Now do a first order Taylor series approximation, meaning you only keep terms up to the first power of n-\bar n:f(n)\approx f(\bar n)+2\sigma \bar n (n-\bar n)Finally, solve for T_{Kepler}:T_{Kepler}=2\pi \bar n ^3