- #1

AwesomeTrains

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**Hey everyone!**

1. Homework Statement

1. Homework Statement

I've been giving the equation for a gaussian wave packet and from that I have to derive this formula:

[itex]T_{Kepler}=2\pi \bar n ^3[/itex] by doing a first order taylor series approximation at [itex]\bar n[/itex] of the phase:

[itex]f(x)=f(\bar n)+\frac{df}{dx}|_{\bar n}(x-\bar n)+\sigma(x^2)[/itex]

The given wave packet:

[itex]\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)[/itex]

## Homework Equations

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

## The Attempt at a Solution

Why is the dimension of [itex]T_{Kepler}[/itex] not seconds?

Anyways I tried doing the taylor approximation with t as the differentiation variable and got this (assuming [itex]i\frac{t}{2n^2}[/itex] is the phase):

[itex]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}[/itex] (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

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