# Minimising for Lambda; Differentiation?

1. Mar 30, 2010

Hi, so first post on this forum so I hope I'm doing everything good-as-gold!

I've been twisting my head around a derivation of the uncertainty principle and I'm a little stumped on something that I feel I really should know. Hope someone would like to help.

Certain things are defined at the start,
$$\hat{Q}'=\hat{Q}-\left\langle\hat{Q}\right\rangle$$
$$\hat{R}'=\hat{R}-\left\langle\hat{R}\right\rangle$$
$$\phi(x)=(\hat{Q}'+\textit{i}\lambda\hat{R}')\psi(x)$$
$$I(\lambda)=\intdx\phi^{*}(x)\phi(x)\geq0$$

I've followed the manipulation of $$I(\lambda)$$ all the way to the following line;
$$I(\lambda)=(\Delta Q)^{2}+\lambda^{2}(\Delta R)^{2}+\textit{i}\lambda\left\langle\left[\hat{Q},\hat{R}\right]\right\rangle\geq0$$

That's all fine. In the notes he then says "Minimise for $$\lambda$$" following it with;
$$2 \lambda(\Delta R)^{2}+\textit{i}\left\langle\left[\hat{Q},\hat{R}\right]\right\rangle=0$$

He then subs it back into $$I(\lambda)$$
It looks like he's differentiated wrt lambda. But I'm not entirely sure what he means when he says minimise for lambda.
Also, I haven't seen anywhere where 'I' has been defined. Not sure what it is!
Has anyone got any ideas?

MUCH appreciated!