# Hamiltonian of oscillators quantized proof

1. Sep 30, 2012

A matrix $N$ has all entries except diagnoal ones zero, and first row, first column also zero, other diagonal entries depending on its column, row place; for example, second row, second column has one as its entry, third row, third column has two as its entry and so on.

It is known that $H = \hbar\omega (N + \frac{1}{2})$.

The question is, my textbook says that if there are a continuous range of possible values, then for some states, $\langle (N - \langle N \rangle )^2 \rangle \leq [\frac{1}{2} (n - \langle N \rangle)]^2$ , where $n = 0,1,2,3,...,$ then progresses to say that since $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)$ then $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2$.

Then it says that this proves that there cannot be a continuous range of possible values, as $(0-\langle N \rangle)^2$, $(1 - \langle N \rangle)^2$ and so on cannot be smaller than $(n - \langle N \rangle)^2$.

I am not sure how one gets $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)$. Can anyone show me the proof?

Also, how does one get the part - $(0-\langle N \rangle)^2$, $(1 - \langle N \rangle)^2$ and so on cannot be smaller than $(n - \langle N \rangle)^2$?