Finding Possible Measurement Results of an Observable

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1. Feb 19, 2015

wgrenard

1. The problem statement, all variables and given/known data

I am trying to find the possible measurement results if a measurement of a given observable $Q=I-\left|u\right\rangle\left\langle u\right|$ is made on a system described by the density operator $\rho={1 \over 4}\left|u\right\rangle\left\langle u\right|+{3 \over 4}\left|v\right\rangle\left\langle v\right|$.

I am given that $\left|u\right\rangle$ and $\left|v\right\rangle$ are normalized states, and that $\left\langle u|v\right\rangle=cos(\theta)$. The definition of $\theta$ is unstated but I assume it is some angle in real space.

2. Relevant equations

3. The attempt at a solution

I know that the possible measurement results are given by the eigenvalues of $Q$, but in this instance, I am unsure of how to determine these. First, I attempted a resolution of the identity to rewrite $Q$ in a different form. Because $\left|u\right\rangle$ and $\left|v\right\rangle$ aren't orthogonal, however, they do not constitute a basis. To resolve the identity operator, you need to find a complete basis. If you create a one by choosing it to consist of $\left|u\right\rangle$ as well as a proper number of other vectors $\left|1\right\rangle ,\left|2\right\rangle ,\left|3\right\rangle ,...$ all orthogonal to each other, then $Q$ is:

$Q=\left|u\right\rangle\left\langle u\right|+\sum \left|i\right\rangle\left\langle i\right|-\left|u\right\rangle\left\langle u\right|=\sum \left|i\right\rangle\left\langle i\right|$

So, I succeeded in writing $Q$ in a more compact form. However, I am unsure how much good this has done me. Because I don't know the eigenstates of $Q$ I cannot say that the possible measurement results are the eigenvalues corresponding to the states $\left|i\right\rangle$, which was originally what I was after, because all I know about the $\left|i\right\rangle$ states is that they are orthogonal, not that they are eigenstates.

Am I on the right track with this direction of reasoning, or should I be attempting this a different way?

2. Feb 20, 2015

Orodruin

Staff Emeritus
What are the eigenvalues of the identity operator?