- #1
wgrenard
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Homework Statement
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.
Homework Equations
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?
