Question

We measured a 3-level quantum system: \[\frac{1} {\sqrt{3}} |0>+\frac{1} {\sqrt{3}} |1>+\frac{1} {\sqrt{3}} |2>\] with respect to the observable: \[\left[\begin{matrix}1 & 0 & 0\\ 0 & 4 & 0 \\ 0 & 0 & 4\end{matrix}\right]\] and saw a 4. What is the state of the new system?

Answer 1

@UnkleRhaukus

Answer 2

So I know we would get the vectors:
\[\frac{1}{\sqrt{3}} |1>+\frac{1}{\sqrt{3}} |2>\]
but now I'm suppose to renormalize it into a proper quantum state and somehow end up with:
\[\frac{1}{\sqrt{2}} |1>+\frac{1}{\sqrt{2}} |2>\]
How do I renormalize it?

Answer 3

Alright, I believe it's because the new state is |+> which translates to the 1/sqrt(2) part. If anyone knows for sure, please let me know.

Answer 4

the vector \(X\) is normalised \(\hat X\) , when its of unit length

\[X=\frac{1}{\sqrt{3}} |1\rangle +\frac{1}{\sqrt{2}} |2\rangle\qquad\qquad|X|=\sqrt{\left(\frac1{\sqrt 3}\right)^2+\left( \frac{2}{\sqrt 2}\right)^2}=\sqrt{\frac{7}{3}}\]

\[\hat X=\frac X{|X|}\]

check the vector is normalized
\[\langle \hat X|\hat X\rangle =1\]