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## Homework Statement

Given a system initially in a state

$$| \psi \rangle = \frac{1}{\sqrt{6}} \left(|1\rangle + 2 |2\rangle + |3\rangle \right)$$

where ##{|n\rangle}_{n=1}^{8} ##form an ON basis.

If we perform a measurement of an observable corresponding to an operator

$$\hat{A}=|2\rangle\langle3|+|3\rangle\langle2|+|3\rangle 2 \langle3|.$$

What are the possible outcomes and probabilities of this measurement?

## Homework Equations

## The Attempt at a Solution

I know that a measurement will yield an eigenvalue corresponding to an eigenvector of the operator. The probability is given by ##P(n)=|\langle n |\psi \rangle|^2## if ##|\psi\rangle## is normalized. Now if I act with ##\hat{A}## on ##|\psi\rangle## I get:

$$ \hat{A} |\psi\rangle = \frac{1}{\sqrt{6}} \left(| 2 \rangle + 4 |3\rangle \right).$$ So the only possible outcomes are eigenvalues to ##| 2 \rangle ## and ##| 3 \rangle##, right?

If so, how do I calculate the probabilities?

If I normalize ##|\psi\rangle## and then calculate ##|\langle 2 | |\psi\rangle |^2 ## and ##|\langle 3 | |\psi\rangle |^2 ## it seems I miss some information about the probable outcomes contained in the operator.

This is how far I got.