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## Homework Statement

Find the probability, [itex]P_{2a}(t)[/itex], that a measurement of the quantity [itex]A[/itex] in

the state [itex]|\varphi (t)\rangle\right[/itex] will yield the value [itex]2a[/itex].

## Homework Equations

[tex]\hat{A}|1\rangle\right = a(|1\rangle\right - i|2\rangle\right[/tex]

[tex]\hat{A}|2\rangle\right = a(i|1\rangle\right + |2\rangle\right[/tex]

[tex]\hat{A}|3\rangle\right = -2a(|3\rangle\right[/tex]

[tex]A = \[ \left( \begin{array}{ccc}

a & ia & 0 \\

-ia & a & 0 \\

0 & 0 & -2a\end{array} \right)\][/tex]

[tex]|\varphi (t)\rangle\right = \[ \left( \begin{array}{ccc}

cos(wt) \\

0 \\

-isin(wt) \end{array} \right)\][/tex]

## The Attempt at a Solution

Well, I kinda suck at finding these probabilities. So I'm not sure what to do, since it asks for [itex]2a[/itex]. Is it just:

[tex]P(2a) = \left|\langle\psi_j|\Psi\rangle\right|^2,[/tex]

where [itex]\psi_j = \varphi[/itex] and [itex]\Psi = A|3\rangle\right[/itex], or am I just not getting it ?

Regards.