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In Quantum Mechanics, McIntyre states the projection postulate like so:

After a measurement of $A$ that yields the result $a_n$, the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement:

$$\left|\psi^\prime\right> = \frac{P_n\left|\psi\right>}{\sqrt{\left<\psi\right|P_n\left|\psi\right>}}.$$

Doesn't this then mean that $$\left|\psi^\prime\right> = \frac{P_n\left|\psi\right>}{\sqrt{\left<\psi\right|P_n\left|\psi\right>}} = \frac{\left|a_n\right>\left<a_n\mid\psi\right>}{\sqrt{\left<\psi\right|P_n\left|\psi\right>}} = \textrm{(some number)}\left|a_n\right>\;?$$

Because if it's so, then that number must be some phase (right?) and then we might as well say that the new state is just the eigenstate $\left|a_n\right>$ corresponding to the result $a_n$. So why isn't the postulate stated that way?

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The key point is

[...] onto the ket (or kets) [...]

When you measure observable $A$ and get result $a_n$, the corresponding eigenspace may have more dimensions than just one, i.e. you cannot speak of "the eigenstate $\left|a_n\right>$ corresponding to the result $a_n$. Hence, you really need to project the original state onto the full eigenspace.

For example, the states of the hydrogen atom are usually labelled as $\left|nlm\right>$, and if you measure energy eigenvalue $E_n$, you still have quite a number of states indexed my $l$ and $m$ to project onto.

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For a single particle I believe what you wrote is perfectly true, and the resulting state of the measure is indeed a phase times the eigenvector (although this phase does matter if your state is in a superposition, so you can't immediatly trash it).

I think maybe the principle was written this way however because it still works directly when you write a multi particle state and you only project on one particle, though your way of expressing the principle still holds true if one is careful in considering the eigenvectors of the operator acting on the whole system.

Still, both those formulations should be correct.

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