Finding the probability of a mixed ensemble of particles

Question

I have a question regarding finding the probability of measuring a specific particle in a mixed ensemble. Say you have a mixed ensemble containing 30% +z particles and 70% -z.

Then the probability of taking a measurement and observing a +z particle is 30%. This makes perfect sense given the ratios of particles. But what I am confused about is the state could be represented by $$|\psi\rangle=\frac{3}{10}|+z\rangle+\frac{7}{10}|-z\rangle$$ using $|\psi\rangle=P_1|+z\rangle+P_2|-z\rangle$. But this is not a normalized function. I was under the impression that to take the probability of a quantum state the state must first be normalized. But if we normalize this state and then take the probability then we get the wrong probabilities. I am very confused about where my misunderstanding is coming from. Any help would be appreciated!

Cheers

Answer 1

You have made a simple mistake. When writing out a quantum state as a linear superposition of orthonormal states $$|\psi\rangle=\sum_na_n|n\rangle$$ The normalization condition is $$\langle\psi|\psi\rangle=\sum_n|a_n|^2=1$$ not $$\sum_n a_n=1$$

In other words, the probability of measuring state $|n\rangle$ is $P_n=|\langle n|\psi\rangle|^2$, not $P_n=\langle n|\psi\rangle$.

Therefore, your state could be $$|\psi\rangle=\sqrt{\frac{3}{10}}|+z\rangle+\sqrt{\frac{7}{10}}|-z\rangle$$

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I am a little wary of how this applies to the ensemble though. Approaching the system this way makes it seem like once you make a measurement all particles will become either spin up or spin down. But at least the probability issue you raised has been fixed here.