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Question

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imagine that you have a box that emits quantons which have a

definite but unknown spin state. if we run quantons from this box

through an SGz device , we find that 9/13 of the electrons come out

the (+) channel and 4/13 from the (-) channel . if we run quantons

from the same box through an SGy device, we find that 1/26 of the

electrons come out of the (+) channel and 25/26 out of the (-)

channel. if we run quantons from the box through an SGx device , we

find that 1/2 come out of each channel.

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a) find a quantom state vector for quantons emerging from the box

that is constant with these data. (Just find any vector that is

normalized and consistant with the experimental data

above)

b) breafly explain why the state

vector you got for (a) is not the only possible state vector. (In

other words, show that you cannot determine the state completely

(uniquely) from above experimental data.)

Explanation / Answer

Even if it is not said, these are spin 1/2 particles, since we only have 2 possible outcomes for a spin measurement, so the Hilbert space of the problem is simply 2-dimensional.

I will use Dirac's notation.

We want to find the state vector |psi>. Let |+z> and |-z> be the 2 two spin eigenstates for a measurement along z. Of course, they form a complete basis for the Hilbert space, so we can decompose |psi> on them:
|psi> = |+z> <+z|psi> + |-z> <-z|psi>

But we know that the probability of measuring a spin up along z (that is corresponding to |+z>) is just the modulus square of the amplitude:
|<+z|psi>|^2 = 20/100 = 1/5.
Similarly
|<-z|psi>|^2 = 80/100 = 4/5.

Then (up to an undetermined global phase factor which doesn't affect any prediction) we have
|psi> = (|+z> + 2 |-z>)/sqrt(5)

Now consider that for a measurement along x we have eigenstates:
|+x> = (|+z> + |-z>)/sqrt(2)
|-x> = (|+z> - |-z>)/sqrt(2)

You can verify that the probabilities are
|<+x|psi>|^2 = 90/100
|<-x|psi>|^2 = 10/100

Similarly for the measurement along y:
|+y> = (|+z> +i |-z>)/sqrt(2)
|-y> = (|+z> -i |-z>)/sqrt(2)
|<+y|psi>|^2 = 50/100
|<-y|psi>|^2 = 50/100

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