imagine that you have a box that emits quantons that have a definite but unknown

Question

imagine that you have a box that emits quantons that have a definite but unknown spin state. if we run quantons from this box through an SGz device , we fin that 20% of the electrons come out the (+) channel and 80% from the (-) channel . if we run quantons from the same box through an SGx device, we find that 50% of the electrons come out of each channel . if we run quantons from the box through an SGy device, we find that 90% come out the (+)channel and 10% out of the (-)channel . Find a quantum state vector for quantons emerging from the box that is consistent with these data. make sure your vector is normalized.

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

I think you switched y and x in your question.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.