How Do Spin Measurements Influence Particle States in Quantum Mechanics?

[WendysRules]

Homework Statement

A beam of spin $\frac{1}{2}$ particles is prepared in the state: $|\psi> = \frac{3}{\sqrt{34}}|+> + \frac{5i}{\sqrt{34}}|->$

a) What are the possible results of a measurement of the spin component $S_z$, and with what probabilities would they occur?
b) Suppose that the $S_z$ measurement yields the result $S_z = -\frac{\hbar}{2}$. Subsequent to that result a second measurement is performed to measure the spin component $S_x$. What are the possible results of that measurement, and with what probabilities would they occur?

Homework Equations

$P_a= |<a|\psi>|^2$

The Attempt at a Solution

For a), the possibilities are spin up, or spin down AKA $\pm \frac{\hbar}{2}$
The probability to measure it in spin up is $|<+|\psi>|^2 = (\frac{3}{\sqrt{34}})^2 = \frac{9}{34}$
The probability to measure it spin down is $|<-|\psi>|^2 =(\frac{5}{\sqrt{34}})^2 = \frac{25}{34}$

b)
The measurement in the Z-axis has no affect on the measurement on the spin for the X-axis due to them being incompatible observables. So, my thought process would be to say there is a 50% of being up/down, giving us the probability to be in spin up $\frac{25}{68}$ and spin down $\frac{25}{68}$ but I'm not if maybe they want me to use the projection postulate? But I'm not sure how to tie it in here.

Thanks for the help.

 

[kuruman]

You are correct that the probabilities in (b) should be equal, but should they not add to 1 instead of $\frac{50}{68}$?

 

[WendysRules]

You are correct that the probabilities in (b) should be equal, but should they not add to 1 instead of $\frac{50}{68}$?

The way I look at it, I have a beam that goes into an analyzer on the Z axis, then I take the spin down of that beam and send it to the analyzer on the X axis, so I can't forget that $\frac{9}{34}$ probability of being in the spin up on the Z axis. So if I include that in my calculations, I get unity ($\frac{9}{34}+\frac{50}{68}=1$ ).
Essentially what I did was, I took the probability of being spin down, and since I think it'll be a 50-50 on the x axis, I just divided my spin down probability by two to get $\frac{25}{68}$

 

[PeroK]

The way I look at it, I have a beam that goes into an analyzer on the Z axis, then I take the spin down of that beam and send it to the analyzer on the X axis, so I can't forget that $\frac{9}{34}$ probability of being in the spin up on the Z axis. So if I include that in my calculations, I get unity ($\frac{9}{34}+\frac{50}{68}=1$ ).
Essentially what I did was, I took the probability of being spin down, and since I think it'll be a 50-50 on the x axis, I just divided my spin down probability by two to get $\frac{25}{68}$

That's an interesting way to look at things! You could say, regarding any particle, there are three possibilities:

1) First measurement is z-up: probability $18/68$

2) First measurement is z-down; second measurement is x-up: $25/68$

3) First measurement is z-down; second measurement is x-down: $25/68$

And that would be a complete analysis of the experiment.

However, normally when you talk about a second measurement and ask about its probabilities, you are talking about the conditional probability given that the first measurement took a certain value.

Formally, you are resetting your sample space to only the particles that met the first criterion. That means focusing on the second experiment only. In this case, you have only two possibilities:

1) Second measurement is x-up: $1/2$

2) Second measurement is x-down: $1/2$

In any case, you'll need to understand when questioners are talking about a conditional probability, even if they do not say so explicitly.

 

[PeroK]

b)
The measurement in the Z-axis has no affect on the measurement on the spin for the X-axis due to them being incompatible observables.

I would take issue with this. For example, if the particles were initially in a state where x-up was more likely than x-down, then a measurement about the z-axis would affect the probabilities of subsequent measurements in the x-direction.

A better statement would be that a measurement about the z-axis leaves the particle in a state where a measurement of the x-spin is up or down with equal probability.

 

[WendysRules]

I would take issue with this. For example, if the particles were initially in a state where x-up was more likely than x-down, then a measurement about the z-axis would affect the probabilities of subsequent measurements in the x-direction.

A better statement would be that a measurement about the z-axis leaves the particle in a state where a measurement of the x-spin is up or down with equal probability.

Thank you for your help!