Sequential Stern-Gerlach experiment to differentiate between two samples

[felicity87]

Homework Statement

Compare the following situations (a) and (b):
(a) A beam of atoms, half of which are preselected as “+ in z”, the
other half as “− in z”, is sent through a Stern-Gerlach apparatus
that
...(A) sorts in the z direction.
...(B) sorts in the x direction.

(b) A beam of atoms, all of which are preselected as “+ in x”, is sent
through a Stern-Gerlach apparatus that
...(A) sorts in the z direction.
...(B) sorts in the x direction.

Homework Equations

| x+ > = $$\frac{1}{\sqrt{2}}$$ | z+ > + $$\frac{1}{\sqrt{2}}$$ | z- >

| z+ > = $$\frac{1}{\sqrt{2}}$$ | x+ > + $$\frac{1}{\sqrt{2}}$$ | x- >

| z- > = $$\frac{1}{\sqrt{2}}$$ | x+ > - $$\frac{1}{\sqrt{2}}$$ | x- >

\\Edit :The Question is whether we can differentiate between situation (a) and (b) by knowing the results of the experiment alone. \\

The Attempt at a Solution

I would think that the measurement results do not enable us to tell the situation apart.
In (a), (A) will results in 50:50 selection and x-selector (B) will sort +/- x atoms 50:50.
In (b), (A) will results in 50:50 selection and x-selector (B) will sort now +/- z atoms 50:50.

However, I found the answer to be 'yes'. I am thinking that we can only tell the situation apart when there was no measurement done in (A) stage.
Can someone explain?

With Thanks,
Felicity

 

[muppet]

Sorry... having described your two experimental set-ups, what's the actual question? What will the outcomes of the two arrangements be?

 

[felicity87]

Sorry. The Question is whether we can differentiate between situation (a) and (b) by knowing the results of the experiment alone.

 

[muppet]

Hmm. Well I've slept on the question... and have come to the conclusion that I don't know the answer There's one suggestion I can make, however, that sounds sensible to me but shouldn't be accepted uncritically. (Hence, I'll be less oblique than I usually am in homework help, or else I might send you looking in completely the wrong direction!)
The problem might really be about the time evolution of the system. You don't have to have a 50:50 sum of eigenstates in order to be in a superposition of states; and all that an eigenstate of z tells you about the x- and y- components is that they aren't eigenstates. Thus, it might be the case that the probability of getting spin -up or -down in the x-direction given that they were previously in an up-eigenstate of x is different to the case in which nothing about the history of the x-component is known.
Sorry I can't be more conclusive, but I hope that helps!

 

[felicity87]

Muppet,
I somehow see the point about probabilites being different. I have not learned about the superposition of state yet so I am still lost.
Thank you all the same!

 

[muppet]

From your post above I can see that you have- you just don't know it!
Terminology: An eigenfunction of an operator is a function that, when operated upon by that operator, gives you itself mulitplied by some constant number called an eigenvalue. If a wave function is an eigenfunction of an operator corresponding to a physical quantity (an observable), then it is said to be in an eigenstate of that quantity.
If a system is described by a wavefunction that is a linear combination of eigenstates (like your spin states above) then it is said to be in a superposition of states.

 

[felicity87]

Okay. Ah, well..

 

[muppet]

Anything else I can help with as I'm online??
The other suggestion I could make is that if you need help with a conceptual point you don't properly understand, you can always raise it in the QM forum. A lot more people check that.

 

[felicity87]

Hihi..
Thanks for your offer! It is much better now :D
Btw, I'd like to close my question. I found out that the question was rather misleading.

I initially thought
"(a) A beam of atoms, half of which are preselected as “+ in z”, the
other half as “− in z”, is sent through a Stern-Gerlach apparatus
that
...(A) sorts in the z direction.
...(B) sorts in the x direction."
means
Source ---> [ (A) + (B) ] ---- measurement result since it is a Stern-Gerlach apparatus.

Apparently, the correct interpretation is
Source ---> [A]
Source ---> ---- two measurement results.

Then, we can tell the situation (a) and (b) apart.