Stern-Gerlach: Measurements of spin

[WWCY]

Homework Statement

Hi all, here's the problem I need help with:

Compare the following situations :

1. A beam of atoms has half of them preselected having spin up along z and the other half having spin down along z. This beam is sent through a Stern-Gerlach (SG) apparatus that sorts in the z direction. Another beam of atoms prepared in the same way is sent through an SG apparatus that sorts in the x direction.
2. Two identical beams of atoms, all of which are preselected having spin up along x axis. As in (a), one beam is sent through an SG apparatus that sorts in z while the other beam is sent through an SG apparatus that sorts in x. Do the measurement results enable you to tell apart situations (a) and (b)?

Homework Equations

The Attempt at a Solution

I'd like to know if my understanding of the problems are correct, and if not, how I should have thought about the problem (especially that of part 1). Many thanks in advance!

1) Since beam is prepared such that $P$(z spin +/-) $= 0.5$, the beam can be described by
$$\frac{1}{\sqrt{2}} (1, -1) = \frac{1}{\sqrt{2}} (1, 0) + \frac{1}{\sqrt{2}} (0 , -1)$$
which incidentally is a vector that describes particles with spin $-$ along $x$ with $P = 1$

2) Beam is prepared to be in state of spin $+$ along $x$ with $P = 1$. Thus it can be described with
$$\frac{1}{\sqrt{2}} (1, 1)$$
which means there is a $0.5$ probability of either being detected in a state of spin $+/-$ in $z$.

The difference between these situations is therefore the state in 1) being spin $-$ in $x$ and 2) being spin $+$ in $x$.

 

[kuruman]

I interpret the statement of the problem to say that in beam 1 50% of the atoms are in pure state $|+>$ and 50% in pure state $| - >$.

 

[WWCY]

Thanks for your response.

However I can't quite understand the difference between your example and mine. If we fired our beams through a detector, would we both not observe -/+ states half the time?

Also, how does the beam being separated into 2 pure states affect the x-state probability?

Thank you.

 

[kuruman]

However I can't quite understand the difference between your example and mine. If we fired our beams through a detector, would we both not observe -/+ states half the time?

I assume you are talking about part (a). It's not a matter of time. We would both observe half the atoms deflected one way and the other half the other way. However the wavefunctions of the atoms emerging from the machine in my interpretation of the wavefunctions would be different from yours. Do you see why?

Don't forget that your task here is to answer the question "Do the measurement results enable you to tell apart situations (a) and (b)?" How do you propose to proceed in order to answer it?

 

[WWCY]

I assume you are talking about part (a). It's not a matter of time. We would both observe half the atoms deflected one way and the other half the other way. However the wavefunctions of the atoms emerging from the machine in my interpretation of the wavefunctions would be different from yours. Do you see why?

I get the feeling that they are different, but I'm not sure I quantitatively understand why, could you elaborate?

Don't forget that your task here is to answer the question "Do the measurement results enable you to tell apart situations (a) and (b)?" How do you propose to proceed in order to answer it?

I'll try with the "+ in z" state. The pure + state is described by the vector $(1,0)$ . I can write this as a weighted sum for "+/- in x" states, which will give me their corresponding probability amplitudes $\alpha$ and $\beta$ in the form
$$
\alpha \frac{1}{\sqrt{2}} (1, 1) + \beta \frac{1}{\sqrt{2}} (1, -1)
$$

But then again, the beam has been "split" beforehand into pure + and - states, how do I factor in the additional "probabilities" that arise out of this fact?

Thank you for your assistance.

 

[kuruman]

get the feeling that they are different, but I'm not sure I quantitatively understand why, could you elaborate?

You have a beam consisting of 50% atoms in the $|+>$ (up) state and 50% in the $|->$ (down) state. How would you write the wavefunction of one of these atoms in terms of $|+>$ or $|->$ or both? Also, I am not sure I understand your notation. When you say an atom is in state $\frac{1}{\sqrt{2}} (1, 1)$ how would you write that in terms of $|+>$ or $|->$ or both? What about state $\frac{1}{\sqrt{2}} (1, -1)$? I need to understand what you mean by these before continuing on.

 

[WWCY]

You have a beam consisting of 50% atoms in the $|+>$ (up) state and 50% in the $|->$ (down) state. How would you write the wavefunction of one of these atoms in terms of $|+>$ or $|->$ or both? Also, I am not sure I understand your notation. When you say an atom is in state $\frac{1}{\sqrt{2}} (1, 1)$ how would you write that in terms of $|+>$ or $|->$ or both? What about state $\frac{1}{\sqrt{2}} (1, -1)$? I need to understand what you mean by these before continuing on.

From what I understand, the vector $(1,0)$ describes atoms in the $|z+>$ state, while $(0,1)$ describes atoms in the $|z->$ state.

To find out the fractions of $|z+>$ atoms that are either + or - in $x$, I would write
$$(1,0) = \frac{1}{\sqrt{2}} [\frac{1}{\sqrt{2}} (1,1) + \frac{1}{\sqrt{2}} [\frac{1}{\sqrt{2}} (1,-1) ] $$
where $\frac{1}{\sqrt{2}}(1,\pm 1)$ denotes states $|x\pm >$

This means that there is a 50-50 chance of finding $|z+>$ atoms in either $|x+>$ or $|x->$ states.

I could then do the same calculation for the $|z->$ states, which gives
$$(0,1) = \frac{1}{\sqrt{2}} [\frac{1}{\sqrt{2}} (1,1) - \frac{1}{\sqrt{2}} [\frac{1}{\sqrt{2}} (1,-1) ]$$

This means that there is a 50-50 chance of finding $|z->$ atoms in either $|x+>$ or $|x->$ states.

So from the initial beam prepared such that .5 of the atoms were either in $|z+>$ or $|z->$, 50% of $|z+>$ atoms will be found in a $|x+>$ state. 50% of $|z->$ atoms will also be found in a $|x+>$ state

This my understanding thus far.

Thank you for your patience!

 

[kuruman]

OK, thanks, I understand your notation.

This means that there is a 50-50 chance of finding $|z+>$ atoms in either $|x+>$ or $|x->$ states.

"Finding" them how?
1. Consider case 1 and let's say you allow only one of the atoms in state $|z+>$ to go through a SG machine.
What are the possible outcomes and their associated probabilities when
(a) the SG machine sorts along the z-direction?
(b) the SG machine sorts along the x-direction?

2. Now consider case 2 and let's say you allow only one of the atoms in state $|x+>$ to go through a SG machine.
What are the possible outcomes and their associated probabilities when
(c) the SG machine sorts along the z-direction?
(d) the SG machine sorts along the x-direction?

Note: The SG machine in this idealized context has a single port on one end through which all particles enter and two ports at the other end through which particles exit. One of the exit ports is marked "+" and the other is marked "-". A display next to each port shows the percentage of incoming particles that exit through that port. Naturally what goes in must come out, so that the sum of the displayed percentages must be 100% at all times.

Once you understand how this works, all you have to do is consider what will happen when you have beams of many atoms entering the SG machine.

 

[WWCY]

OK, thanks, I understand your notation.
1. Consider case 1 and let's say you allow only one of the atoms in state $|z+>$ to go through a SG machine.
What are the possible outcomes and their associated probabilities when
(a) the SG machine sorts along the z-direction?
(b) the SG machine sorts along the x-direction?

2. Now consider case 2 and let's say you allow only one of the atoms in state $|x+>$ to go through a SG machine.
What are the possible outcomes and their associated probabilities when
(c) the SG machine sorts along the z-direction?
(d) the SG machine sorts along the x-direction?

1a) The atom will exit out of of the "+" port
1b) Probability of atom exiting out of either port is .5

2a) The vector representing $|x+>$ can be decomposed into
$$
\frac{1}{\sqrt{2}} (1,1) = \frac{1}{\sqrt{2}}(1,0) + \frac{1}{\sqrt{2}}(0,1)
$$
Thus probability of atom exiting out of either port is .5
2b) Atom exits out of "+" port.

Is this right so far?

Also, going back to this in post #4,

However the wavefunctions of the atoms emerging from the machine in my interpretation of the wavefunctions would be different from yours

Was my example different from yours because $\frac{1}{\sqrt{2}} (1, -1) = \frac{1}{\sqrt{2}} (1, 0) + \frac{1}{\sqrt{2}} (0 , -1)$ represents an atom in a superposition of "+z" and "-z" states rather than being purely in either?

Thank you for your guidance

 

[kuruman]

Is this right so far?

This is all correct. It seems you have a good grasp of what's going on.

Was my example different from yours because $\frac{1}{\sqrt{2}} (1, -1) = \frac{1}{\sqrt{2}} (1, 0) + \frac{1}{\sqrt{2}} (0 , -1)$ represents an atom in a superposition of "+z" and "-z" states rather than being purely in either?

Yes, that's the difference.

So now can you answer the original question that you posted? You have two "experiments" in case 1 and two in case 2, a total of four. For each of the four experiments figure out what percentage come out of the "+" port (no need to figure out the "-" port) and see what's different (if anything) between the two cases in terms of these percentages.

 

[WWCY]

So now can you answer the original question that you posted? You have two "experiments" in case 1 and two in case 2, a total of four. For each of the four experiments figure out what percentage come out of the "+" port (no need to figure out the "-" port) and see what's different (if anything) between the two cases in terms of these percentages.

For the first experiment, the fraction of atoms exiting from the "+x" port is 0.5. For the second, all the atoms exit from the "+x" port.

In the z direction however, we won't be able to tell the difference.

Is this right?

Thank you for your assistance.

 

[kuruman]

For the first experiment, the fraction of atoms exiting from the "+x" port is 0.5. For the second, all the atoms exit from the "+x" port.

In the z direction however, we won't be able to tell the difference.

The SG machine used here as an analyzer has no +x or +z port because it does not know the alphabet. It has a single axis (like a polarizer for light) with two holes along it, one marked "+" and the other marked "-". Particles emerging from the + port have magnetic moments parallel to the axis and particles emerging from the - port have magnetic moments antiparallel to the axis. Let's say you choose for reference to paint a z-axis in your lab in the up and down direction as defined by gravity. You also paint an x-axis perpendicular to that direction. Once you have defined x and z, the y-axis follows from the right hand rule. Now let's look at the cases. Note that you have to use a "polarizer" SG machine other than the 'analyzer" SG machine mentioned in the question in order to prepare the beam as specified by the question.

Case 1
To prepare a beam as needed for case 1, you use a "polarizer" SG machine with its axis parallel to the z-axis.. The atoms that emerge from both the + and minus ports are recombined into a single beam and sent through the analyzer. In part (a) of case 1, the analyzer is along the z-axis and in part (b) of case 1 the analyzer is along the x-axis.
Case 2
To prepare a beam as needed for case 2, you use a "polarizer" SG machine with its axis parallel to the x-axis. The atoms that emerge from the + port only are sent through the analyzer. In part (c) of case 2, the analyzer is along the z-axis and in part (d) of case 2 the analyzer is along the x-axis.

So my question requires you to list the percentages of particles emerging from the + port in each of the cases a - d. This means providing four numbers. Then we will look at the results and decide whether cases 1 and 2 are different or the same.

 

[WWCY]

The SG machine used here as an analyzer has no +x or +z port because it does not know the alphabet. It has a single axis (like a polarizer for light) with two holes along it, one marked "+" and the other marked "-". Particles emerging from the + port have magnetic moments parallel to the axis and particles emerging from the - port have magnetic moments antiparallel to the axis. Let's say you choose for reference to paint a z-axis in your lab in the up and down direction as defined by gravity. You also paint an x-axis perpendicular to that direction. Once you have defined x and z, the y-axis follows from the right hand rule. Now let's look at the cases. Note that you have to use a "polarizer" SG machine other than the 'analyzer" SG machine mentioned in the question in order to prepare the beam as specified by the question.

Case 1
To prepare a beam as needed for case 1, you use a "polarizer" SG machine with its axis parallel to the z-axis.. The atoms that emerge from both the + and minus ports are recombined into a single beam and sent through the analyzer. In part (a) of case 1, the analyzer is along the z-axis and in part (b) of case 1 the analyzer is along the x-axis.
Case 2
To prepare a beam as needed for case 2, you use a "polarizer" SG machine with its axis parallel to the x-axis. The atoms that emerge from the + port only are sent through the analyzer. In part (c) of case 2, the analyzer is along the z-axis and in part (d) of case 2 the analyzer is along the x-axis.

So my question requires you to list the percentages of particles emerging from the + port in each of the cases a - d. This means providing four numbers. Then we will look at the results and decide whether cases 1 and 2 are different or the same.

I'm not sure I fully understand what you mean, but I'll try to answer anyway.

a) 50% of atoms pass through "+" , analyser along z
b) 50% of atoms pass through "+" , analyser along x

c) 50% through "+", analyser along z
d) 100% through "+", analyser along x

Is this what you meant?

Thank you for your patience.

 

[kuruman]

Is this what you meant?

This is exactly what I meant and your analysis is 100% correct. Now can you answer the original question, Do the measurement results enable you to tell apart case 1 and case 2?

 

[WWCY]

This is exactly what I meant and your analysis is 100% correct. Now can you answer the original question, Do the measurement results enable you to tell apart case 1 and case 2?

Yes, the results are distinct. But I still find something you say quite confusing.

The SG machine used here as an analyzer has no +x or +z port because it does not know the alphabet.

When you say this, do you mean that there isn't an all-in-one detector that sorts out "+x" or "+z" atoms, but rather we need to manually align the analyser's axis according to the axes we (properly) pre-defined as x, y and z?

Thank you!

 

[kuruman]

When you say this, do you mean that there isn't an all-in-one detector that sorts out "+x" or "+z" atoms, but rather we need to manually align the analyser's axis according to the axes we (properly) pre-defined as x, y and z?

That is correct. The atomic moments line up parallel or antiparallel to the magnetic field (its direction being the "axis" of the machine") whilst a gradient in the magnetic field pulls the "up" moments in one direction and the "down" moments in the opposite direction. The poles of the magnet are shaped so that the dominant component of the field gradient is in the same direction as the field itself. Thus, the machine has only one preferred direction.

You will find a wonderful SG simulation here
https://phet.colorado.edu/en/simulation/legacy/stern-gerlach
It is what I had in mind when I described what the idealized version looks like and what it does. Play with it and enjoy.

 

[WWCY]

That is correct. The atomic moments line up parallel or antiparallel to the magnetic field (its direction being the "axis" of the machine") whilst a gradient in the magnetic field pulls the "up" moments in one direction and the "down" moments in the opposite direction. The poles of the magnet are shaped so that the dominant component of the field gradient is in the same direction as the field itself. Thus, the machine has only one preferred direction.

You will find a wonderful SG simulation here
https://phet.colorado.edu/en/simulation/legacy/stern-gerlach
It is what I had in mind when I described what the idealized version looks like and what it does. Play with it and enjoy.

Thank you very much for your guidance and the link to the sim, really appreciate it!