Another thought experiment about spin 1/2

[Christian Thom]

TL;DR Summary

Let's take a beam of spin 1/2 particles prepared in the state |up> in the Z direction, let's pass it through a Stern& Gerlach apparatus in the X direction and then directly in another inverted S&G to reunite the two beams. My question is : will we get only spin |up> particles at the exit of the system ?

Let's take a beam of spin 1/2 particles prepared in the state |up> in the Z direction, let's pass it through a Stern& Gerlach apparatus in the X direction to get two beams of spin |right> and spin |left>, and then redirect these two beams directly in another inverted S&G to reunite them in one unique beam. My question is : will we get only spin |up> particles in this beam ?

 

[PeroK]

Summary:: Let's take a beam of spin 1/2 particles prepared in the state |up> in the Z direction, let's pass it through a Stern& Gerlach apparatus in the X direction and then directly in another inverted S&G to reunite the two beams. My question is : will we get only spin |up> particles at the exit of the system ?

Let's take a beam of spin 1/2 particles prepared in the state |up> in the Z direction, let's pass it through a Stern& Gerlach apparatus in the X direction to get two beams of spin |right> and spin |left>, and then redirect these two beams directly in another inverted S&G to reunite them in one unique beam. My question is : will we get only spin |up> particles in this beam ?

Yes.

 

[Christian Thom]

Thank you.

 

[sandy stone]

Yes.

Would you mind elaborating on this just a bit? It was always my understanding that a beam of left- (or right-) spin particles would be a 50-50 mix of up- and down- spin particles, regardless of the preparation before the left-right split. What am I missing?

 

[PeroK]

Would you mind elaborating on this just a bit? It was always my understanding that a beam of left- (or right-) spin particles would be a 50-50 mix of up- and down- spin particles, regardless of the preparation before the left-right split. What am I missing?

It's not a mix, it's a superposition of spin states that equate to z-spin-up. If the spatially separated superposition is perfectly recombined, then you still have a beam of z-spin-up particles.

If, however, you pass either spatially separated component of the superposition through a z-aligned SG magnet and measure the outcome, you will find a 50-50 split representing z-spin-up and z-spin-down (i.e. either x-spin up in one component and x-spin-down in the other).

 

[sandy stone]

Thanks for the clarification. Back to QM kindergarten I go.

 

[PeroK]

Thanks for the clarification. Back to QM kindergarten I go.

There's an excellent presentation of these ideas here (see chapter 6 and section 6.5 in particular):

http://physics.mq.edu.au/~jcresser/Phys304/Handouts/QuantumPhysicsNotes.pdf

 

[Christian Thom]

Yes.

How this result depends on the equality of the path lengths of the two separated beams ?

 

[PeroK]

How this result depends on the equality of the path lengths of the two separated beams ?

It's a thought experiment because the recombination would have to be perfect. I don't believe it's a experiment that is practically achievable.

 

[vanhees71]

That reminds me of the "Humpty Dumpty Papers" by Schwinger et al.

https://link.springer.com/article/10.1007/BF01909939
https://link.springer.com/article/10.1007/BF01384847
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.40.1775

Of course you cannot do such experiments with the original SGE, because it's impossible to make the magnets as accurate as needed, but in a sense a setup like a Penning trap is a kind of SGE too.

Another modern realization of a coherent-spin experiment is here:

https://www.nature.com/articles/ncomms3424?origin=ppub

 

[Christian Thom]

It's a thought experiment because the recombination would have to be perfect. I don't believe it's a experiment that is practically achievable.

I would think that there must be some leeway, otherwise free beams of definite spin particles would loose more or less quickly their spin definiteness due to spurious magnetic gradients that are always present in actual experiments.

 

[PeroK]

I would think that there must be some leeway, otherwise free beams of definite spin particles would loose more or less quickly their spin definiteness due to spurious magnetic gradients that are always present in actual experiments.

You'll need to explain what you mean by that.

Note that the SG experiment itself is fuzzy rather than precisely clear-cut.

 

[Christian Thom]

You'll need to explain what you mean by that.

Note that the SG experiment itself is fuzzy rather than precisely clear-cut.

If you consider a beam like the one at the origin of the experiment above, in absence of S&G devices, the slightest natural magnetic gradients would play the same role as the S&G device, and if the recombination was so difficult to obtain, that would result in the loosing of the spin purity of the original beam after some propagation. This effect would have surely already been observed (maybe it has ?).

 

[PeroK]

If you consider a beam like the one at the origin of the experiment above, in absence of K&G devices, the slightest natural magnetic gradients would play the same role as the K&E device, and if the recombination was so difficult to obtain, that would result in the loosing of the spin purity of the original beam after some propagation. This effect would have surely already been observed (maybe it has ?).

What you are describing is simply state evolution. You would gradually lose the spatial coherence of the beam.

For SG, you need a powerful magnet with a specific varying magnetic field to create a superposition of spatially distinct wavefunctions. Putting those together by perfectly reversing that separation process is theoretically possible but practically very difficult.

 

[PeroK]

The SG experiment requires a very specific and controlled magnetic field. Just any weak background field won't do.

 

[Christian Thom]

Thank you all.