# Stern-Gerlach "interferometry"

• I
• WWCY
In summary: The former will happen with a probability of ##p_1## and the latter with a probability of ##p_2 = 1-p_1##.So, the question is: What is ##p_1##?Your guess is that it is 1/2. That would be the case if the particle were in a fully superposed state as it emerged from the S-G device. Since it is not, you need to calculate the probabilities from the wavefunction. Since you have not specified the state of the particle at the S-G device, this is impossible.But, since the state of the particle is known after the S-G device, you can calculate the
WWCY
TL;DR Summary
State passing through Stern-Gerlach magnet w/o projective measurement. How to treat it?
Suppose we have a state ##|\psi\rangle \sim |\uparrow z\rangle + |\downarrow z\rangle## passing through a Stern-Gerlach magnetic field oriented in the ##\hat{z}## direction such that ##|\uparrow z\rangle##'s are pulled up and ##|\downarrow z\rangle##'s are pulled down. Then suppose we place some operation ##M_\uparrow## on the top path and ##M_\downarrow## on the bottom path and both paths are later recombined.

If we treated the whole set-up as some kind of black-box where we don't obtain read-outs for ##M_{\uparrow/\downarrow}##, what state would we observe post-recombination?

Would it be ##|\psi\rangle \sim M_\uparrow|\uparrow z\rangle + M_\downarrow|\downarrow z\rangle##? Or even ##|\psi\rangle \sim M_\uparrow|\uparrow z\rangle|\text{u}\rangle + M_\downarrow|\downarrow z\rangle|\text{d}\rangle##, where ##u,d## represent "path degrees-of-freedom".

My guess is that it's the former since ##|\uparrow z\rangle## "intrinsically" specifies the upwards path in such a setup. But I don't really have anything concrete to check this guess against. Assistance would be greatly appreciated!

For your operations ##M_{\uparrow}## and ##M_{\downarrow}## you can take some Stern-Gerlach experiment with the field oriented in some other direction.

What happens by the magnet in the SGE is that the spin component in direction of the field gets entangled with the component of the particle's position. Dynamically this happens, because the magnetic field leads to the rapid precession of the spin components perpendicular to this field, so that their dynamical effect during the particle motion is (almost) irrelevant. That's why in the two partial beams the spin component in direction of the field is (almost) perfectly determined, while the two perpendicular components are maximally indetermined. This is independent of the initial state the particle enters the field.

Now the same holds true for a 2nd SGE performed on one of the beams with determined ##s_z##-component. Say, you direct its field in ##x##-direction. Again the magnet will separate the beam in two partial beams with determined spin components, but this time of course ##s_x## is determined. All you can say from the preparation in a given ##s_z=\pm \hbar/2## state is that the probability with which a particular particle will end up with ##s_x=\hbar/2## is 1/2 and to end up with ##s_z=-\hbar/2## is also 1/2, but now the ##s_x## component is entangled with the ##x## component of position and now the ##s_z## component is completely indetermined.

Thanks for the response.

When you say
vanhees71 said:
What happens by the magnet in the SGE is that the spin component in direction of the field gets entangled with the component of the particle's position
Do you mean that any initial state expanded in the ##\hat{e}## basis: ##|\psi\rangle = \alpha_1 |\uparrow e\rangle + \alpha_2 |\downarrow e\rangle## is mapped to the state* ##|\psi '\rangle =\alpha_1 |\uparrow e\rangle|+\hat{e}\rangle + \alpha_2 |\downarrow e\rangle|-\hat{e}\rangle ## after it passes through a SG magnet with a field along the ##\hat{e}## axis? Would your suggestion of applying a successive SG magnet oriented along axis ##\hat{x}## (say on the ##+\hat{e}## path) then be tantamount to applying this map again but in terms of the particle's spin components in the ##\hat{x}## direction?

*I am using ##\pm \hat{e}## to refer to position components.

Cheers.

I get the feeling from both of your posts that you are fundamentally misunderstanding something about QM here.

For example, a state is a vector. In your OP you say:

WWCY said:
Suppose we have a state ##|\psi\rangle \sim |\uparrow z\rangle + |\downarrow z\rangle##

That's like saying:

Suppose we have a vector ##\vec{v} \sim \hat{x} + \hat{y}##

Which doesn't make a lot of sense to me.

And, when you say:

WWCY said:
*I am using ##\pm \hat{e}## to refer to position components.

What are "position components" in the context of quantum spin?

Thanks for the response. Perhaps I should have worded my question better:

PeroK said:
Suppose we have a vector ##\vec{v} \sim \hat{x} + \hat{y}##

I should have wrote it as ##|\psi\rangle = a_1|\uparrow \rangle + a_2|\downarrow \rangle##

PeroK said:
What are "position components" in the context of quantum spin?

By this I meant the path the particle takes after it passes through the SG magnet, not a position-representation of spin. So ##\{ |+\hat{e}\rangle, |-\hat{e}\rangle \}## are states representing the path the particle takes after the SG magnet. If ##|+\hat{e}\rangle##, the particle is deflected upwards along ##\hat{e}##, if ##|-\hat{e}\rangle## it is deflected downwards. My question in my preceding post would then be whether or not the spin components are entangled with the path "states" in the manner I wrote.

Cheers.

WWCY said:
My question in my preceding post would then be whether or not the spin components are entangled with the path "states" in the manner I wrote.

Cheers.

That's not what entanglement means. The full wave-function of an electron is a combination of its position-space wave-function and its spin state.

In the case of SG the electron is either in state ##\psi_1 \chi_+## or state ##\psi_2 \chi_-##. Where ##\psi## is a position-space wave-function and ##\chi## is a spin state.

PeroK said:
That's not what entanglement means. The full wave-function of an electron is a combination of its position-space wave-function and its spin state.

In the case of SG the electron is either in state ##\psi_1 \chi_+## or state ##\psi_2 \chi_-##. Where ##\psi## is a position-space wave-function and ##\chi## is a spin state.
Do ##\psi_{1/2}## refer to distinct positions?

WWCY said:
Do ##\psi_{1/2}## refer to distinct positions?

They are the two position-space wave-functions associated with the two trajectories out of the SG apparatus.

WWCY said:
If we treated the whole set-up as some kind of black-box where we don't obtain read-outs for ##M_{\uparrow/\downarrow}##...
That bit about not obtaining a readout suggests that you are thinking of the ##M## devices as macroscopic devices that could in principle tell us which path the particle takes. If so...

As @vanhees71 says above, the interaction with the S-G device entangles the particle spin and the particle position. Thus, the the particle emerges in a state ##|\psi_{\uparrow} \chi_+\rangle+|\psi_\downarrow \chi_-\rangle## where the two ##\chi## functions are "positive spin" and "negative spin" and the two ##\psi## functions are "deflected up" and "deflected down". (There are some pitfalls in treating a continuous observable such as position in this manner, but we can get away with ignoring them here).

You have placed the two ##M## devices at different positions, so the interaction with either one will be a position measurement. This will collapse the wavefunction to one of "spin up on the upper path and affected by ##M_{\uparrow}##" and "spin down on the lower path and and affected by ##M_\downarrow##". As with all quantum mechanical observations, whether we actually get a readout from the ##M## is irrelevant - the interaction itself collapses the wave function.

Now when you recombine the beams you'll have prepared a mixed state (a bunch of particles half in one state and half in the other) instead of a superposition. This mixed state cannot be written as a state vector at all - you'll need a density matrix.

This thought experiment is somewhat analogous to a double-slit experiment with detectors at the slits; the superposition of paths is eliminated by the interaction with the detectors.

WWCY, PeroK and vanhees71
Thanks a lot for the comments, I'm starting to see the bigger picture now.

I still have a question regarding the position-spin wavefunction as I've never really learned it properly.

If I start with the initial electron wavefunction ##\phi = \psi_1 \chi_+##, how would I write the Hamiltonian that governs the time-evolution of the electron as it passes through the SG magnet? i.e. How do I show that ##\phi## propagates in the ##+z## direction (and ##-z## for a state with opposite spin)?Also,

Nugatory said:
Now when you recombine the beams you'll have prepared a mixed state (a bunch of particles half in one state and half in the other) instead of a superposition. This mixed state cannot be written as a state vector at all - you'll need a density matrix.

Does this mean that entire setup can be represented by a completely positive (CP), trace-preserving map that is a sum of two CP maps?

Cheers.

It's
$$\hat{H}=\frac{1}{2M} \hat{\vec{p}}^2 + \frac{g \mu_{\text{B}}}{\hbar} \hat{\vec{s}} \cdot \vec{B}(\hat{\vec{x}}),$$
where
$$\mu_{\text{B}}=\frac{e}{2m} \hbar>0$$
is the Bohr magneton (the additional - sign in front of the potential energy is due to the fact that in the silver atom the magnetic moment comes from an electron with a negative charge ##q_{\text{e}}=-e##). The gyro-factor of the electron ##g \simeq 2## (with small deviations due to radiative QED corrections. The precise value of the gyro-factor of the electron is in fact among the most accurate predictions of the Standard Model).

## What is Stern-Gerlach interferometry?

Stern-Gerlach interferometry is an experimental technique used to measure the magnetic moment of an atom or molecule. It involves passing a beam of particles through a non-uniform magnetic field and observing the resulting deflection of the beam.

## How does Stern-Gerlach interferometry work?

In Stern-Gerlach interferometry, a beam of particles is passed through a series of magnets with alternating magnetic fields. The particles are deflected by the magnetic fields, and the resulting pattern of deflection can be used to determine the magnetic moment of the particles.

## What is the significance of Stern-Gerlach interferometry?

Stern-Gerlach interferometry is significant because it allows for the precise measurement of the magnetic moment of particles, which is an important property in quantum mechanics. It has also been used to study the quantum behavior of atoms and molecules.

## What are some applications of Stern-Gerlach interferometry?

Stern-Gerlach interferometry has been used in a variety of applications, including studying the magnetic properties of materials, measuring the spin of particles, and investigating quantum effects in atomic and molecular systems.

## What are the limitations of Stern-Gerlach interferometry?

One limitation of Stern-Gerlach interferometry is that it can only measure the magnetic moment of particles that have a non-zero magnetic moment. It also requires precise alignment of the magnetic fields, which can be challenging to achieve. Additionally, the technique may be affected by external factors such as temperature and pressure.

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