Experimental realization of Stern-Gerlach SGy filter

[SpectraCat]

I have been reviewing Sakurai's treatment of the SG experiment by analogy with polarized light, and I realized that I am not sure that I really understand how to construct in a laboratory the [tex]SG_{y}[/tex] filter to produce and separate the [tex]\left|S_{y};\pm\right\rangle[/tex] states that are analogous to right and left circularly polarized light.

Taking Sakurai's definitions of the lab frame axes (which I believe are standard), the z and x directions are perpendicular to the direction of the beam of atoms, which must therefore define the y-axis in the lab frame. I can imagine creating a magnetic field gradient along the beam direction that will interact with the atoms by drilling holes through the poles of a normal SG magnet pair. What I am unclear on is what precisely happens when a previously prepared beam of atoms in the [tex]\left|S_{z};+\right\rangle[/tex] state is passed through this filter. I guess that the two [tex]\left|S_{y};\pm\right\rangle[/tex] components of the beam are retarded and accelerated along the beam direction? This would be tricky to observe with a continuous beam, but one could imagine using a pulsed beam, or measuring differential transit times for single atoms to observe this behavior.

So, does anyone know of a reference that describes this experiment (I assume it has been conducted in some form)? Or have I made a mistake somewhere in my predictions of what would happen?

 

[Maxim Zh]

I guess that the two [tex]\left|S_{y};\pm\right\rangle[/tex] components of the beam are retarded and accelerated along the beam direction?

It looks like different refraction coefficients. So oblique incidence could help.

Let's consider a particle with magnetic moment

[tex]
\mathbf{m} = (0, m_y, 0),
[/tex]

which is in nonhomogeneous magnetic field

[tex]
\mathbf{H} = (H_0(x+y), H_0(x+y), 0).
[/tex]

Then the force will be

[tex]
\matbf{F} = (\mathbf{m} \nabla)\matbf{H} = m_y\frac{\partial}{\partial y}\matbf{H} =
H_0 m_y(1, 1, 0).
[/tex]

The beam deviation along the x-axis depends on magnetic moment direction.

But such magnetic field will also turn the dipoles. In quantum mechanics it's equivalent to a perturbation which can stimulate a spin state change.

 

[Entropia]

I am happy to provide a response to your inquiry about the experimental realization of the Stern-Gerlach SGy filter.

Firstly, it is important to understand that the Stern-Gerlach experiment is a fundamental experiment in quantum mechanics that demonstrates the concept of spin and its quantization. In this experiment, a beam of particles with a specific spin is passed through a magnetic field gradient, which causes the particles to be deflected in different directions depending on their spin state.

To construct a SGy filter in a laboratory, one would need to create a magnetic field gradient along the direction of the beam of atoms. This can be achieved by using a pair of magnets with a gap in between, as you have correctly mentioned. The direction of the magnetic field gradient would determine the direction of the y-axis in the lab frame.

When a beam of atoms in the \left|S_{z};+\right\rangle state is passed through this filter, the atoms will experience a force in the y-direction, depending on their spin state. This is because the magnetic field gradient will interact with the magnetic moment of the atoms, which is proportional to their spin.

In this case, the \left|S_{z};+\right\rangle state will experience a force in the positive y-direction, while the \left|S_{z};-\right\rangle state will experience a force in the negative y-direction. This will result in the separation of the two spin states, with the \left|S_{z};+\right\rangle state being deflected upwards and the \left|S_{z};-\right\rangle state being deflected downwards.

To observe this behavior, one could use a pulsed beam of atoms or measure the differential transit times for single atoms, as you have suggested. This experiment has indeed been conducted in various forms, and there are several references that describe it in detail.

I would recommend looking into the original Stern-Gerlach experiment and its variations, as well as the work of other scientists who have conducted similar experiments. This will provide you with a better understanding of the construction and operation of the SGy filter in a laboratory setting.

In conclusion, the SGy filter is a fundamental tool in quantum mechanics that allows us to separate spin states of particles. Its experimental realization has been extensively studied and documented, and I am confident that with further research, you will be able to gain a deeper understanding of its construction and operation.