Realistic Stern-Gerlach experiment
One of the most puzzling properties of quantum spin, the intrinsic angular momentum of a particle, is the fact that, once a spatial orientation of spin is measured, it can take only some discrete values. This is in contrast with classical angular momentum that can have a continuous distribution. People say that spin is quantized.
The most common measurement procedure is probably by using a Stern-Gerlach apparatus. This measurement consists in sending a beam of particles through an inhomogeneous magnetic field and observing their deflection. With spin-1/2 particles, the result will be that two beams are formed, with some particles deflecting in one direction of the inhomogeinity axis, the others in the opposite direction.
This behavior is described within the formulation standard of quantum mechanics by using matricial spin operators. We have already claimed in a previous post that our local-realistic formulation of QM of spin-1/2 particles contains all the elements to correctly reproduce the interactions between spin and a magnetic field, although we have discussed so far only interactions with an homogeneous field.
Let us now consider a SG measurement. Inside the apparatus, the field has a prevalent magnitude along a constant direction β and some small inhomogenity inducing a magnetic force along the constant direction ν (here we use the same notation for a direction and its three direction cosines).
First, what does QM say? QM does not know about our "polarizations" and "spin propensity". It uses spin matrices and eigenspinors. The SG is represented by a spin matrix that is a linear combination of the three Pauli matrices, with coefficients the components of the field along the three Cartesian coordinates, i.e., our direction cosines{βi}, i = 1,2,3.
The two eigenvectors of such matrix (one for spin up and one for spin down), the eigenspinors, can be calculated as a function of β,
A spin state is expressed as a linear combination of the eigenspinors. We have seen in a previous post that QM spin states are equivalent to our "polarization", through
Expanding this spinor into the two eigenspinors, we obtain the two eigenvalues, whose square moduli represent the probabilities of spins up and down:
=\frac{1+s(\mu_1\beta_1+\mu_2\beta_2+\mu_3\beta_3)}{2} "\Pr(s=\pm 1)=\frac{1+s(\mu_1\beta_1+\mu_2\beta_2+\mu_3\beta_3)}{2}")
So far so good. Now, let us see how our model describes this process, without matrices, eigenvectors and complex numbers, but with a realistic underlying mechanism that describes spin dynamics and makes QM results emerge.
Outside the SG, the field is null and the polarization is μ (again a three-valued quantity). According to the spin rules of this post, the spin propensity M is thus null. That means that spins up and down are equally distributed in the ensemble of particles emitted toward the SG and entering it. At the very iteration when a particle enters the SG, the field manifests itself and the spin propensity becomes M0 = cos(μ,β). A new spin distribution results, according to the second equation displayed in this post. However, the particle also experiences a magnetic force due to the inhomogeneity that, besides changing the particle's momentum, activates an External Reset (the last equation in this post). The spin propensity jumps to the value s that the spin has taken. At successive iterations, since M = s, the spin won't change throughout the whole SG apparatus. At the exit of the SG, the probability of having spins up or down is thus
=\frac{1+sM_0}{2}=\frac{1+s(\mu_1\beta_1+\mu_2\beta_2+\mu_3\beta_3)}{2} "\Pr(s)=\frac{1+sM_0}{2}=\frac{1+s(\mu_1\beta_1+\mu_2\beta_2+\mu_3\beta_3)}{2}")
which is precisely the QM prediction.
In a future post, we will show how our underlying spin mechanism is able to reproduce more complex experiments, such as sequential SG.
The most common measurement procedure is probably by using a Stern-Gerlach apparatus. This measurement consists in sending a beam of particles through an inhomogeneous magnetic field and observing their deflection. With spin-1/2 particles, the result will be that two beams are formed, with some particles deflecting in one direction of the inhomogeinity axis, the others in the opposite direction.
This behavior is described within the formulation standard of quantum mechanics by using matricial spin operators. We have already claimed in a previous post that our local-realistic formulation of QM of spin-1/2 particles contains all the elements to correctly reproduce the interactions between spin and a magnetic field, although we have discussed so far only interactions with an homogeneous field.
Let us now consider a SG measurement. Inside the apparatus, the field has a prevalent magnitude along a constant direction β and some small inhomogenity inducing a magnetic force along the constant direction ν (here we use the same notation for a direction and its three direction cosines).
First, what does QM say? QM does not know about our "polarizations" and "spin propensity". It uses spin matrices and eigenspinors. The SG is represented by a spin matrix that is a linear combination of the three Pauli matrices, with coefficients the components of the field along the three Cartesian coordinates, i.e., our direction cosines{βi}, i = 1,2,3.
The two eigenvectors of such matrix (one for spin up and one for spin down), the eigenspinors, can be calculated as a function of β,
A spin state is expressed as a linear combination of the eigenspinors. We have seen in a previous post that QM spin states are equivalent to our "polarization", through
Expanding this spinor into the two eigenspinors, we obtain the two eigenvalues, whose square moduli represent the probabilities of spins up and down:
So far so good. Now, let us see how our model describes this process, without matrices, eigenvectors and complex numbers, but with a realistic underlying mechanism that describes spin dynamics and makes QM results emerge.
Outside the SG, the field is null and the polarization is μ (again a three-valued quantity). According to the spin rules of this post, the spin propensity M is thus null. That means that spins up and down are equally distributed in the ensemble of particles emitted toward the SG and entering it. At the very iteration when a particle enters the SG, the field manifests itself and the spin propensity becomes M0 = cos(μ,β). A new spin distribution results, according to the second equation displayed in this post. However, the particle also experiences a magnetic force due to the inhomogeneity that, besides changing the particle's momentum, activates an External Reset (the last equation in this post). The spin propensity jumps to the value s that the spin has taken. At successive iterations, since M = s, the spin won't change throughout the whole SG apparatus. At the exit of the SG, the probability of having spins up or down is thus
which is precisely the QM prediction.
In a future post, we will show how our underlying spin mechanism is able to reproduce more complex experiments, such as sequential SG.
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