Revisiting the spin states
While in the standard formulation of Quantum Mechanics spin is described by a matricial operator, acting on vector quantities (spinors) that describe spin states, in our quest for a local and realist model we have introduced (in an earlier post) spin as a (2S+1)-valued attribute of particles.
For S = 1/2, our spin s will take values +1 and -1, according to the value of the particle's intrinsic source spin s0 and its spin propensity M, which in turn results from the interaction of the intrinsic spin "polarization" and an external magnetic field. We have also described the Reset of spin quantities that occurs when a particle experiences an external force, notably, a magnetic force.
Alone, the rules shown in that post are sufficient to explain typical quantum processes such as quantization of spin, the apparent "collapse" of spin states, and generally spin measurement outcomes. This subject is trated in detail in my 2020 ArXiV paper.
Let us consider first an ensemble of spin-1/2 particles emitted from a single source in a homogeneous magnetic field. Let us consider a preparation that attributes the same polarization μ0 to all particles of the ensemble. As we shall see later, this preparation corresponds to a pure quantum state. Since the magnetic field is homogeneous in space, no magnetic force arise. If the field is also constant, the spin propensity M does not change with respect to its initial value
(remember that the field is a three-component quantity where Bd = BMβd for d = 1,2,3, and BM is the field's scalar intensity). The spin of each particle thus only depends on its intrinsic values of s0 and M0, both attributed at the source, according to the fundamental rule already discussed. Since the source spin is uniformly distributed between -1 and +1, the ensemble probability of spins up and down is
=\frac{1\pm&space;M_0}{2}&space;=&space;\frac{1\pm&space;\cos(\mu_0,\beta)}{2} "\Pr(s=\pm 1)=\frac{1\pm M_0}{2} = \frac{1\pm \cos(\mu_0,\beta)}{2}")
where the cosine argument means the angle between the ordinary space vectors whose components are the same as those of μ0 and β. This result can be easily generalized to the case of variable-in-time, homogenoeus-in-space field.
Now, the relation between "spin polarization" and standard QM variables can be clarified. If the field is directed along one particular direction d, then βd = 1 and M = μd. Consequently, the ensemble probability of spins up and down is (1±μd)/2 and the expectation of spin is found by its definition,
&space;\Pr(s=+1)&space;+&space;(-1)&space;\Pr(s=-1)&space;=&space;\frac{1+\mu_d}{2}-\frac{1-\mu_d}{2}=\mu_d "\langle s \rangle=(+1) \Pr(s=+1) + (-1) \Pr(s=-1) = \frac{1+\mu_d}{2}-\frac{1-\mu_d}{2}=\mu_d")
Thus the d-component of the polarization represents the standard QM quantity usually expressed as〈Sd〉that is, the expected value of spin measured along the d-direction.
It should also be apparent that the standard QM representation of a spin state as a two-component spinor ψ corresponds to our polarization according to the following equivalence,
)}}&space;\end{pmatrix} "\psi = \begin{pmatrix} \displaystyle\sqrt{\frac{1+\mu_3}{2}}\\ \displaystyle\frac{\mu_1+i\mu_2}{\sqrt{2(1+\mu_3))}} \end{pmatrix}")
Thus,
This equivalence confirms that a pure state indeed corresponds to a definite polarization. The polarization components thus correspond to the cartesian coordinates on a Bloch sphere (see Figure).
For S = 1/2, our spin s will take values +1 and -1, according to the value of the particle's intrinsic source spin s0 and its spin propensity M, which in turn results from the interaction of the intrinsic spin "polarization" and an external magnetic field. We have also described the Reset of spin quantities that occurs when a particle experiences an external force, notably, a magnetic force.
Alone, the rules shown in that post are sufficient to explain typical quantum processes such as quantization of spin, the apparent "collapse" of spin states, and generally spin measurement outcomes. This subject is trated in detail in my 2020 ArXiV paper.
Let us consider first an ensemble of spin-1/2 particles emitted from a single source in a homogeneous magnetic field. Let us consider a preparation that attributes the same polarization μ0 to all particles of the ensemble. As we shall see later, this preparation corresponds to a pure quantum state. Since the magnetic field is homogeneous in space, no magnetic force arise. If the field is also constant, the spin propensity M does not change with respect to its initial value
(remember that the field is a three-component quantity where Bd = BMβd for d = 1,2,3, and BM is the field's scalar intensity). The spin of each particle thus only depends on its intrinsic values of s0 and M0, both attributed at the source, according to the fundamental rule already discussed. Since the source spin is uniformly distributed between -1 and +1, the ensemble probability of spins up and down is
where the cosine argument means the angle between the ordinary space vectors whose components are the same as those of μ0 and β. This result can be easily generalized to the case of variable-in-time, homogenoeus-in-space field.
Now, the relation between "spin polarization" and standard QM variables can be clarified. If the field is directed along one particular direction d, then βd = 1 and M = μd. Consequently, the ensemble probability of spins up and down is (1±μd)/2 and the expectation of spin is found by its definition,
Thus the d-component of the polarization represents the standard QM quantity usually expressed as〈Sd〉that is, the expected value of spin measured along the d-direction.
It should also be apparent that the standard QM representation of a spin state as a two-component spinor ψ corresponds to our polarization according to the following equivalence,
Thus,
- polarization along direction z (μ3 = 1) corresponds to a state (1,0),
- polarization along direction x (μ1 = 1) corresponds to a state 1/√2(1,1),
- polarization along direction y (μ2 = 1) corresponds to a state 1/√2(1,i),
This equivalence confirms that a pure state indeed corresponds to a definite polarization. The polarization components thus correspond to the cartesian coordinates on a Bloch sphere (see Figure).
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