The study shall look to the group of generators SU(4). From these generators, a new group spin operator will be constructed. We will classify these groups into right handed groups and left handed groups. These two groups will satisfy all the properties of Pauli spin operators Sx, Sy and Sz with respect to the frame xyz. The analysis shows that the number of groups spin operators depends on the order of the group. This leads us to construct the theorem which defines the number of the groups spin operators. The analysis also leads to two kinds of frames: left handed frame (LHF) and right handed frame (RHF). The right handed operators will act on the RHF, and left hand operators act on the LHF. The study shall discuss the notion of spin squeezing for pure spin 3/2 system by using our new frames and new spin operators. It will show that our calculation is equivalent to the calculation by using Pauli spin operators.
The study shall look to the group of generators SU(4). From these generators, a new group spin operator will be constructed. We will classify these groups into right handed groups and left handed groups. These two groups will satisfy all the properties of Pauli spin operators Sx, Sy and Sz with respect to the frame xyz. The analysis shows that the number of groups spin operators depends on the order of the group. This leads us to construct the theorem which defines the number of the groups spin operators. The analysis also leads to two kinds of frames: left handed frame (LHF) and right handed frame (RHF). The right handed operators will act on the RHF, and left hand operators act on the LHF. The study shall discuss the notion of spin squeezing for pure spin 3/2 system by using our new frames and new spin operators. It will show that our calculation is equivalent to the calculation by using Pauli spin operators.
In this paper, we shall establish the connection between the group theory and quantum mechanics by showing how the group theory helps us to construct the spin operators. We look to the group generators SU (3). From these generators, new spin 1 operators will be constructed. These operators S-x, S-y and S-z satisfy all the properties of Pauli spin operators S-x, S-y and S-z. We shall discuss the notion of spin squeezing and correlations for pure spin 1 system using our spin operators S-x, S-y and S-z.
For the fermion transformation in the space all books of quantum mechanics propose to use the unitary operator Un(φ)=exp(−i2φ(σ⋅n)), where φ is angle of rotation around the axis n. But this operator turns the spin in inverse direction presenting the rotation to the left. The error of defining of Un(φ) action is caused because the spin supposed as simple vector which is independent from σ-operator a priori. In this work it is shown that each fermion marked by number i has own Pauli-vector σi and both of them change together. If we suppose the global σ-operator and using the Bloch Sphere approach define for all fermions the common quantization axis z the spin transformation will be the same: the right hand rotation around the axis n is performed by the operator Un+(φ)=exp(+i2φ(σ⋅n)).
In spin-1 collective atomic systems, the spin and nematic-tensor operators constitute the su(3) Lie algebra, whose su(2) subalgebras are shown to give two distinct classes of squeezing which are unitarily equivalent to spin squeezing and spin-nematic squeezing. We explicitly construct a unitary operator that generates an arbitrary squeezed spin-nematic state from an arbitrary spin-1 condensate state. In particular, we demonstrate that squeezed spin states can be generated from a polar state and that squeezed spin-nematic states can be generated from a fully spin-polarized state.
Spin ½, which follows from Dirac’s linearization of the Klein-Gordon equation, is associated with a fermion which in turn is linked to an antisymmetric spin wavefunction under the permutation of particles if other quantum numbers are the same, e.g. energy etc. We argue that the Klein-Gordon equation with a rest mass mo implies a rest frame in which one may observe rotations If one wishes to linearize E, one may write: E = +/- sqrt( p dot p +momo) (c=1). Here p dot p is rotationally invariant, but E and px, py and pz are treated in a different manner even though E and (px,py,pz) appear equally as vector entries of a 4-vector. As a result, one may attempt to linearize the Klein-Gordon equation -EE + pp = -momo to have E and px,py,pz appear on the same footing. This, however, cannot be done using numbers, but requires 4x4 matrices containing 2x2 Pauli matrices as shown by Dirac. We consider the simpler case here of the Pauli 2x2 matrices such that: A = sx p(x) + sy p(y) + sz p(z) and AA = -EE+p dot p. The Pauli matrices are 2x2 and ensure that there is orthogonality among p(x), p(y), p(z) so no cross terms appear in AA. The 2x2 Pauli matrices, however, are linked with more than orthogonality of p(x), p(y), p(z). Given that sxsx=sysy=szsz=1, they carry the same feature of the sqrt function, namely that -1 or 1 squared yields 1. In this case, if one considers the two eigenvectors of sz, namely (1,0) and (0,1), the second has an eigenvalue of -1. One must operate twice with sz i.e. sz sz in order to restore 1. Thus sz is a kind of square root rotation. So far, there has been no mention of antisymmetry which requires at least two particles. If one considers W(spin)= |+>|-> + a |->|+>, where |+>=(1,0) and |->=(0,1), (where a could be +1 or -1), one may ask: Is there a condition on an operator which permutes the first particle and second? Here |n>|m> applies to the first particle in state n and the second in m. We consider the operator sz(1) sz(2), where 1 and 2 denote particles 1 and 2, acting on W(spin). For a single particle, sz acting on |-> returns a -1 eigenvalue, and so for a mixed |+.>|->, an eigenvalue of -1 is also returned. We argue that this represents a physical situation and so the permutation operator should also return the same physical state because Permutation operator * Permutation operator = 1. Thus, we suggest that a permutation should have the same effect as sz(1)sz(2). This suggests that a=-1 i.e. one has |+>|->-|->|+> in order that the permutation operator have the same effect as sz(1)sz(2).
Recently, we have shown how the interpretation of quantum mechanics due to Lande' can be used to derive from first principles generalized formulas for the operators and some eigenvectors for spin 1/2 Though we gave the operators for all the components of the spin, we did not give the eigenvectors of the operators for the x and y components of the spin. We now give these vectors. In addition, we present a new and simple method of deriving the operators for the x and y components of the spin as well as their vectors from those for the z component. We give a general proof that the operator for the square of the spin is the unit matrix multiplied by the value of the square of the spin.
Physics is based on experimental observables. Some of these observables are functions of other observables. For example, mass is an observable associated with inertia or gravity and velocity is classically measured, but the product mov yields nonrelativistic momentum which may be measured independently through an impulse hit with a target. A further issue with measurement is the case of vectors. For a vector pointing in one direction, one may orient the z axis to lie along this direction. If one always does this, momentum appears as a scalar, but one has to continuously rotate the co-ordinate system to account for different orientations. In this note, we are interested in spin which is linked to a kind of rotation and so we focus on the momentum vector, but only consider a function of momentum which yields a linear E, free particle energy. Thus, for the Lorentz invariant: -EE + p dot p = -momo, we consider the nonrelativistic limit: E = p dot p /2mo. In quantum mechanics, a free particle is represented by the complex probability (wavefunction) exp(ip dot r).This function has physical implications as it manifests itself in n-slit interference experiments so it is physical (not a mathematical convenience), but at the same time is responsible for conservation of momentum in x,y,z because Integral dx exp(ip1x) exp(ip2x) = 0 if p1 is not equal to p2. A physical function (property) through orthogonality creates an important relationship, namely conservation of momentum. We suggest that a similar idea may be at work in a relationship linking a linear E (free particle energy) to a function of momentum. We start with a linear E related to a function of p and only focus on the p vector because if it is treated as a number, the co-ordinate axis must rotate to account for its orientation. We only consider a situation in which p(x), p(y) and p(z) behave in an independent manner in the function of p (i.e. in the same manner as p itself) and suggest that instead of using vector analysis, there should exist a physical co-ordinate axis for each projection p(x), p(y) and p(z) if they are independent. We argue that p dot p = p(x)p(x) + p(y)p(y)+p(z)p(z) does not contain cross terms such as p(x)p(y). This is a kind of orthogonality which we associate with a rotational kind of operator linked with the p(i) operator -id/dx(i). In other words, we seek: p vector = generalized number = p(x)sx+p(y)sy+p(z)sz, where s(i) are generalized numbers or matrices, so that when this multiplied by itself, it yields p dot p. It turns out that the s(i) are the 2x2 Pauli matrices. In the case of a photon, there is no nonrelativistic limit and a linear E equation related to p is: E= |p|c. This equation is not independent in p(x), p(y), p(z) i.e. one cannot write p(x)sx+p(y)sy+p(z)sz an obtain E = {p(x)sx+p(y)sy+p(z)sz} c. Thus we look for another equation containing a linear E and a function of p. In this case, we use energy and momentum densities as given by the continuity equation: d/dt partial energy density + d grad dot ElxB =0 where d is a constant and El and B electric and magnetic 3-vectors. As in the above analysis, one wishes to have a momentum operator linked with a kind of rotational operator which allows one to eliminate vector math and use generalized numbers. In such a case, grad in the continuity equation becomes the momentum vector and eijk (Levi-Civita symbol in the cross product) becomes the spin matrix. Thus in both cases, spin governs the form of the function containing momentum which yields a linear E. In other words, it is not momentum alone which yields the relationship between a linear E (free particle energy) and momentum.