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arXiv:1801.06129

Right sign of spin rotation operator

R. A. Shindin, D. K. Guriev, Alexey N. Livanov, Ivan P. Yudin

Published Jan 15, 2018• 0 citations
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Abstract

For the fermion transformation in the space all books of quantum mechanics propose to use the unitary operator $\widehat{U}_{\vec n}(φ)=\exp{(-i\frac\varphi2(\widehatσ\cdot\vec n))}$, where $φ$ is angle of rotation around the axis $\vec{n}$. But this operator turns the spin in inverse direction presenting the rotation to the left. The error of defining of $\widehat{U}_{\vec n}(φ)$ action is caused because the spin supposed as simple vector which is independent from $\widehatσ$-operator a priori. In this work it is shown that each fermion marked by number $i$ has own Pauli-vector $\widehatσ_i$ and both of them change together. If we suppose the global $\widehatσ$-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 $\vec{n}$ is performed by the operator $\widehat{U}^+_{\vec n}(φ)=\exp{(+i\frac\varphi2(\widehatσ\cdot\vec n))}$.

DOI
10.48550/arxiv.1801.06129

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New degrees of freedom having the form of a unit vector are identified for characterizing the spin polarization of free electron. It is shown that when only the spin is considered, the non-commutativity of the Cartesian components of the Pauli vector allows us to use the azimuthal angle of a second direction, denoted by unit vector I\mathbf II, with respect to the quantization direction to characterize the spin polarization. The rotation of I\mathbf II through an angle about the quantization axis leads to a rotation of the spin polarization vector through twice the angle about the same axis. Discussions are also made in Heisenberg picture as well. Upon utilizing this approach to a free electron and letting the quantization direction for each plane wave be the wave vector, we arrive at a representation in which the unit vector I\mathbf II functions as an independent index to characterize the spin polarization.
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The Bloch sphere concept is a useful tool to represent the actions of various quantum-mechanical operators on a spinor. Most important, it provides a link between the rather abstract concept of a spinor and the more intuitive way (although not rigorous) of thinking of the spin of an electron as being associated with a magnetic moment along the axis of self-rotation. The Bloch sphere is particularly useful when describing the action of a spatially uniform external (including time-dependent) magnetic field on the spin [1]. This will be illustrated in the next chapter through the derivation of the Rabi formula. Bloch sphere concepts are also frequently invoked in discussions of spin-based quantum computing, so that a sound understanding of this notion is imperative to comprehend the literature in that field.
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