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3

Your matrix is wrong. It should be det {$\begin{pmatrix} -\lambda & \hbar/2 (1-i) \\ \hbar/2 (1+i) & -\lambda \end{pmatrix}$} = 0 Now when you compute the determinant, it should be okay.

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Indeed as was commented before usually in physics these derivatives are 'derived' by postulating how the object transforms. Obviously there are more rigorous definitions, as is usually the case. In this particular case you could try to find the structure which gives you these 'covariant derivatives' in the first place. In books they are often referred to ...

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The eigenvalues (or measurable values) of each spin operator are ±1, corresponding to the only two mutually exclusive outcomes that could occur after a measurement of an electron's spin has been made. The mutual exclusivity is an experimental fact, but follows from the definition of the eigenstates above.In classical mechanics an electron has a magnetic ...

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The principle behind STT is conservation of total angular momentum of the system. The equation that describes spin dynamics is called the LLG equation, named after Landau, Lifshitz and Gilbert. I will try to illustrate its physical meaning by using a general example. Consider an s-d model, where the sp-band electrons are itinerant and thus contribute to ...

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A two component spinor can be geometrically interpreted as representing a point on the Riemann sphere, defined by the ratio of its two complex components, and its stereographic projection onto the xy-plane. Similarly, a four component spinor can be interpreted, by a more complicated ratio defined by its four complex components, as a point on the Riemann ...

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The spin of a quantum field is related to the representation of the Lorentz group they transform under: scalar fields transform under the trivial representation, spinors transform under the spinorial representation, gauge bosons under the vectorial representation, gravitons (if they exist) under the second-rank tensorial representation... If you restrict to ...

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