We have some arbitrary quantum state, lets say $$\vert\Psi\rangle=\alpha_{1}\vert\uparrow\rangle+\alpha_{2}\vert\downarrow\rangle= \begin{pmatrix} \alpha_{1} \\ \alpha_{2} \\ \end{pmatrix}$$. And we act on it with some operator, whatever is appropriate math for it, lets say in this case particular case a linear combination of Pauli matrices $$ \sigma_{n} = \begin{pmatrix} n_z & n_x-in_y \\ n_x+in_y & -n_z \\ \end{pmatrix}$$ My understanding of physical process, that is happening here, is as follows. We have some arbitrary electron, with operator we put on magnetic field in arbitrary direction and eigenvalues make sure we calculate expectation value correctly. So eigenvalues are measurement result. (?)
On the other hand operator acts on a vector and we get a new state. (Rotates the state on a Bloch sphere?) So eigenvalues change basis vectors and we rotate a state.
How does the operators relate to rotations on Bloch sphere? In my interpretation eigenvalue existence mess up probabilities.
Addition: It seems I can't connect the idea of an observable (mathematical construction for expectation value - to get measurement results) and idea of an unitary operator which changes state of a system (due to magnetic field). How are they related? Are they one and the same?
Answer
I would re-state the problem slightly to conform to conventional notation. In any discussion of magnetic resonance -- of spin 1/2 particles, be they electrons, or say, protons-- the direction of the magnetic (polarizing) field is always chosen as +z in a laboratory coordinate frame. Then the two stationary states, which (for convenience) I write as |+> and |->, correspond to the spin aligned with its z-component of angular momentum parallel or anti-parallel to the polarizing field.
The stationary states are in fact eigenstates of the z component Pauli matrix, with respect to the current laboratory coordinate frame.
Then the operator you illustrate is a linear combination of all three Pauli matrices in this frame. Any Pauli matrix (or linear combination thereof) acts mathematically as the generator of an infinitesimal rotation of the spin. Your conventional x, y, and z Pauli matrices each individually generate infinitesimal rotations of the spin about these axes. The best discussion I know of this is Messiah's book (as noted in an earlier response, above); you may also want to consult M. E. Rose 'Theory of Angular Momentum,' which is available as a Dover reprint.
Rotating an eigenket gives you a new state, but it doesn't change your basis, which is defined essentially by the direction of the polarizing magnetic field. You can rotate the spins any way you like, but your result will still be expressed as a linear combination of those same eigenkets, as long as you don't physically change the direction of the field.
So far we have dealt exclusively in quantum mechanical rotation.
Once we bring in the 'Bloch sphere', we need to introduce the Bloch equations, which are classical. In fact, the Bloch equations without relaxation are exactly generators of infinitesimal rotations, but for 3 dimensional classical vectors, specifically (in the case of magnetic resonance) the magnetization. I usually (for convenience) try to keep the classical and quantum views separate in my mind-- so I think of the Pauli matrices as rotating an individual spin, and the Bloch matrices as rotating a bulk magnetization vector comprising the resultant of many millions of individual spins.
However, I emphasize that the there is no logical requirement to view things this way. The simple view is that the Pauli matrices give a quantum rotation and the Bloch matrices give a classical rotation. Nonetheless, you will see purely quantum discussions which refer also to the Bloch sphere-- I don't think this should cause you any confusion.
I hope this helps a bit; do not hesitate to request clarification.
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