Confuse about the spin and pauli matrices

[KFC]

In the textbook, it uses the pauli matrices to describe the spin and it will also form a vector

$$\vec{\sigma} = \sigma_1 \hat{x} + \sigma_2\hat{y} + \sigma_3\hat{z}$$

But each component, $$\sigma_i, i=1,2,3$$ is a 2x2 matrix. I am really confuse about the relation between $$\sigma_i$$ and the component of the spin (I mean the magnetic moment) along each cartesian coordinate. For example, why people use $$\sigma_2$$ to refer to a spin along y direction while $$\sigma_2$$ is a 2x2 matrix instead of a column/row vector?

 

[malawi_glenn]

pauli matrices are OPERATORS, not states.

The eigenvectors of those operators in matrix form (2x2-matricies ) gives you the states of spin.

 

[Avodyne]

The spin state is a two-component column vector (often called a "spinor") that is acted on by the Pauli matrices, which are operators; multiplied by (1/2)hbar, they are the three angular momentum operators. A spinor that is an eigenvector of (say) sigma_3 with eigenvalue +1 is a state in which the z component of angular momentum has a definite value, in this case +(1/2)hbar. When the z component has a definite value, the x and y components do not (because the three angular momentum operators do not commute).

 

[RandallB]

Can the x and y components be defined together (including both using definite values), to permit considering them to represent something like a “right ascension” on a star chart, using that as a single operator to create a “two angular momentum operators system” for the spinor?

Remembering that as part of a two operator description the “declination” component piece, from the z component, would be indefinite when the “right ascension” component has a definite value. (Because the “angular momentum operators”, only two in this case, do not commute).

Or would operators defined that way just not be compatable with pauli matrics?

 

[sirchasm]

I think you have to keep a close track of the abstractions being used, and the real physical objects that atoms and electrons actually are. It's kind of important to be able to relate the properties of electrons as groups of particles, to electrons as single particles.

Group and phase velocities explain the behaviour of charge and spin as waves (of current or 'electron flux', and magnetic fields with curl, or 'spin flux'). The Maxwellian views are reflected in a close-up view, but at the quantum level there aren't any 'solid particles' or atoms, positions and 'energies' smear out over a space that's ruled by spin precession and the motion of charge. Quantum logic is about how to mix or separate signals from fundamental waveforms, and how magnetic potential can be a 'switch' - since it's the equivalent of a solid angle in an abstract spherical volume.

It gets harder to maintain a grip on phases and differences between them because the dimensions are 'fundamental', and very small; we can apply a magnetic potential, or polarize a space with electric potential to 'control' the precession angles of a spin component or momentum of a charge component.
All fundamental particles have a spin component, even the 'spinless' ones like photons have a tangent connection to a spherical surface; another way to say: "photon polarization has to 'find' an angle, as a vector normal to a tangent on a spherical surface, as its wordline evolves in linear time". We can 'measure' the electric and magnetic moments of any particles that interact with these fields, by fixing some direction on or in this spherical space.

Pauli's algebra is a direct consequence of the properties of "unitary objects". What you do with the Taylor expansion is see what happens when you 'insert' a phase into exponential representations of the number 1 (you multiply to get a 'phase product').
The expansion that 'generates' the approximation shows that it's fundamentally asymmetrical; small angles mean the approximation approaches 1, this approach is dominated by the second term, the third order term has a much lower effect on the 'gap'; large angles mean the approximation expands quickly - which reflects a quantum 'event', like an electron leaving an orbital since its precession angle is too large to allow it to stay where it is.

The expansion 'expands' then, or oscillates. It 'drives' something towards a value, which is unity.
Taylor's series is ubiquitous, and appears to be universal, like the way e is universal.