Pauli matrix Definition and 12 Threads

If we define Si=(1/2)× (reduced Planck's const)×sigma Then what will be (sigma dot vect{A})multiplied by (Sigma dot vect{B}) Here (sigma)i is Pauli matrix. Next one is, what will we get from simplifying <Alpha|vect{S}|Alpha> where vect{S} is spin vector & |Apha>is equal to " exp[{i×(vect{S} dot...

The spin exchange operator would have the property $$\begin{align*}P\mid \chi_{\uparrow\downarrow} \rangle = \mid\chi_{\downarrow\uparrow} \rangle & &P\mid \chi_{\downarrow\uparrow} \rangle =\mid \chi_{\uparrow\downarrow} \rangle \end{align*}$$ This also implies ##P\mid \chi_{\text{sym.}}...

A nice discussion of the density operator for a qubit can be found here: http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf

I'm not sure I have the right approach here: Using the three 2 X 2 Pauli spin matrices, let $ \vec{\sigma} = \hat{x} \sigma_1 + \hat{y} \sigma_2 +\hat{z} \sigma_3 $ and $\vec{a}, \vec{b}$ are ordinary vectors, Show that $ \left( \vec{\sigma} \cdot \vec{a} \right) \left( \vec{\sigma} \cdot...

I have had no problem while finding the eigen vectors for the x and y components of pauli matrix. However, while solving for the z- component, I got stuck. The eigen values are 1 and -1. While solving for the eigen vector corresponding to the eigen value 1 using (\sigma _z-\lambda I)X=0, I got...

Homework Statement Find the eigenvectors and eigenvalues of exp(iπσx/2) where σx is the x pauli matrix: 10 01 Homework Equations I know that σxn = σx for odd n I also know that σxn is for even n: 01 10 I also know that the exponential of a matrix is defined as Σ(1/n!)xn where the sum runs...

I want to find a matrix such that it takes a spin z ket in the z basis, | \; S_z + >_z and operates on it, giving me a spin z ket in the x basis, U \; | \; S_z + >_z = | \; S_z + >_x I would have thought that I could find this transformation operator matrix simply by using the...

Homework Statement In the Pauli theory of the electron, one encounters the expresion: (p - eA)X(p - eA)ψ where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ...

Homework Statement Pauli Spin matrices (math methods in physics question) Show that D can be expressed as: D=d_1\sigma_1+d_2\sigma_2+d_3\sigma_3 and write the d_i in terms of D's elements, let D also be Unitary Homework Equations - Any 2x2 complex matrix can be written as ...

Why is norm of (pauli matrix)/sqrt(2)=1

I'm completely lost and need some advice on how to continue. I need to prove the 1st line on the link http://upload.wikimedia.org/math/0/f/8/0f873eaca989ffa1af9a323c6e62f3ed.png

i am learning Quaternion now for my EM course. Can someone enlighten me on the correspondence between Quaternion and Pauli Matrix algebra?