What is Pauli matrix: Definition and 12 Discussions
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.
These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field.
Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices.
This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.
Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space R3.
The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for the real Lie algebra
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
, which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of R3, and the (unital associative) algebra generated by iσ1, iσ2, iσ3 is isomorphic to that of quaternions.
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.}}...
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 ...
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