Pauli Matrices

Three Matrix which are defined as

σ0=σ0=I2=(1001)σ^x=σ1=σ1=(0110)σ^y=σ2=σ2=(0−ii0)σ^z=σ3=σ3=(100−1)\begin{gather*} \sigma^0=\sigma_0=I_2=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\\ \hat{\sigma}_x=\sigma^1=\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\\ \hat{\sigma}_y=\sigma^2=\sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\\ \hat{\sigma}_z=\sigma^3=\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{gather*}

Properties

  • spacing
σ^i2=I2∀i∈{x,y,z}\begin{gather*} \hat{\sigma}^2_i=I_2\quad \forall \quad i\in\{x,y,z\} \end{gather*}
  • commutation

note that

The symbol εijk\varepsilon_{ijk} is the Levi-Civita symbol, an antisymmetric tensor that encodes signs based on the order of its indices.

εijk={+1if (i,j,k) is an even permutation of (1,2,3)−1if (i,j,k) is an odd permutation of (1,2,3)0if any index repeats\begin{gather*} \varepsilon_{ijk}=\begin{cases} +1\quad \text{if $(i,j,k)$ is an even permutation of (1,2,3)} \\ -1\quad \text{if $(i,j,k)$ is an odd permutation of (1,2,3)} \\ 0\quad\text{if any index repeats} \end{cases} \end{gather*}
  • in 2D (only two indices, written εij\varepsilon_{ij})
εij=(01−10)\begin{gather*} \varepsilon_{ij}=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} \end{gather*} [σi,σj]=2iεijkσk\begin{gather*} [\sigma_i,\sigma_j]=2i\varepsilon_{ijk}\sigma_k \end{gather*}

where an “even” permutation is (1,2,3), (2,3,1), (3,1,2) and an “odd” is (2,1,3), (1,3,2), (3,2,1)

  • anti commutation

Note that the anticommutator is

{A,B}=AB+BA\begin{gather*} \{A,B\}=AB+BA \end{gather*} {σi,σj}=2δijI2\begin{gather*} \{\sigma_i,\sigma_j\}=2\delta_{ij}I_2 \end{gather*} σ0=I2=(1001)\begin{gather*} \sigma_0=I_2=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} \end{gather*}

the Set {σ0,σx,σy,σz}\{\sigma_0,\sigma_x,\sigma_y,\sigma_z\} spans all 2x2 complex matrices

{I,X,Y,Z}form a basiss for operator L(C)2\begin{gather*} \{I,X,Y,Z\}\quad\text{form a basiss for operator }L(\mathbb{C})^2 \end{gather*}

where

forms a basis for all 2x2 complex matrices. Any 2x2 matrix

M=∑μ=03cμσμ\begin{gather*} M=\sum_{\mu=0}^3c_\mu\sigma_\mu \end{gather*}
  • whatever this is
σ⃗=(X,Y,Z)are Hermitianσ1=σx=Xσ2=σy=Yσ3=σz=ZX2+Y2+Z2=Itr(X)=tr(Y)=tr(Z)=0XY=iZ,YZ=iX,ZX=iY[X,Y]=iℏZ,[Y,Z]=iℏX,[Z,X]=iℏYYX=(XY)+=(iZ)+=−iZ\begin{gather*} \vec{\sigma}=(X,Y,Z)\quad\text{are Hermitian}\\ \sigma^1=\sigma^x=X\\ \sigma^2=\sigma^y=Y\\ \sigma^3=\sigma^z=Z\\ X^2+Y^2+Z^2=I\\ tr(X)=tr(Y)=tr(Z)=0\\ XY=iZ, YZ=iX, ZX=iY\\ [X, Y] = i\hbar Z, \quad [Y, Z] = i\hbar X, \quad [Z, X] = i\hbar Y\\ YX=(XY)^+=(iZ)^+=-iZ \end{gather*}

and

A^=a0σ0+a1σx+a2σy+a3σzai∈C\begin{gather*} \hat{A}=a_0\sigma_0+a_1\sigma_x+a_2\sigma_y+a_3\sigma_z\quad a_i\in \mathbb{C} \end{gather*}

A^⋅B^\hat{A}\cdot\hat{B} can be expanded as

A^B^=∑i=03ciσi\begin{gather*} \hat{A}\hat{B}=\sum_{i=0}^3c_i\sigma^i \end{gather*}

If ci∈Rc_i\in\mathbb{R}, then A^\hat{A} is Hermitian