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\heading{Ex102: Pauli Matrixes} 
\auname{Nir Stein}


{\bf The problem:}

\Dn

The Pauli Matrixes are defined as:

\[
  \sigma_0 = \left( \amatrix { 1 & 0 \cr 0 & 1}\right) \\ 	
  \sigma_1 = \left( \amatrix { 0 & 1 \cr 1 & 0}\right) \\    
  \sigma_2 = \left( \amatrix { 0 & -i \cr i & 0}\right) \\ 
  \sigma_3 = \left( \amatrix { 1 & 0 \cr 0 & -1}\right)
\]

where $\sigma_0$ is the unit matrix.
\Dn

(1) Write these matrixes using Dirac notation, choose a base in which $\sigma_3$ is diagonaled.

\Dn

 We defined: $X = \sum_{i=0}^3 a_i \sigma_i$

(2) Express Tr($\sigma_i$X) and Tr(X) with $a_i$.

(3) Show (with the help of the Dirac formalizm) that the pauli matrixes along with the unit matrix can be used as a base for the $2\times2$ Hermitian matrixes.




\Dn\Dn


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{\bf The solution:}

\Dn

(1)
The $2\times2$  matrixes are two diamentional matrixes. Therefor, we define two ortogonal states ( vectors): 

\[ |1 \rangle = \left( \amatrix { 0 \cr 1 }\right)  \\ |2 \rangle = \left( \amatrix { 1 \cr 0}\right) \]


\Dn

To write these matrixes in Dirac notation we will use:

\[ A = \sum_{i,j} |i \rangle A_{ij} \langle j| \]

This formula is good for any matrix A and also for all $\sigma_k$, therefor:

\[ \sigma_0 =  |1 \rangle 0 \langle 2| + |2 \rangle 0 \langle 1| + |1 \rangle 1 \langle 1| + |2 \rangle 1 \langle 2| = |1 \rangle \langle 1| + |2 \rangle \langle 2| \]

\[ \sigma_1 =  |1 \rangle 1 \langle 2| + |2 \rangle 1 \langle 1| + |1 \rangle 0 \langle 1| + |2 \rangle 0 \langle 2| = |1 \rangle \langle 2| + |2 \rangle \langle 1| \]

\[ \sigma_2 =  |1 \rangle -i \langle 2| + |2 \rangle i \langle 1| + |1 \rangle 0 \langle 1| + |2 \rangle 0 \langle 2| = -i|1 \rangle \langle 2| +i |2 \rangle \langle 1| \]

\[ \sigma_3 =  |1 \rangle 1 \langle 1| + |2 \rangle -1 \langle 2| + |1 \rangle 0 \langle 2| + |2 \rangle 0 \langle 1| = |1 \rangle \langle 1| - |2 \rangle \langle 2| \]

\Dn

Also, we can see that $\sigma_3$ is already diagonaled so the base for it is the same base we chose before. which is:

\[ |1 \rangle = \left( \amatrix { 0 \cr 1 }\right)  \\ |2 \rangle = \left( \amatrix { 1 \cr 0}\right) \]




%%%%%%%%%%%%%%%
\Dn

(2)
First, the multiplication table of the $\sigma_i$  (i=1,2,3) is:

\[ \sigma_i\sigma_j = I\delta_{ij} + i\epsilon_{ijk}\sigma_k \]

which means (for example):

\[ \sigma_1\sigma_2 = \sigma_3 \\ ; \\ \sigma_i^2 = I \\ ; \\ \sigma_0\sigma_i = \sigma_i \]

Its also easy to see that 

\[ Tr(\sigma_0) = 2 \\ ; \\ Tr(\sigma_i) = 0 \\ , \\ i=1,2,3 \]

Now, with the definition of X we get:

\[ Tr ( \sigma_0X ) = Tr(\sigma_0\sum_{i=0}^3 a_i \sigma_i) = Tr[\sigma_0(a_0\sigma_0 + a_1\sigma_1 + a_2\sigma_2 + a_3\sigma_3)] = Tr(a_0\sigma_0\sigma_0) = a_0Tr(\sigma_0) = 2a_0 \] 

We will find all others in the same way. So shortly:

\[ Tr(\sigma_iX) = a_iTr(\sigma_i\sigma_i) = a_iTr(\sigma_0) = 2a_i \\ , \\ i=0,1,2,3  \]



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\Dn

(3)
We know that all Hermitian $2\times2$ matrixes can be writen as a linear combination of four ortogonal matrixes. Also, the Pauli matrixes $\sigma_i$ and the unit $2\times2$ matrix are all ortogonal. Therefor, all four matrixes can be used as a base for the $2\times2$ hermitian matrix. 

\Dn

Just for checking, we will take a general hermitian $2\times2$ matrix and check its Coefficients and whether we can write that matrix as a linear combination of $\sigma_i$ and the unit matrix:

\Dn

The general hermitian $2\times2$ matrix is:

\[ \left( \amatrix { a & b + ic \cr b - ic & d}\right) \] 

where a,b,c and d real.

Also we have 
 
\[ \left( \amatrix { a & 0 \cr 0 & 0}\right) = \frac{a}2(I + \sigma_3) \\ , \\ \left( \amatrix { 0 & 0 \cr 0 & d}\right) = \frac{d}2(I - \sigma_3) \\ ,\\ \left( \amatrix { 0 & b + ic \cr b - ic & 0}\right) = \frac{1}2(b\sigma_1 + c\sigma_2) \]

Therefor, we can write the general $2\times2$ Hermitian matrix as follows:

\[ \left( \amatrix { a & b + ic \cr b - ic & d}\right) = \frac {1}2 (a + d)I + \frac {1}2 b\sigma_1 + \frac {1}2 c\sigma_2 + \frac {1}2 (a-d) \sigma_3 \]




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