The Pauli matrices are a set of three \(2 \times 2\) complex matrices that are Hermitian, unitary, and traceless. They are fundamental to Quantum Mechanics, specifically in the description of spin-1/2 particles, and form the basis for the \(SU(2)\) group.
Derivation of Pauli Sigma Matrices
To derive the Pauli matrices, we look for a matrix representation of the angular momentum operators for a particle with spin \(s = 1/2\). We define the spin operator \(\vec{S}\) related to the Pauli vector \(\vec{\sigma}\) by:
\[ \vec{S} = \frac{\hbar}{2} \vec{\sigma} \]Our goal is to determine the matrix forms of \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) such that they satisfy the commutation relations of angular momentum.
1. Derivation of \(\sigma_z\)
We begin by choosing a basis where the operator \(\hat{S}_z\) is diagonal. For a spin-1/2 system, there are two eigenstates: spin up (\(\ket{\uparrow}\)) and spin down (\(\ket{\downarrow}\)). The eigenvalues are \(+\hbar/2\) and \(-\hbar/2\) respectively.
In matrix notation, we represent the basis vectors as:
\[ \ket{\uparrow} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \ket{\downarrow} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]The operator \(\hat{S}_z\) acts on these states to produce the eigenvalues:
\[ \hat{S}_z \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{\hbar}{2} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \] \[ \hat{S}_z \begin{pmatrix} 0 \\ 1 \end{pmatrix} = -\frac{\hbar}{2} \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]This implies that \(\hat{S}_z\) is a diagonal matrix with these eigenvalues on the diagonal:
\[ \hat{S}_z = \begin{pmatrix} \hbar/2 & 0 \\ 0 & -\hbar/2 \end{pmatrix} = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]Comparing this to \(\hat{S}_z = \frac{\hbar}{2} \sigma_z\), we obtain:
2. Using Ladder Operators for \(\sigma_x\) and \(\sigma_y\)
To find the x and y components, we utilize the "Ladder Operators" (raising and lowering operators), defined as:
\[ \hat{S}_+ = \hat{S}_x + i\hat{S}_y \] \[ \hat{S}_- = \hat{S}_x - i\hat{S}_y \]The action of these operators on the spin states is well defined in quantum mechanics. For \(s=1/2\):
- The raising operator \(\hat{S}_+\) turns spin down to spin up: \(\hat{S}_+ \ket{\downarrow} = \hbar \ket{\uparrow}\)
- It destroys the spin up state: \(\hat{S}_+ \ket{\uparrow} = 0\)
- The lowering operator \(\hat{S}_-\) turns spin up to spin down: \(\hat{S}_- \ket{\uparrow} = \hbar \ket{\downarrow}\)
- It destroys the spin down state: \(\hat{S}_- \ket{\downarrow} = 0\)
Matrix Form of Ladder Operators
Based on the actions above, we can write the matrices for \(\hat{S}_+\) and \(\hat{S}_-\):
\[ \hat{S}_+ = \begin{pmatrix} 0 & \hbar \\ 0 & 0 \end{pmatrix}, \quad \hat{S}_- = \begin{pmatrix} 0 & 0 \\ \hbar & 0 \end{pmatrix} \]3. Solving for \(\sigma_x\)
We can express \(\hat{S}_x\) in terms of the ladder operators:
\[ \hat{S}_x = \frac{1}{2}(\hat{S}_+ + \hat{S}_-) \]Substituting the matrices:
\[ \hat{S}_x = \frac{1}{2} \left[ \begin{pmatrix} 0 & \hbar \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ \hbar & 0 \end{pmatrix} \right] = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]Since \(\hat{S}_x = \frac{\hbar}{2} \sigma_x\), we get:
4. Solving for \(\sigma_y\)
Similarly, we express \(\hat{S}_y\) as:
\[ \hat{S}_y = \frac{1}{2i}(\hat{S}_+ - \hat{S}_-) \]Substituting the matrices:
\[ \hat{S}_y = \frac{1}{2i} \left[ \begin{pmatrix} 0 & \hbar \\ 0 & 0 \end{pmatrix} - \begin{pmatrix} 0 & 0 \\ \hbar & 0 \end{pmatrix} \right] = \frac{\hbar}{2i} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]Using the identity \(1/i = -i\):
\[ \hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \]Which gives us the final Pauli matrix:
Summary
The three Pauli Sigma matrices are:
\[ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]These matrices are crucial because they satisfy the condition \(\sigma_i \sigma_j = \delta_{ij}I + i\epsilon_{ijk}\sigma_k\), which encapsulates the algebra of spin-1/2 systems.