Question

If ${\sigma }_x,{\sigma }_y$, and ${\sigma }_z$ denote the Pauli spin matrices (16-109), show that $$\text{Missing left or extra right}$$ where $\text{Missing left or extra right}$ is the anticommutator of ${\sigma }_i$ and ${\sigma }_j$.

Short answer

\left\{\sigma_i, \sigma_j\right\} = 2 \delta_{ij\).

Step-by-step solution

Define the Pauli Spin Matrices

The Pauli spin matrices are defined as: ${\sigma }_x=\left(\begin{array}{ccc}0& 11& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{ccc}0& -ii& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{ccc}1& 00& -1\end{array}\right)$.

Define the Anticommutator

The anticommutator of two matrices ${\sigma }_i$ and ${\sigma }_j$ is $\{{\sigma }_i,{\sigma }_j\}$ and is given by: $$\{{\sigma }_i,{\sigma }_j\}={\sigma }_i{\sigma }_j+{\sigma }_j{\sigma }_i$$.

Calculate the Anticommutators for Different Combinations

Compute ${\sigma }_i{\sigma }_j+{\sigma }_j{\sigma }_i$ for different combinations of ${\sigma }_i$ and ${\sigma }_j$.

Case 1 - i ≠ j

For $i\ne j$, using the properties of Pauli matrices: ${\sigma }_x{\sigma }_y=i{\sigma }_z$,\ ${\sigma }_y{\sigma }_x=-i{\sigma }_z$,\ ${\sigma }_x{\sigma }_z=-i{\sigma }_y$,\ and\ ${\sigma }_z{\sigma }_x=i{\sigma }_y$, we have:\ \ $\{{\sigma }_x,{\sigma }_y\}={\sigma }_x{\sigma }_y+{\sigma }_y{\sigma }_x=i{\sigma }_z-i{\sigma }_z=0$.\This result holds similarly for other combinations of ${\sigma }_i{\sigma }_j$.\ When\ $ieqj$,\ $\{{\sigma }_i,{\sigma }_j\}=0$.

Case 2 - i = j

For $i=j$:\ Using the properties of Pauli matrices: ${\sigma }_x^2={\sigma }_y^2={\sigma }_z^2=I$, where I is the identity matrix. Thus:\ \ $\{{\sigma }_x,{\sigma }_x\}={\sigma }_x{\sigma }_x+{\sigma }_x{\sigma }_x=2{\sigma }_x^2=2I$.\ Similarly for ${\sigma }_y\text{and}{\sigma }_z$\ we\ have:\ \ $\{{\sigma }_i,{\sigma }_i\}=2I$.

Combine Results

Combine the two cases: $\{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}$, where ${\delta }_{ij}$ is the Kronecker delta.

Key concepts

Pauli Matrices

The Pauli matrices are a set of three 2x2 complex matrices. They are fundamental in the study of quantum mechanics. Each matrix is named ${\sigma }_x,{\sigma }_{y}and{\sigma }_z$. They are defined as follows:
${\sigma }_x=\left(\begin{array}{ccc}0& 11& 0\end{array}\right),\partial {\sigma }_y=\left(\begin{array}{ccc}0& -ii& 0\end{array}\right),\partial {\sigma }_z=\left(\begin{array}{ccc}1& 00& -1\end{array}\right)$. These matrices are important because they form a basis for the Lie algebra su(2), which is essential in describing spin in quantum mechanics. Each matrix has unique properties that characterize rotations in the spin space.

Anticommutators

The anticommutator is a specific way to combine two matrices. It is defined as follows:
$\{\partial {\sigma }_i,{\sigma }_j\}={\sigma }_i{\sigma }_j+{\sigma }_j{\sigma }_i$.
This formula essentially sums two products of the matrices in both possible orders. Unlike the commutator, which measures the difference in order, the anticommutator measures their similarity. Understanding anticommutators is crucial in matrix algebra and quantum mechanics because of their unique symmetry properties.

Quantum Mechanics

The principles of quantum mechanics are essential for understanding phenomena at a microscopic scale. In this field, Pauli matrices play a central role in describing the spin of quantum particles. Spin is an intrinsic form of angular momentum carried by elementary particles. Pauli matrices act as the mathematical tools to describe this property.
Quantum mechanics often involves solving complex equations that describe the behavior of particles. Here, the Pauli matrices and their anticommutators give us the necessary framework to understand and predict these behaviors.

Matrix Algebra

Matrix algebra involves various operations on matrices, including addition, subtraction, and multiplication. It also explores properties such as commutation and anticommutation relationships. Pauli matrices serve as fundamental examples in this context.
In the given problem, we calculate the anticommutator for Pauli matrices, which is crucial for validating the relationship $\{\partial {\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}$:
\ \bullet When \mathbf{i \eq j}\, the anticommutator is zero.
\ \bullet When \mathbf{i = j}\, it equals twice the identity matrix, 2I.
This showcases how matrix algebra can be applied to confirm theoretical results in quantum mechanics.