Problem 17 - Differential Equations and Linear Algebra

Question

The Pauli spin matrices \(\sigma_{1}, \sigma_{2},\) and \(\sigma_{3}\) are defined by $$ \sigma_{1}=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right], \sigma_{2}=\left[\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right] $$ and $$ \sigma_{3}=\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right] $$ Verify that they satisfy $$ \sigma_{1} \sigma_{2}=i \sigma_{3}, \sigma_{2} \sigma_{3}=i \sigma_{1}, \sigma_{3} \sigma_{1}=i \sigma_{2} $$ If \(A\) and \(B\) are \(n \times n\) matrices, we define their commutator, denoted \([A, B],\) by $$ [A, B]=A B-B A $$ Thus, \([A, B]=0\) if and only if \(A\) and \(B\) commute. That is, \(A B=B A .\) Problems \(19-22\) require the commutator.

Step-by-Step Solution

Answer

We have verified that the Pauli spin matrices satisfy the following relations: \(\sigma_{1} \sigma_{2} = i \sigma_{3}\), \(\sigma_{2} \sigma_{3} = i \sigma_{1}\), and \(\sigma_{3} \sigma_{1} = i \sigma_{2}\).

Step 1: Pauli Spin Matrices and Commutator Definition

The Pauli spin matrices are given as: \(\sigma_{1}=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_{2}=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_{3}=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) The commutator is defined as: \([A, B] = A B - B A\) Now, let's verify the relations one by one.

Step 2: Verify \(\sigma_{1} \sigma_{2} = i \sigma_{3}\)

Perform the multiplication between \(\sigma_{1}\) and \(\sigma_{2}\): \(\sigma_{1}\sigma_{2} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} = i\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = i\sigma_{3}\) This verifies that \(\sigma_{1} \sigma_{2} = i \sigma_{3}\).

Step 3: Verify \(\sigma_{2} \sigma_{3} = i \sigma_{1}\)

Perform the multiplication between \(\sigma_{2}\) and \(\sigma_{3}\): \(\sigma_{2}\sigma_{3} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} = i\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = i\sigma_{1}\) This verifies that \(\sigma_{2} \sigma_{3} = i \sigma_{1}\).

Step 4: Verify \(\sigma_{3} \sigma_{1} = i \sigma_{2}\)

Perform the multiplication between \(\sigma_{3}\) and \(\sigma_{1}\): \(\sigma_{3}\sigma_{1} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = i\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = i\sigma_{2}\) This verifies that \(\sigma_{3} \sigma_{1} = i \sigma_{2}\). Thus, all the relations have been verified.

Key Concepts

Linear Algebra

Linear Algebra is a fundamental field of mathematics that is concerned with vector spaces and linear mappings between these spaces. This includes the study of lines, planes, and subspaces, but extends to higher dimensions as necessary. It provides a way of systematically constructing and manipulating matrices and vectors, which can represent a variety of mathematical objects.

At its core, Linear Algebra focuses on solving systems of linear equations, which are equations involving linear combinations of variables. In the case of the Pauli spin matrices, we deal with 2x2 matrices and vectors in a two-dimensional space. These matrices are essential when analyzing quantum states and behaviors in quantum mechanics, especially in the field of quantum computing and spintronics.

Matrix Multiplication

Matrix multiplication is a critical operation in Linear Algebra that combines two matrices to form a new matrix. This operation is not element-wise—like multiplication of individual numbers—but is instead a rule-based combination where the entry in the resulting matrix is the sum of the products of elements from the rows of the first matrix and the columns of the second matrix.

To perform matrix multiplication successfully, the number of columns in the first matrix must match the number of rows in the second matrix. In the case of the Pauli spin matrices, we multiply 2x2 matrices together. The result of multiplying these special matrices is another 2x2 matrix that has significance in quantum mechanics due to its corresponding spin states and transitions.

Commutator

The commutator is an operation used to measure the degree to which two matrices, or operators, fail to commute with each other. Essentially, the commutator of two matrices \(A\) and \(B\), denoted as \([A, B]\), is the difference between the product of \(A\) and \(B\), and the product of \(B\) and \(A\). It is a central concept in quantum mechanics, as it helps in understanding uncertainties and the compatibility of different physical quantities.

The commutativity of the Pauli matrices is a key property and understanding it can reveal much about underlying symmetries and conservation laws within a physical system. The commutator is particularly important in quantum physics because it signifies whether two operators, or observables, can be simultaneously known or are subject to Heisenberg's uncertainty principle.