Question

The Pauli spin matrices in quantum mechanics are given by the following matrices: \(\sigma_{1}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right), \sigma_{2}=\left(\begin{array}{cc}0 & -i \\ i & 0\end{array}\right)\), and \(\sigma_{3}=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right) .\) Show that a. \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=I\). b. \(\left\\{\sigma_{i}, \sigma_{j}\right\\} \equiv \sigma_{i} \sigma_{j}+\sigma_{j} \sigma_{i}=2 \delta_{i j} I\), for \(i, j=1,2,3\) and \(I\) the \(2 \times 2\) identity matrix. \(\\{,\),\(} is the anti-commutation operation.\) c. \(\left[\sigma_{1}, \sigma_{2}\right] \equiv \sigma_{1} \sigma_{2}-\sigma_{2} \sigma_{1}=2 i \sigma_{3}\) and similarly for the other pairs. \([,\), is the commutation operation. d. Show that an arbitrary \(2 \times 2\) matrix \(M\) can be written as a linear combination of Pauli matrices, \(M=a_{0} I+\sum_{j=1}^{3} a_{j} \sigma_{j}\), where the \(a_{j}^{\prime}\) s are complex numbers.

Short answer

Solving this problem involves a deep understanding of matrix operations, specifically matrix multiplication, commutation, anti-commutation, as well as the ability to express a given matrix in terms of other predefined matrices.

Step-by-step solution

Matrix Squaring

To show that \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=I\), we need to square each Pauli matrix and check whether each product is the 2x2 identity matrix \(I\). This involves typical matrix multiplication.

Anti-Commutations

We have to show that \(\{\sigma_{i}, \sigma_{j}\} = 2 \delta_{i j} I\), where \(\{, \}\) is the anti-commutation operation. Here, \(\delta_{i j}\) is the Kronecker delta. This boils down to multiplying matrices in both orders (i.e. \(\sigma_{i} \sigma_{j}\) and \(\sigma_{j} \sigma_{i}\)) and adding the results. The Kronecker delta is 1 if \(i=j\) and 0 if \(i \neq j\).

Commutations

We also need to compute the commutation \([\sigma_{1}, \sigma_{2}] = 2 i \sigma_{3}\]. Here, \([,]\) denotes the commutation operation. This involves finding the difference between the product of \(\sigma_{1}\) and \(\sigma_{2}\) in two orders (i.e. \(\sigma_{1} \sigma_{2}\) and \(\sigma_{2} \sigma_{1}\)). Similarly, the other pairs must be calculated.

Expressing a Matrix as a Linear Combination of Pauli Matrices

Lastly, we must show that any arbitrary 2x2 matrix \(M\) can be written as \(M=a_{0} I+\sum_{j=1}^{3} a_{j} \sigma_{j}\), where the \(a_{j}\)s are complex numbers. This involves expressing a 2x2 matrix in terms of Pauli matrices and the identity matrix.

Key concepts

Quantum Mechanics and Pauli Spin Matrices

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. A critical aspect of quantum mechanics is the use of mathematical constructs called operators to describe observable quantities such as the spin of particles. The Pauli spin matrices, denoted as \( \sigma_1 \) , \( \sigma_2 \), and \( \sigma_3 \), are a set of matrices that represent the spin operators for a spin-1/2 particle in three orthogonal directions.

These matrices have distinct properties that enable them to describe the intrinsic angular momentum, or spin, of elementary particles such as electrons. The spins can take on values of +1/2 or -1/2, often illustrated as 'spin up' and 'spin down.' The Pauli matrices are vital in several quantum mechanics applications, including quantum computing and the formulation of the spin-statistics theorem.

Understanding how these matrices function individually and in combination is an essential part of mastering quantum mechanics and is well highlighted in the exercise we're exploring. By squaring the Pauli matrices and showing their commutation and anti-commutation relationships, we delve deeper into their unique properties within the quantum framework.

Matrix Multiplication in Quantum Calculations

In quantum mechanics, matrix multiplication is a central operation used to manipulate and understand various physical systems. When we talk about squaring the Pauli spin matrices, as in the first part of the exercise, we are essentially performing matrix multiplication of a matrix by itself.

Matrix multiplication involves summing the products of the rows of the first matrix with the columns of the second matrix. It is important to note that matrix multiplication is not commutative, meaning that in general, \( AB \) does not equal \( BA \). However, the resulting identity matrices from squaring the Pauli spin matrices signify that these special matrices do behave in a way akin to numbers when squared, emphasizing their uniqueness in the quantum context.

Commutation and Anti-Commutation

Commutators in Quantum Mechanics

Commutation in quantum mechanics refers to the operation of finding the difference between the product of two matrices in opposite orders: \( [A, B] = AB - BA \). When two operators commute, the result is zero (\( [A, B] = 0 \)), indicating that the physical quantities represented by the operators can be simultaneously measured precisely.

On the other hand, the exercise demonstrates the anti-commutation relationship using the Pauli matrices, where \( \{\sigma_i, \sigma_j\} = 2 \delta_{ij} I \). This anti-commutation property indicates that the spin measurements in perpendicular directions are independent of each other, an important principle in quantum mechanics.

Anti-Commutators and Spin States

The anti-commutation relations reflect the underlying quantum reality that spin states along orthogonal axes cannot be determined simultaneously. In other words, knowledge of a particle's spin in one direction erases any certainty about its spin in the other two perpendicular directions, resonating with the Heisenberg uncertainty principle, one of the cornerstones of quantum mechanics.

Linear Combination of Matrices

The concept of a linear combination of matrices arises frequently in quantum mechanics, particularly in the representation of quantum states and operators. A linear combination involves adding together multiple matrices, each multiplied by their respective scalar coefficients. This method is powerful as it allows for the composition of new matrices from a set of existing matrices.

In the context of the Pauli matrices, any arbitrary 2x2 matrix \( M \) can be represented as a linear combination of the Pauli matrices and the 2x2 identity matrix \( I \). This is a fundamental result because it shows that these matrices form a basis for the space of 2x2 matrices over the complex numbers. The coefficients in this linear combination (\( a_0, a_1, a_2, a_3 \) in the exercise) can be viewed as weights that determine the contribution of each Pauli matrix to the final matrix \( M \).

Through the process outlined in the exercise, we learn that the identity matrix and the three Pauli matrices together can construct any 2x2 matrix by adjusting these weights. This speaks to the generality and the complete nature of the Pauli matrices in the realm of quantum operations.