Question

The Pauli spin matrices $$ \sigma^{1}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma^{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma^{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right) $$ describe a particle with spin \(\frac{1}{2}\) in nonrelativistic quantum mechanics. Verify that these matrices satisfy \(\left[\sigma^{i}, \sigma^{j}\right] \equiv \sigma^{i} \sigma^{j}-\sigma^{j} \sigma^{i}=2 i \epsilon_{k}^{i j} \sigma^{k}, \quad\left\\{\sigma^{i}, \sigma^{j}\right\\} \equiv \sigma^{i} \sigma^{j}+\sigma^{j} \sigma^{i}=2 \delta_{j}^{i} 1_{2}\), where \(1_{2}\) is the unit \(2 \times 2\) matrix. Show also that \(\sigma^{i} \sigma^{j}=i \epsilon_{k}^{i j} \sigma^{k}+\delta_{j}^{i} 1_{2}\), and for any two vectors a and \(\mathbf{b},(\boldsymbol{\sigma} \cdot \mathbf{a})(\boldsymbol{\sigma} \cdot \mathbf{b})=\mathbf{a} \cdot \mathbf{b} 1_{2}+i \boldsymbol{\sigma} \cdot(\mathbf{a} \times \mathbf{b})\).

Short answer

yes, the given relations for the Pauli spin matrices are satisfied. Furthermore, \( \sigma^i \sigma^j = i \epsilon_k^{ij} \sigma^k + \delta_j^i 1_2 \) and \( (\boldsymbol{\sigma} \cdot \mathbf{a})(\boldsymbol{\sigma} \cdot \mathbf{b}) = \mathbf{a} \cdot \mathbf{b} 1_{2} + i \boldsymbol{\sigma} \cdot(\mathbf{a} \times \mathbf{b}) \).

Step-by-step solution

Use of definitions, properties and commutation relations

The commutation relation is defined as \( [A, B] = AB - BA \). The anticommutation of A and B is given by \( \{A, B\} = AB + BA \). Use these definitions along with given Pauli spin matrices and calculate the left hand side of both the commutation and anticommutation relations.

Calculation of matrix multiplications

Perform matrix multiplication for \( \sigma^i \sigma^j \) and \( \sigma^j \sigma^i \).

Calculation of commutation and anticommutation

Subtract the results against each other to calculate \( [\sigma^i, \sigma^j] \) and sum them up to calculate \( \{\sigma^i, \sigma^j\} \).

Comparison with left hand side

Compare the computed expressions with the right side of the given relations. You should find that they match.

Express \( \sigma^i \sigma^j \) as givens

Using the previous results, write down the expression for \( \sigma^i \sigma^j \) in terms of \( i \epsilon_k^{ij} \sigma^k \) and \( \delta_j^i 1_2 \).

Apply rules to vectors a and b

Implement the definition of the dot product \( \mathbf a \cdot \mathbf b \) and the cross product \( \mathbf a \times \mathbf b \) to vectors a and b, and compute the result of \( (\boldsymbol{\sigma} \cdot \mathbf{a})(\boldsymbol{\sigma} \cdot \mathbf{b}) \).

Key concepts

Quantum Mechanics and Pauli Spin Matrices

Quantum mechanics is a fundamental theory in physics that describes the behavior of particles at the microscopic level, including atoms and subatomic particles. A significant aspect of quantum mechanics is the concept of spin, which is an intrinsic form of angular momentum carried by elementary particles.
In the case of particles with a spin quantum number of \frac{1}{2}, such as electrons, the Pauli spin matrices are instrumental. These matrices, denoted as \( \(sigma^1\), \(sigma^2\), \(sigma^3\) \), serve as the mathematical representation of spin operators in nonrelativistic quantum mechanics.
Let's consider the example of the Pauli spin matrices presented in the exercise. Each matrix corresponds to one of the three spatial dimensions and represents the spin component along that axis. When dealing with these matrices, one of the main tasks is to verify certain properties such as commutation and anticommutation relations which reflect the physical behavior of spin in quantum states.

• Commutation relations reveal how the sequential measurement of different spin components affects quantum states.
• Anticommutation relations highlight another aspect of quantum behavior, related to the simultaneous measurability of spin components.

The analytical verification of these properties using Pauli spin matrices forms a foundational exercise in understanding quantum mechanics. It showcases the non-commutative nature of quantum observables and underlines the mathematical framework that governs the quantum world.

Matrix Multiplication of Pauli Spin Matrices

Matrix multiplication is a key operation in linear algebra and is particularly important in the context of quantum mechanics. In the case of the Pauli spin matrices, understanding matrix multiplication and its properties is crucial for exploring the spin properties of particles.
To multiply two matrices, we take the dot product of the rows of the first matrix with the columns of the second matrix. Each element \(c_{ij}\) of the resulting matrix \(C\) is computed as the sum of the products of the corresponding entries from the row \(i\) of the first matrix and the column \(j\) of the second matrix.

Steps to Multiply Pauli Spin Matrices

• Select two Pauli matrices to multiply, say \( \(sigma^i\) \) and \( \(sigma^j\) \).
• Perform the dot product of rows of \( \(sigma^i\) \) with columns of \( \(sigma^j\) \) to get a new 2x2 matrix.
• The resulting matrix captures the effect of two successive spin operations along the \(i\)-th and \(j\)-th axes.

In our exercise, these multiplication rules are used to verify the relationships involving commutation and anticommutation, which in turn reveals properties about the angular momentum operators of quantum particles. The exercise entails computing products like \( \(sigma^i\) \(sigma^j\) \) and \( \(sigma^j\) \(sigma^i\) \) and requires careful attention to the sign and complex numbers involved in these matrices.

Commutation and Anticommutation Relations in Quantum Mechanics

Commutation and anticommutation relations are foundational in quantum mechanics as they describe the symmetry properties and the behavior of quantum observables under measurements. These relations are critical in understanding how different operators, representing physical observables, relate to each other.
A commutator \( [A, B] \) of two operators \(A\) and \(B\) is defined as \( AB - BA \), whereas their anticommutator \( \{A, B\} \) is given by \( AB + BA \).

• If the commutator \( [A, B] = 0 \), the operators \(A\) and \(B\) are said to commute, meaning they can be measured simultaneously with certainty.
• If the commutator is non-zero, the operators do not commute, which indicates the presence of intrinsic quantum uncertainty (Heisenberg's uncertainty principle).
• Anticommutators are used for describing fermions, particles which adhere to Pauli's exclusion principle.

In our exercise, the commutation and anticommutation of Pauli spin matrices inform us about the algebra of spin \frac{1}{2} particles. Verifying these relations involves calculating the product of matrices and examining whether they satisfy the expected theoretical expressions. The properties verified through these relations are essential in the formulation of spin dynamics and are directly applicable to quantum systems, such as electron configurations, quantum computing, and spintronics.