Problem 24 - Advanced Engineering Mathematics

Question

(a) Two \(n \times n\) matrices \(\mathbf{A}\) and \(\mathbf{B}\) are said to anticommute if \(\mathbf{A B}=-\mathbf{B A}\). Show that each of the Pauli spin matrices $$ \sigma_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \sigma_{y}=\left(\begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right) \quad \sigma_{z}=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) $$ where \(i^{2}=-1\), anticommutes with the others. Pauli spin matrices are used in quantum mechanics. (b) The matrix \(\mathbf{C}=\mathbf{A B}-\mathbf{B A}\) is said to be the commutator of the \(n \times n\) matrices \(\mathbf{A}\) and \(\mathbf{B}\). Find the commutators of \(\sigma_{x}\) and \(\sigma_{y}, \sigma_{y}\) and \(\sigma_{z}\), and \(\sigma_{z}\) and \(\sigma_{x}\).

Step-by-Step Solution

Answer

(a) Each pair of Pauli matrices anticommutes. (b) Commutators: \([\sigma_x, \sigma_y] = 2i\sigma_z\), \([\sigma_y, \sigma_z] = 2i\sigma_x\), \([\sigma_z, \sigma_x] = 2i\sigma_y\).

Step 1: Verify Anticommutation for \( \sigma_x \) and \( \sigma_y \)

To prove that \( \sigma_x \) and \( \sigma_y \) anticommute, calculate \( \sigma_x \sigma_y \) and \( \sigma_y \sigma_x \). First, compute \( \sigma_x \sigma_y \):\[ \sigma_x \sigma_y = \left(\begin{array}{cc} 0 & 1 \ 1 & 0 \end{array}\right) \left(\begin{array}{cc} 0 & -i \ i & 0 \end{array}\right) = \left(\begin{array}{cc} i & 0 \ 0 & -i \end{array}\right) \]Next, compute \( \sigma_y \sigma_x \):\[ \sigma_y \sigma_x = \left(\begin{array}{cc} 0 & -i \ i & 0 \end{array}\right) \left(\begin{array}{cc} 0 & 1 \ 1 & 0 \end{array}\right) = \left(\begin{array}{cc} -i & 0 \ 0 & i \end{array}\right) \]These two results show \( \sigma_x \sigma_y = -\sigma_y \sigma_x \), confirming they anticommute.

Step 2: Verify Anticommutation for \( \sigma_y \) and \( \sigma_z \)

To verify that \( \sigma_y \) and \( \sigma_z \) anticommute, calculate \( \sigma_y \sigma_z \) and \( \sigma_z \sigma_y \). First, compute \( \sigma_y \sigma_z \):\[ \sigma_y \sigma_z = \left(\begin{array}{cc} 0 & -i \ i & 0 \end{array}\right) \left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right) = \left(\begin{array}{cc} 0 & i \ i & 0 \end{array}\right) \]Next, compute \( \sigma_z \sigma_y \):\[ \sigma_z \sigma_y = \left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right) \left(\begin{array}{cc} 0 & -i \ i & 0 \end{array}\right) = \left(\begin{array}{cc} 0 & -i \ -i & 0 \end{array}\right) \]The result \( \sigma_y \sigma_z = -\sigma_z \sigma_y \) confirms they anticommute.

Step 3: Verify Anticommutation for \( \sigma_z \) and \( \sigma_x \)

To verify that \( \sigma_z \) and \( \sigma_x \) anticommute, calculate \( \sigma_z \sigma_x \) and \( \sigma_x \sigma_z \). First, compute \( \sigma_z \sigma_x \):\[ \sigma_z \sigma_x = \left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right) \left(\begin{array}{cc} 0 & 1 \ 1 & 0 \end{array}\right) = \left(\begin{array}{cc} 0 & 1 \ -1 & 0 \end{array}\right) \]Next, compute \( \sigma_x \sigma_z \):\[ \sigma_x \sigma_z = \left(\begin{array}{cc} 0 & 1 \ 1 & 0 \end{array}\right) \left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right) = \left(\begin{array}{cc} 0 & -1 \ 1 & 0 \end{array}\right) \]The equality \( \sigma_z \sigma_x = -\sigma_x \sigma_z \) confirms they anticommute.

Step 4: Calculate the Commutator \( [\sigma_x, \sigma_y] \)

The commutator is defined as \( [\sigma_x, \sigma_y] = \sigma_x \sigma_y - \sigma_y \sigma_x \). Using previous results:\[ \sigma_x \sigma_y = \left(\begin{array}{cc} i & 0 \ 0 & -i \end{array}\right), \quad \sigma_y \sigma_x = \left(\begin{array}{cc} -i & 0 \ 0 & i \end{array}\right) \]Substitute and simplify:\[ [\sigma_x, \sigma_y] = \left(\begin{array}{cc} i & 0 \ 0 & -i \end{array}\right) - \left(\begin{array}{cc} -i & 0 \ 0 & i \end{array}\right) = \left(\begin{array}{cc} 2i & 0 \ 0 & -2i \end{array}\right) \]

Step 5: Calculate the Commutator \( [\sigma_y, \sigma_z] \)

The commutator \( [\sigma_y, \sigma_z] = \sigma_y \sigma_z - \sigma_z \sigma_y \). Use previous results:\[ \sigma_y \sigma_z = \left(\begin{array}{cc} 0 & i \ i & 0 \end{array}\right), \quad \sigma_z \sigma_y = \left(\begin{array}{cc} 0 & -i \ -i & 0 \end{array}\right) \]Substitute and simplify:\[ [\sigma_y, \sigma_z] = \left(\begin{array}{cc} 0 & i \ i & 0 \end{array}\right) - \left(\begin{array}{cc} 0 & -i \ -i & 0 \end{array}\right) = \left(\begin{array}{cc} 0 & 2i \ 2i & 0 \end{array}\right) \]

Step 6: Calculate the Commutator \( [\sigma_z, \sigma_x] \)

The commutator \( [\sigma_z, \sigma_x] = \sigma_z \sigma_x - \sigma_x \sigma_z \). Use previous results:\[ \sigma_z \sigma_x = \left(\begin{array}{cc} 0 & 1 \ -1 & 0 \end{array}\right), \quad \sigma_x \sigma_z = \left(\begin{array}{cc} 0 & -1 \ 1 & 0 \end{array}\right) \]Substitute and simplify:\[ [\sigma_z, \sigma_x] = \left(\begin{array}{cc} 0 & 1 \ -1 & 0 \end{array}\right) - \left(\begin{array}{cc} 0 & -1 \ 1 & 0 \end{array}\right) = \left(\begin{array}{cc} 0 & 2 \ -2 & 0 \end{array}\right) \]

Key Concepts

Anticommutation

In mathematics, particularly in the field of quantum mechanics, anticommutation describes a special relationship between two matrices. Two matrices are said to anticommute if their product results in the negative of the reverse order product. For matrices \( \mathbf{A} \) and \( \mathbf{B} \), they anticommute if \( \mathbf{A} \mathbf{B} = - \mathbf{B} \mathbf{A} \). This unique property is vital in various quantum mechanical calculations.

The Pauli matrices, which are fundamental in quantum mechanics, provide an excellent example of anticommutation. The matrices \( \sigma_x \), \( \sigma_y \), and \( \sigma_z \) demonstrate this property when multiplied with each other, for example, \( \sigma_x \sigma_y = -\sigma_y \sigma_x \). This property is not just a mathematical curiosity but reflects the underlying characteristics of quantum systems, such as spin in quantum mechanics.

Commutator

The concept of a commutator is central in both quantum mechanics and matrix algebra. For two matrices \( \mathbf{A} \) and \( \mathbf{B} \), the commutator is defined as \( [\mathbf{A}, \mathbf{B}] = \mathbf{A} \mathbf{B} - \mathbf{B} \mathbf{A} \). It essentially measures the degree to which two matrices fail to commute.

This concept is critical in quantum mechanics, as commutators are tied to the fundamental uncertainty principle. If the commutator of two observables (represented by matrices) is zero, it means they can be measured precisely at the same time. For Pauli matrices, computing commutators like \( [\sigma_x, \sigma_y] \) helps in understanding spin interactions and their incompatibilities.

Quantum Mechanics

Quantum mechanics is the branch of physics dealing with the behavior of very small particles, like atoms and subatomic particles. In quantum mechanics, the properties of these particles are described using mathematics, more specifically, matrix mathematics. The state of a quantum system is represented by vectors, and observables (like position, momentum, and spin) by matrices, often known as operators.

The Pauli matrices are crucial in describing quantum systems involving spin. These matrices help represent spin states and are involved in calculations of electron spin in magnetic fields. Understanding how these matrices work, including their commutation and anticommutation properties, allows physicists to predict and explain the behavior of quantum systems.

Matrix Algebra

Matrix algebra is a field of mathematics used to handle and manipulate arrays of numbers. A matrix is essentially a rectangular array of numbers organized in rows and columns which can be used to solve systems of linear equations, among other things.

In quantum physics, matrix algebra becomes indispensable due to its ability to represent complex systems in compact forms. The Pauli matrices are 2x2 matrices used extensively in quantum mechanics for representing the spin operators. Familiarity with matrix multiplication and properties like commutation and anticommutation is imperative. These operations have a deep connection to the physical properties they are used to model in quantum mechanics.