Question

The Pauli spin matrices in quantum mechanics are $$ A=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) . \quad B=\left(\begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right), \quad C=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) $$ Show that \(A^{2}=B^{2}=C^{2}=1\). (Note carefully that this 1 means the 2 by 2 unit matrix and not the number 1 ; this notation is customary in quantum mechanics.) Also show that any pair of these matrices anticommute, that is, \(A B=-B A\), etc. Show that the commutator of \(A\) and \(B\), that is, \(A B-B A\), is \(2 i C\), and similarly for other pairs in cyclic order.

Short answer

To show: A² = B² = C² = I, AB = -BA and similar for pairs, [A, B] = 2iC and similarly for other pairs.

Step-by-step solution

Understanding the Matrices

The given Pauli spin matrices are: \[ A = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \] We need to show two main results. First, the square of each matrix is the identity matrix. Second, any pair of these matrices anticommute, and the commutator follows specific rules.

Calculating \( A^2 \)

Calculate \( A^2 \): \[ A^2 = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + 1 \cdot 1 & 0 \cdot 1 + 1 \cdot 0 \ 1 \cdot 0 + 0 \cdot 1 & 1 \cdot 1 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = I \]

Calculating \( B^2 \)

Calculate \( B^2 \): \[ B^2 = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + (-i) \cdot i & 0 \cdot (-i) + (-i) \cdot 0 \ i \cdot 0 + 0 \cdot i & i \cdot (-i) + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = I \]

Calculating \( C^2 \)

Calculate \( C^2 \): \[ C^2 = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 0 \cdot 0 & 1 \cdot 0 + 0 \cdot -1 \ 0 \cdot 1 + (-1) \cdot 0 & 0 \cdot 0 + (-1) \cdot -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = I \]

Showing Anticommutation

Check the anticommutation of matrices. Calculate \( AB \): \[ AB = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + 1 \cdot i & 0 \cdot (-i) + 1 \cdot 0 \ 1 \cdot 0 + 0 \cdot i & 1 \cdot (-i) + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} \] Calculate \( BA \): \[ BA = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + (-i) \cdot 1 & 0 \cdot 1 + (-i) \cdot 0 \ i \cdot 0 + 0 \cdot 1 & i \cdot 1 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} =-\begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} \] Hence, \( AB = -BA \). A similar approach shows that other pairs of these matrices also anticommute.

Commutator Calculation

Calculate the commutator \( [A, B] = AB - BA \): \[ AB - BA = \begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} - -\begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} = \begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} + \begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} = 2 \begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} \] Noting that this is equal to \(2iC\), the calculation for other pairs will be similarly straightforward.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental branch of physics. It describes physical phenomena at the smallest scales, such as the behavior of electrons and photons. Unlike classical mechanics, it uses complex numbers, probability, and wave-particle duality.

One of the key aspects is the concept of 'quantum states' which can be represented using vectors. These vectors live in a mathematical space called a Hilbert space. In this context, operators act on the state vectors to extract physical information.

Pauli spin matrices are critical in quantum mechanics. They describe the spin property of particles like electrons. Spin is an intrinsic form of angular momentum carried by elementary particles. In the exercise, the Pauli matrices are represented as \( A, B, \) and \( C \). They are central to understanding the dynamics and interactions of quantum systems.

Commutator

A commutator is an important operation in matrix algebra, especially in quantum mechanics. For two matrices \(A\) and \(B\), the commutator is defined as \[ [A, B] = AB - BA \]

In simple terms, it measures how much two matrices fail to commute. Commuting operations are those where changing the order of operations does not change the result. If \([A, B] = 0\), then the matrices commute. In quantum mechanics, commuting operators have common eigenstates; thus, their physical quantities can be measured simultaneously.

In the provided solution, we showed that \[ [A, B] = 2iC \] This result is crucial because it showcases the non-commutative nature of Pauli spin matrices, and points to deeper principles about the structure of quantum theory.

Anticommutation

Anticommutation is another important concept. For matrices \( A \) and \( B \), they anticommute if \[ AB = -BA \] Unlike commutation, anticommutation involves a negative sign.

Anticommutators are prevalent in quantum mechanics. They often surface in the context of fermions, particles that follow Fermi-Dirac statistics. Pauli spin matrices, which represent spin-1/2 particles, demonstrate anticommutation relations.

For example, in the solution, we showed that: \[ AB = -BA \] and similar for other pairs. This property is key to the algebraic structure of quantum mechanics because it plays a role in defining the behavior of electrons and other fermions in various quantum systems.

Matrix Algebra

Matrix algebra is not just about numbers; it includes operations like addition, multiplication, and finding inverses. In quantum mechanics, matrices like the Pauli spin matrices encapsulate transformations and states.

The Pauli spin matrices \( A, B, \) and \( C \) have special properties. One such property is that their squares equal the identity matrix, \[ A^2 = B^2 = C^2 = I \] This is not trivial; it means each matrix, when squared, returns to the state of no transformation.

Moreover, learning how to compute commutators \[ [A, B] = AB - BA \] and understanding anticommutation \[ AB = -BA \] requires solid grounding in matrix algebra. These operations allow us to explore deeper quantum characteristics and behaviors. This is why mastering matrix operations is fundamental for any quantum physics student.