Question

The Pauli spin matrices in quantum mechanics are $$\mathrm{A}=\left(\begin{array}{ll} 0 & 1 \\\1 & 0\end{array}\right), \quad \mathrm{B}=\left(\begin{array}{cc}0 & -i \\\i & 0 \end{array}\right), \quad \mathrm{C}=\left(\begin{array}{rr}1 & 0 \\\0 & -1\end{array}\right)$$ (You will probably find these called \(\sigma_{x}, \sigma_{y}, \sigma_{z}\) in your quantum mechanics texts.) Show that \(\mathrm{A}^{2}=\mathrm{B}^{2}=\mathrm{C}^{2}=\mathrm{a}\) unit matrix. Also show that any two of these matrices anticommute, that is, \(\mathrm{AB}=-\mathrm{BA},\) etc. Show that the commutator of \(\mathrm{A}\) and \(\mathrm{B},\) that is, \(\mathrm{AB}-\mathrm{BA},\) is \(2 i \mathrm{C},\) and similarly for other pairs in cyclic order.

Short answer

The square of each Pauli matrix is the unit matrix. Any two Pauli matrices anticommute, and the commutator of any two is given by a third matrix multiplied by 2i.

Step-by-step solution

- Matrix Squaring

First, calculate the square of matrix A.\( A^2 = \left(\begin{array}{cc} 0 & 1 \ 1 & 0\end{array}\right) \left(\begin{array}{cc} 0 & 1 \ 1 & 0\end{array}\right) \)

- Multiplication of A

\( A^2 = \left(\begin{array}{cc} 0 & 1 \ 1 & 0\end{array}\right) \left(\begin{array}{cc} 0 & 1 \ 1 & 0\end{array}\right) = \left(\begin{array}{cc} 1 & 0 \ 0 & 1\end{array}\right) \)

- Calculate B squared

\( B^2 = \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) \)

- Multiplication of B

\( B^2 = \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) = \left(\begin{array}{cc} 1 & 0 \ 0 & 1\end{array}\right) \)

- Calculate C squared

Last, calculate the square of matrix C.\( C^2 = \left(\begin{array}{cc} 1 & 0 \ 0 & -1\end{array}\right) \left(\begin{array}{cc} 1 & 0 \ 0 & -1\end{array}\right) \)

- Multiplication of C

\( C^2 = \left(\begin{array}{cc} 1 & 0 \ 0 & -1\end{array}\right) \left(\begin{array}{cc} 1 & 0 \ 0 & -1\end{array}\right) = \left(\begin{array}{cc} 1 & 0 \ 0 & 1\end{array}\right) \)

- Show anticommutation of A and B

Next, show that two matrices anticommute by calculating \( A B \) and \( B A \).\( A B = \left(\begin{array}{cc} 0 & 1 \ 1 & 0\end{array}\right) \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) = \left(\begin{array}{cc} i & 0 \ 0 & -i\end{array}\right) \).\( B A = \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) \left(\begin{array}{cc} 0 & 1 \ 1 & 0\end{array}\right) = \left(\begin{array}{cc} -i & 0 \ 0 & i\end{array}\right) \). Clearly, \( A B = - B A \).

- Show anticommutation of B and C

Calculate \( B C \) and \( C B \).\( B C = \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) \left(\begin{array}{cc} 1 & 0 \ 0 & -1\end{array}\right) = \left(\begin{array}{cc} 0 & i \ i & 0\end{array}\right) \).\( C B = \left(\begin{array}{cc} 1 & 0 \ 0 & -1\end{array}\right) \left(\begin{array}{cc} 0 & -i \ i & 0\end{array}\right) = \left(\begin{array}{cc} 0 & -i \ -i & 0\end{array}\right) \). Clearly, \( B C = - C B \).

- Show commutation of A and B

Lastly, find the commutator of A and B. \[ [A, B] = AB - BA = \left(\begin{array}{cc} 0 & i \ i & 0 \end{array}\right) - \left(\begin{array}{cc} -i & 0 \ 0 & i \end{array}\right) = 2i \left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{array}\right) = 2iC \]

Key concepts

quantum mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at small scales, such as those of molecules and atoms. Unlike classical mechanics, which uses deterministic principles, quantum mechanics is inherently probabilistic.

One of the core elements of quantum mechanics is the concept of spin. Spin is a type of intrinsic angular momentum carried by elementary particles such as electrons. The Pauli spin matrices are mathematical tools that represent spin-1/2 particles. These are essential in understanding the quantum behavior of particles as they interact with magnetic fields, among other things.

Here are the Pauli spin matrices:

• \( \text{A} = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \)
• \( \text{B} = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} \)
• \( \text{C} = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \)

These matrices facilitate the description and analysis of spin states, which is crucial for a deeper understanding of quantum mechanics.

matrix anticommutation

Matrix anticommutation is a mathematical property wherein the product of two matrices yields a result that is opposite in sign when the order of multiplication is reversed. Mathematically, two matrices \(A\) and \(B\) anticommute if:

\( AB = -BA \)

This property is significant in quantum mechanics. The Pauli matrices demonstrate this property, as seen by the following example:

• Calculate \( AB \):

\[ A B = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} = \begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} \]

• Calculate \( BA \):

\[ B A = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \begin{pmatrix} -i & 0 \ 0 & i \end{pmatrix} \]

Clearly, \( AB \) and \( BA \) are negatives of each other, illustrating the anticommutation property.

commutator

The commutator of two matrices \(A\) and \(B\) is a measure of how much these matrices fail to commute. This concept is crucial in quantum mechanics for understanding the incompatibility of certain measurable quantities.

The commutator \[ [A, B] \] is defined as:

\[ [A, B] = AB - BA \]

For the Pauli matrices, let's compute the commutator \[ [A, B] \]:

\[ [A, B] = AB - BA = \begin{pmatrix} 0 & i \ i & 0 \end{pmatrix} - \begin{pmatrix} -i & 0 \ 0 & i \end{pmatrix} = 2i \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \]

This result illustrates that the commutator is not zero, indicating that matrices \(A\) and \(B\) do not commute. It shows a specific relationship between them that can be pivotal for understanding spin and other quantum properties.

linear algebra

Linear algebra is a branch of mathematics concerning linear equations, matrices, vector spaces, and their transformations. It provides the language and framework for understanding many physical phenomena, including those described by quantum mechanics.

Matrices are fundamental objects in linear algebra. They can represent linear transformations, making them essential for physicists and engineers. Understanding how matrices interact—whether they commute, anticommute, or how their commutators behave—provides vital insights.

The Pauli matrices are classic examples used often in quantum mechanics. Their properties, like squaring to the identity matrix, demonstrate key linear algebra concepts:

\[ A^2 = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]

The product equals the identity matrix, a fundamental concept in linear algebra indicating that multiplying the Pauli matrix by itself gives back an unchanged matrix function that acts as the identity.