Question
Show that the Pauli spin matrices (Problem 6.6) are Hermitian.
Step 1
They are given by: \[ \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \] Show more…
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Key Concepts
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(a) Show that the Pauli spin matrices (Problem 6.6) are Hermitian. (b) Show that the Pauli spin matrices satisfy the Jacobi identity $[\mathrm{A},[\mathrm{B}, \mathrm{C}]]+$ $[\mathrm{B},[\mathrm{C}, \mathrm{A}]]+[\mathrm{C},[\mathrm{A}, \mathrm{B}]]=0$ where $[\mathrm{A}, \mathrm{B}]$ is the commutator of $\mathrm{A}, \mathrm{B}[\text {see}(6.3)]$ (c) Generalize (b) to prove the Jacobi identity for any (conformable) matrices A, B, C. Also see Chapter 6, Problem 3.14.
Linear Algebra
Special Matrices and Formulas
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.
LINEAR EQUATIONS; VECTORS, MATRICES, AND DETERMINANTS
Matrix operations
Repeat Problem 30 for the Pauli spin matrix C in Problem 6. Hint: Show that if a matrix is diagonal, say $\mathrm{D}=\left(\begin{array}{cc}a & 0 \\ 0 & b\end{array}\right),$ then $f(\mathrm{D})=\left(\begin{array}{cc}f(a) & 0 \\ 0 & f(b)\end{array}\right).$
Matrix Operations
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