Q26P a) Check that the spin matrices ... [FREE SOLUTION]

91影视

Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 4: Q26P (page 177)

a) Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134.
$S_{z} = \frac{h}{2} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (4.145) . S_{x} = \frac{h}{2} (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), s_{y} = \frac{h}{2} (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) (4.147) . [S_{x}, S_{y}] = i h S_{z}, [S_{y}, S_{z}] = i h S_{x}, [S_{z}, S_{x}] = i h S_{y} (4.134) . (b) Show that the Pauli spin matrices (Equation 4.148) satisfy the product rule 蟽_{x} 鈮� (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), 蟽_{y} 鈮� (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), 蟽_{z} 鈮� (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (4.148) . 蟽_{j} 蟽_{k} = 未_{jk} + i {\underset{I}{鈭� \overset{'}{o}}}_{jkl} 蟽_{I}, (4.153) .$
$Where the indices stand for x, y, or z, and {\overset{'}{o}}_{jkl} is the Levi - Civita symbol : + 1 ifjkl = 123, 231, or 2 = 312; - 1 if jkl = 132, 213, or 321; otherwise .$

Short Answer

Expert verified

$(a) [S_{x}, S_{y}] = i h S_{z} - (b) 蟽_{j} 蟽_{k} = 未_{jk} + i \underset{I}{鈭� \overset{'}{o_{jkl}} 蟽_{I} -}$

Step by step solution

(a) Checking the spin matrices obey the fundamental commutation relations for angular momentum.

The spin matrices are

$S_{z} = \frac{h}{2} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) S_{x} = \frac{h}{2} (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), S_{y} = \frac{h}{2} (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) [S_{x}, S_{y}] = S_{x} S_{y} - S_{y} S_{x} = \frac{h^{2}}{2} [(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) - (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})] . = \frac{h^{2}}{4} [(\begin{matrix} i & 0 \\ 0 & - i \end{matrix}) - (\begin{matrix} - i & 0 \\ 0 & i \end{matrix})] = \frac{h^{2}}{4} (\begin{matrix} 2 i & 0 \\ 0 & - 2 i \end{matrix}) = i h \frac{h^{2}}{2} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) [S_{x}, S_{y}] = i h S_{z}$

(b) Showing the Pauli spin matrices satisfies the product rule.

$蟽_{x} 蟽_{x} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) = 1 = 蟽_{y} 蟽_{y} = 蟽_{z} 蟽_{z}, S o 蟽_{j} 蟽_{j} = 1 f o r j = x, y, o r z . 蟽_{x} 蟽_{y} = (\begin{matrix} i & 0 \\ 0 & - i \end{matrix}) = i 蟽_{z}; 蟽_{y} 蟽_{z} = (\begin{matrix} 0 & i \\ i & 0 \end{matrix}) = i 蟽_{x}; 蟽_{z} 蟽_{x} = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) = i 蟽_{y} . 蟽_{y} 蟽_{x} = (\begin{matrix} - i & 0 \\ 0 & - i \end{matrix}) = i 蟽_{z}; 蟽_{z} 蟽_{y} = (\begin{matrix} 0 & - i \\ - i & 0 \end{matrix}) = - i 蟽_{x}; 蟽_{x} 蟽_{z} = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) = - i 蟽_{y} .$

Equation 4.153 packages all this in a single formula.

$蟽_{j} 蟽_{k} = 未_{jk} + i \underset{I}{鈭�} \overset{'}{o_{jkl} 蟽_{I} .}$

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Short Answer

Step by step solution

(a) Checking the spin matrices obey the fundamental commutation relations for angular momentum.

(b) Showing the Pauli spin matrices satisfies the product rule.

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