Q31P Construct the spin matrices聽(Sx... [FREE SOLUTION]

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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 4: Q31P (page 178)

Construct the spin matrices $(S_{x}, S_{y} 鈥塧苍诲 S_{z})$ , for a particle of spin 1. Hint: How many eigenstates of $S_{z}$ are there? Determine the action of $S_{z}$ , $S_{+}$ , and $S_{鈭�}$ on each of these states. Follow the procedure used in the text for spin $1 / 2$ .

Short Answer

Expert verified

The matrices are $S_{x} = \frac{鈩�}{\sqrt{2}} (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix})$ $S_{y} = i \frac{鈩�}{\sqrt{2}} (\begin{matrix} 0 & 鈭� 1 & 0 \\ 1 & 0 & 鈭� 1 \\ 0 & 1 & 0 \end{matrix})$ and $S_{z} = 鈩� (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 鈭� 1 \end{matrix})$

Step by step solution

Important relations related to the solution

Important relations,

(1) $S_{z}$ is diagonal with its eigenvalues in the diagonal.

(2) Spin 1 particles have $m_{s} = 鈭� 1, 0, 1$ .

(3) The eigenvalue of $S_{z}$ for spin 1 particles are: $鈭� 鈩�, 0, 鈩�$ .

(4) $S_{卤} | s m 鉄� = 鈩� \sqrt{s (s + 1) 鈭� m (m 卤 1)} | s (m 卤 1) 鉄�$

Step 2: Construct spin matrice Sz

For spin 1 particles, the eigen spinors are :

$\begin{array}{l} | 蠂 {鉄�}_{+} = (\begin{array}{l} 1 \\ 0 \\ 0 \end{array}) \\ | 蠂 {鉄�}_{0} = (\begin{array}{l} 0 \\ 1 \\ 0 \end{array}) \\ | 蠂 {鉄�}_{鈭�} = (\begin{array}{l} 0 \\ 0 \\ 1 \end{array}) \end{array}$

$S_{卤} | s m 鉄� = 鈩� \sqrt{s (s + 1) 鈭� m (m 卤 1)} | s (m 卤 1) 鉄�$ ...(4.136)

$\begin{array}{l} S_{z} 蠂_{+} = 鈩� 蠂_{+} \\ S_{z} 蠂_{0} = 0 \\ S_{z} 蠂_{鈭�} = 鈭� 鈩� 蠂_{鈭�} \end{array}$

From Equation 4.136

$S_{z} = 鈩� (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 鈭� 1 \end{matrix})$

Step 3: Construct spin matrice Sx

$\begin{array}{l} S_{+} 蠂_{+} = 0 \\ S_{鈭�} 蠂_{+} = 鈩� \sqrt{2} 蠂_{0} \\ S_{+} 蠂_{0} = 鈩� \sqrt{2} 蠂_{+} \end{array}$ .

$\begin{array}{l} S_{+} 蠂_{鈭�} = 鈩� \sqrt{2} 蠂_{0} \\ S_{鈭�} 蠂_{0} = 鈩� \sqrt{2} 蠂_{鈭�} \\ S_{鈭�} 蠂_{鈭�} = 0 \end{array}$

$\begin{array}{l} 鈬� S_{+} = \sqrt{2} 鈩� (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) \\ S_{鈭�} = \sqrt{2} 鈩� (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) \end{array}$

$\begin{matrix} S_{x} = \frac{1}{2} (S_{+} + S_{鈭�}) \\ = \frac{鈩�}{\sqrt{2}} (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) \end{matrix}$

Construct spin matrice Sy

$\begin{matrix} S_{y} = \frac{1}{2 i} (S_{+} 鈭� S_{鈭�}) \\ = \frac{i 鈩�}{\sqrt{2}} (\begin{matrix} 0 & 鈭� 1 & 0 \\ 1 & 0 & 鈭� 1 \\ 0 & 1 & 0 \end{matrix}) \end{matrix}$

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91影视

Construct the spin matrices $(S_{x}, S_{y} 鈥塧苍诲 S_{z})$ , for a particle of spin 1. Hint: How many eigenstates of $S_{z}$ are there? Determine the action of $S_{z}$ , $S_{+}$ , and $S_{鈭�}$ on each of these states. Follow the procedure used in the text for spin $1 / 2$ .

Short Answer

Step by step solution

Important relations related to the solution

Step 2: Construct spin matrice Sz

Step 3: Construct spin matrice Sx

Construct spin matrice Sy

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Most popular questions from this chapter

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91影视

Construct the spin matrices(Sx,Sy鈥塧苍诲Sz) , for a particle of spin 1. Hint: How many eigenstates ofSz are there? Determine the action of Sz, S+, and S鈭�on each of these states. Follow the procedure used in the text for spin 1/2.

Short Answer

Step by step solution

Important relations related to the solution

Step 2: Construct spin matrice Sz

Step 3: Construct spin matrice Sx

Construct spin matrice Sy

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Famous Physicists

Translational Dynamics

Radiation

Quantum Physics

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Construct the spin matrices $(S_{x}, S_{y} 鈥塧苍诲 S_{z})$ , for a particle of spin 1. Hint: How many eigenstates of $S_{z}$ are there? Determine the action of $S_{z}$ , $S_{+}$ , and $S_{鈭�}$ on each of these states. Follow the procedure used in the text for spin $1 / 2$ .