Chapter 4: Q31P (page 178)

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

Short Answer

Expert verified

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

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 $m_{s} = - 1, 0, 1$ .

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

(4) $S_{\pm} | s m ⟩ = ℏ \sqrt{s (s + 1) - m (m \pm 1)} | s (m \pm 1) ⟩$

Step 2: Construct spin matrice Sz

For spin 1 particles, the eigen spinors are :

$\begin{array}{l} | χ ⟩_{+} = (\begin{array}{l} 1 \\ 0 \\ 0 \end{array}) \\ | χ ⟩_{0} = (\begin{array}{l} 0 \\ 1 \\ 0 \end{array}) \\ | χ ⟩_{-} = (\begin{array}{l} 0 \\ 0 \\ 1 \end{array}) \end{array}$

$S_{\pm} | s m ⟩ = ℏ \sqrt{s (s + 1) - m (m \pm 1)} | s (m \pm 1) ⟩$ ...(4.136)

$\begin{array}{l} S_{z} χ_{+} = ℏ χ_{+} \\ S_{z} χ_{0} = 0 \\ S_{z} χ_{-} = - ℏ χ_{-} \end{array}$

From Equation 4.136

$S_{z} = ℏ (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - 1 \end{matrix})$

Step 3: Construct spin matrice Sx

$\begin{array}{l} S_{+} χ_{+} = 0 \\ S_{-} χ_{+} = ℏ \sqrt{2} χ_{0} \\ S_{+} χ_{0} = ℏ \sqrt{2} χ_{+} \end{array}$ .

$\begin{array}{l} S_{+} χ_{-} = ℏ \sqrt{2} χ_{0} \\ S_{-} χ_{0} = ℏ \sqrt{2} χ_{-} \\ S_{-} χ_{-} = 0 \end{array}$

$\begin{array}{l} \Rightarrow S_{+} = \sqrt{2} ℏ (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) \\ S_{-} = \sqrt{2} ℏ (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) \end{array}$

$\begin{matrix} S_{x} = \frac{1}{2} (S_{+} + S_{-}) \\ = \frac{ℏ}{\sqrt{2}} (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) \end{matrix}$

Construct spin matrice Sy

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

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Construct the spin matrices $(S_{x}, S_{y} and S_{z})$ , for a particle of spin 1. Hint: How many eigenstates of $S_{z}$ are there? Determine the action of $S_{z}$ , $S_{+}$ , and $S_{-}$ 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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Construct the spin matrices(Sx,Sy andSz) , for a particle of spin 1. Hint: How many eigenstates ofSz are there? Determine the action of Sz, S+, and S−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

Modern Physics

Measurements

Conservation of Energy and Momentum

Electricity

Geometrical and Physical Optics

Force

Study anywhere. Anytime. Across all devices.

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