Clebsch–Gordan Coefficients and Pion–Nucleon Amplitudes

Problem Statement

Using the Clebsch–Gordan coefficients for $I = 1/2$ plus $I = 1$, obtain the amplitudes displayed in Equations (1.2)–(1.4).

Step-by-Step Solution

Step 1: Write the Kets Using Clebsch–Gordan Coefficients

Write down the kets for $|\pi p\rangle$ using the Clebsch–Gordan coefficients of Table 1.2. Recall that the C–G coefficients are a real unitary transformation, so the same matrix element goes $|I_1, I_2\rangle \to |I, I_z\rangle$ as $|I, I_z\rangle \to |I_1, I_2\rangle$.

$$|p\pi^+\rangle = |I_1 = 1/2, I_2 = 1; +1/2, +1\rangle = |I = 3/2, I_z = +3/2\rangle$$

$$|p\pi^-\rangle = |I_1 = 1/2, I_2 = 1; +1/2, -1\rangle = \frac{|I = 3/2, I_z = -1/2\rangle - \sqrt{2}\,|I = 1/2, I_z = -1/2\rangle}{\sqrt{3}}$$

$$|n\pi^0\rangle = |I_1 = 1/2, I_2 = 1; -1/2, 0\rangle = \frac{\sqrt{2}\,|I = 3/2, I_z = -1/2\rangle + |I = 1/2, I_z = -1/2\rangle}{\sqrt{3}}$$

Step 2: Calculate the Amplitudes

Calculate the amplitudes keeping in mind that the $I = 3/2$ and $I = 1/2$ states are orthogonal:

$$\langle \pi^+ p|\,H_{3/2} + H_{1/2}\,|\pi^+ p\rangle = A_{3/2}$$

$$\langle \pi^- p|\,H_{3/2} + H_{1/2}\,|\pi^- p\rangle = \frac{A_{3/2} + 2A_{1/2}}{3}$$

$$\langle \pi^- p|\,H_{3/2} + H_{1/2}\,|\pi^0 n\rangle = \frac{\sqrt{2}(A_{3/2} - A_{1/2})}{3}$$

The magnitude squares of these amplitudes give Equations (1.2), (1.3), and (1.4).