Question

l. An electron is in a general spin function ψ-: ca-c2ß where c1 and c2 are constants. Calculate (a)5? ψ , (b) Š , , and (c) (s? + Š) ψ . (d) What is the relationship between G and c2 if we require that ψ is normalized?

Answer 1

psi = c_alpha + c_2 Beta S^x = 1/2 (1/2 + 1) times alpha S^2 Beta = 1/2 (1/2 + 1) h^2 Beta Therefore S^2 psi = S^2 (C_1 alpha + C_2 Beta) = C_1 S^2 alpha + C_2 S^2 Beta = C_1 1/2 (1/2 + 1) h^2 alpha + 1/2 (1/2 + 1) h^2 Beta = 1/2 (1/2 + 1) h^2 (c_1 alpha + c_2 Beta) S^2 psi = 3/4 h^2 psi S^_z alpha = 1/2 h alpha, S^_z Beta = -1/2 h Beta S^2_z psi = S^2_z (C_1 alpha + C_2 Beta) = C_1 [S^z S^_z alpha] + c_2 [S_z S^_z Beta] = c_1 S^_z (1/2 h alpha) + c_2 S^_z (- 1/2 h Beta) = c_1 (1/2 h)^2 alpha + c_2 (- 1/2 K)^2 Beta \r\n S^2_z psi = h^2/4 (c_1 alpha + c_2 Beta) S^2_z psi = h^2/4 psi S^2 = S^2_x + S^2_y + S^2_z