import numpy as np

z_plus_bra = np.array([[1,0]]) z_minus_bra = np.array([[0,1]]) z_plus_ket = z_plus_bra.T z_minus_ket = z_minus_bra.T x_plus_bra = 1/np.sqrt(2)*np.array([[1,1]]) x_minus_bra = 1/np.sqrt(2)*np.array([[1,-1]]) x_plus_ket = x_plus_bra.T x_minus_ket = x_minus_bra.T y_plus_bra = 1/np.sqrt(2)*np.array([[1, 1j]]) y_minus_bra = 1/np.sqrt(2)*np.array([[1, -1j]]) y_plus_ket = np.conj(y_plus_bra.T) y_minus_ket = np.conj(y_minus_bra.T)

psi_ket = (1/2)*z_plus_ket + 1j*(np.sqrt(3)/2)*z_minus_ket # a) Find <S_x> print("Find <S_x>") x_plus = x_plus_bra@psi_ket x_plus_prob = np.conj(x_plus)*x_plus print(x_plus_prob) x_minus = x_minus_bra@psi_ket x_minus_prob = np.conj(x_minus)*x_minus print(x_minus_prob) S_x = x_plus_prob - x_minus_prob print(S_x) # b) Find <S_y> print("Find <S_y>") y_plus = y_plus_bra@psi_ket y_plus_prob = np.conj(y_plus)*y_plus print(y_plus_prob) y_minus = y_minus_bra@psi_ket y_minus_prob = np.conj(y_minus)*y_minus print(y_minus_prob) S_y = y_plus_prob - y_minus_prob print(S_y) # Find <S_z> print("Find <S_z>") z_plus = z_plus_bra@psi_ket z_plus_prob = np.conj(z_plus)*z_plus print(z_plus_prob) z_minus = z_minus_bra@psi_ket z_minus_prob = np.conj(z_minus)*z_minus print(z_minus_prob) S_z = z_plus_prob - z_minus_prob print(S_z)

psi_1_ket = 1/2*z_plus_ket + 1j*np.sqrt(3)/2*z_minus_ket psi_2_ket = -1j/2*z_plus_ket + np.sqrt(3)/2*z_minus_ket # a) Calculate <psi_2|psi_1> print("Calculate <psi_2|psi_1>") psi_2_bra = np.conj(psi_2_ket.T) print(psi_2_bra@psi_1_ket) # b) Calculate |<psi_2|psi_1>|^2 print("Calculate |<psi_2|psi_1>|^2") amplitude = psi_2_bra@psi_1_ket prob = np.conj(amplitude)*amplitude print(prob)

psi_1_ket = 1/2*z_plus_ket + 1j*np.sqrt(3)/2*z_minus_ket psi_1_ket_perp = 2*z_plus_ket - 1j*np.sqrt(3)/2*z_minus_ket psi_1_bra_perp = np.conj(psi_1_ket_perp.T) psi_10_ket = 1/2*z_plus_ket + 1j*np.sqrt(3)/2*z_minus_ket psi_10_perp_ket = 1/(1/2)*z_plus_ket - 1j/(np.sqrt(3)/2)*z_minus_ket psi_10_perp_bra = np.conj(psi_10_perp_ket.T) psi_11_ket = -1j/2*z_plus_ket + np.sqrt(3)/2*z_minus_ket norm_perp = psi_10_perp_bra@psi_10_perp_ket print(norm_perp) amp = psi_10_perp_bra@psi_10_ket print(amp) amp2 = psi_10_perp_bra@psi_11_ket print(amp2)