Ch2 Set 2 solutions

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) Jz_z = np.array([[1,0],[0,-1]]) #Jz operator in z-basis

print("R(" + "theta" + " * k)" + " in the Sz basis can be represented as:") Rotation_theta = np.array([[np.exp(-1j*np.pi/2),0],[0,np.exp(1j*np.pi/2)]]) print(Rotation_theta) Rotation_theta_transverse = np.conj(Rotation_theta.T) print("taking the tranverse(complex conjugate, we obtain") print(Rotation_theta_transverse) print() print("Applying this to find if R is unitary,") Unitary = z_plus_bra@Rotation_theta_transverse@Rotation_theta@z_plus_ket print(Unitary)

x_plus_ket_Sy = np.array([[y_plus_bra@x_plus_ket],[y_minus_bra@x_plus_ket]]) print(x_plus_ket_Sy) x_minus_ket_Sy = np.array([[y_plus_bra@x_minus_ket],[y_plus_bra@x_minus_ket]]) print(x_minus_ket_Sy)

print("We want to find the representation of jz in the sy basis. unlike our value of S before, we are converting from Sz basis to Sy, not Sx") #Sy = np.array([[(z_plus_bra@y_plus_ket), (z_plus_bra@y_minus_ket)],[(z_minus_bra@y_plus_ket),(z_minus_bra@y_minus_ket)]]) Sy = np.array([[1/np.sqrt(2), 1/np.sqrt(2)],[1j/np.sqrt(2),-1j/np.sqrt(2)]]) print(Sy) Jz_y = np.conj(Sy.T)@Jz_z@Sy print(Jz_y)