import numpy as np

# Put cursor in this cell and hit SHIFT+ENTER A = np.array([[1, 2]]) B = np.array([[3], [5]]) print(A) print(B) print(A @ B) #inner product

# Solution print(A.T, B.T) print(B.T@A.T) #innerproduct print(A.T@B.T) #outerproduct

# Solution print(1j*1j) print((1+3j)*(3-7j)) C = 3 + 4j print(C*np.conj(C))

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)

# z-axis print("z_plus normalized " , z_plus_bra@z_plus_ket) print(z_minus_bra@z_minus_ket) print("z_plus and z_minues orthogonal?", z_plus_bra@z_minus_ket) print(z_minus_bra@z_plus_ket) # x-axis print(x_plus_bra@x_plus_ket) print(x_minus_bra@x_minus_ket) print("x_plus and x_minus orthogonal", x_plus_bra@x_minus_ket) print(x_minus_bra@x_plus_ket) # y-axis print(y_plus_bra@y_plus_ket) print(y_minus_bra@y_minus_ket) print("y_plus and y_minus orthogonal", y_plus_bra@y_minus_ket) print(y_minus_bra@y_plus_ket)

A = x_plus_bra@z_plus_ket print(A) probA= A*np.conj(A) print("Probability of measuring spin-up alonf x-axis for +z state", probA) B = x_plus_bra@y_plus_ket print(B) print(B*np.conj(B))

psi_ket = 1j/np.sqrt(3)*z_plus_ket + np.sqrt(2/3)*z_minus_ket print(psi_ket) print(z_minus_bra @ psi_ket) A = z_plus_bra @ psi_ket B = z_minus_bra@psi_ket print(A[0],B[0]) Sz_minus_prob = np.conj(A)*A Sz_plus_prob = np.conj(B)*B print(Sz_plus_prob, Sz_minus_prob) Sz_expect = Sz_plus_prob - Sz_minus_prob print("<S_z> = ", Sz_expect, "hbar/2") Sz_sq_expect = Sz_plus_prob + Sz_minus_prob print("<S_z^2> =", Sz_sq_expect, "hbar^2/4") Delta_Sz = np.sqrt(Sz_sq_expect - Sz_expect**2) print("Delta S_z = ", Delta_Sz, "hbar/2") print(Delta_Sz - 2*np.sqrt(2)/3)