"Exercise 6.10"

u = (1,0)
d = (0,1)

uu = kronecker(u,u)
ud = kronecker(u,d)
du = kronecker(d,u)
dd = kronecker(d,d)

sigmaz = ((1,0),(0,-1))
sigmax = ((0,1),(1,0))
sigmay = ((0,-i),(i,0))

tauz = ((1,0),(0,-1))
taux = ((0,1),(1,0))
tauy = ((0,-i),(i,0))

I = ((1,0),(0,1))

sigmaz = kronecker(sigmaz,I)
sigmax = kronecker(sigmax,I)
sigmay = kronecker(sigmay,I)

tauz = kronecker(I,tauz)
taux = kronecker(I,taux)
tauy = kronecker(I,tauy)

sing = (ud - du) / sqrt(2)

T1 = (ud + du) / sqrt(2)
T2 = (uu + dd) / sqrt(2)
T3 = (uu - dd) / sqrt(2)

H = dot(sigmax,taux) + dot(sigmay,tauy) + dot(sigmaz,tauz)
H = 1/2 omega H
H

I = unit(4)

"Characteristic polynomial"

p = det(H - lambda I)
p

"Verify eigenvalues"

check(eval(p,lambda,-3 omega / 2) == 0)
check(eval(p,lambda,omega / 2) == 0)

"ok"

"Verify eigenvectors"

check(dot(H,sing) == -3/2 omega sing)
check(dot(H,T1) == 1/2 omega T1)
check(dot(H,T2) == 1/2 omega T2)
check(dot(H,T3) == 1/2 omega T3)

"ok"

"Verify other formulas"

check(uu = (T2 + T3) / sqrt(2))
check(ud = (T1 + sing) / sqrt(2))
check(du = (T1 - sing) / sqrt(2))
check(dd = (T2 - T3) / sqrt(2))

check(dot(H,uu) == 1/2 omega uu)
check(dot(H,ud) == omega du - 1/2 omega ud)
check(dot(H,du) == omega ud - 1/2 omega du)
check(dot(H,dd) == 1/2 omega dd)

"ok"