"Angular momentum"

X1(f) = x1 f
X2(f) = x2 f
X3(f) = x3 f

P1(f) = -i hbar d(f,x1)
P2(f) = -i hbar d(f,x2)
P3(f) = -i hbar d(f,x3)

L1(f) = X2(P3(f)) - X3(P2(f))
L2(f) = X3(P1(f)) - X1(P3(f))
L3(f) = X1(P2(f)) - X2(P1(f))

Lsq(f) = L1(L1(f)) + L2(L2(f)) + L3(L3(f))

J1(f) = L1(f) / hbar
J2(f) = L2(f) / hbar
J3(f) = L3(f) / hbar

psi = Psi(x1,x2,x3)

-- 1

check(X1(L3(psi)) - L3(X1(psi)) == -i hbar X2(psi))
check(X2(L3(psi)) - L3(X2(psi)) == i hbar X1(psi))
check(X3(L3(psi)) - L3(X3(psi)) == 0)

-- 2

check(P1(L3(psi)) - L3(P1(psi)) == -i hbar P2(psi))
check(P2(L3(psi)) - L3(P2(psi)) == i hbar P1(psi))
check(P3(L3(psi)) - L3(P3(psi)) == 0)

-- 3

check(L1(Lsq(psi)) - Lsq(L1(psi)) == 0)
check(L2(Lsq(psi)) - Lsq(L2(psi)) == 0)
check(L3(Lsq(psi)) - Lsq(L3(psi)) == 0)

-- 4

check(J2(J3(psi)) - J3(J2(psi)) == i J1(psi))
check(J3(J1(psi)) - J1(J3(psi)) == i J2(psi))
check(J1(J2(psi)) - J2(J1(psi)) == i J3(psi))

-- 5

PxL = (P2(L3(psi)) - P3(L2(psi)),
       P3(L1(psi)) - P1(L3(psi)),
       P1(L2(psi)) - P2(L1(psi)))

LxP = (L2(P3(psi)) - L3(P2(psi)),
       L3(P1(psi)) - L1(P3(psi)),
       L1(P2(psi)) - L2(P1(psi)))

P = (P1(psi),P2(psi),P3(psi))

check(PxL + LxP == 2 i hbar P)

-- 6

check(1/2 (PxL - LxP) == PxL - i hbar P)

-- 7

epsilon = (((0,0,0),(0,0,1),(0,-1,0)),
           ((0,0,-1),(0,0,0),(1,0,0)),
           ((0,1,0),(-1,0,0),(0,0,0)))

S(I1,I2) = epsilon[I1,I2,1] P1(psi) + epsilon[I1,I2,2] P2(psi) + epsilon[I1,I2,3] P3(psi)

check(L1(P1(psi)) - P1(L1(psi)) == i hbar S(1,1))
check(L1(P2(psi)) - P2(L1(psi)) == i hbar S(1,2))
check(L1(P3(psi)) - P3(L1(psi)) == i hbar S(1,3))

check(L2(P1(psi)) - P1(L2(psi)) == i hbar S(2,1))
check(L2(P2(psi)) - P2(L2(psi)) == i hbar S(2,2))
check(L2(P3(psi)) - P3(L2(psi)) == i hbar S(2,3))

check(L3(P1(psi)) - P1(L3(psi)) == i hbar S(3,1))
check(L3(P2(psi)) - P2(L3(psi)) == i hbar S(3,2))
check(L3(P3(psi)) - P3(L3(psi)) == i hbar S(3,3))

"ok"