"Verify quantum LRL operators" R = sqrt(x^2 + y^2 + z^2) -- linear momentum operators Px(f) = -i hbar d(f,x) Py(f) = -i hbar d(f,y) Pz(f) = -i hbar d(f,z) -- angular momentum operators Lx(f) = y Pz(f) - z Py(f) Ly(f) = z Px(f) - x Pz(f) Lz(f) = x Py(f) - y Px(f) -- Laplace-Runge-Lenz (LRL) operators Ax(f) = 1 / (2 m) * (Py(Lz(f)) - Pz(Ly(f)) - Ly(Pz(f)) + Lz(Py(f))) - Z / R x f Ay(f) = 1 / (2 m) * (Pz(Lx(f)) - Px(Lz(f)) - Lz(Px(f)) + Lx(Pz(f))) - Z / R y f Az(f) = 1 / (2 m) * (Px(Ly(f)) - Py(Lx(f)) - Lx(Py(f)) + Ly(Px(f))) - Z / R z f -- squared operators P2(f) = Px(Px(f)) + Py(Py(f)) + Pz(Pz(f)) L2(f) = Lx(Lx(f)) + Ly(Ly(f)) + Lz(Lz(f)) A2(f) = Ax(Ax(f)) + Ay(Ay(f)) + Az(Az(f)) -- hamiltonian operator H(f) = 1 / (2 m) P2(f) - Z / R f -- psi is a generic function of x, y, and z psi = f(x,y,z) check(Ax(Lx(psi)) + Ay(Ly(psi)) + Az(Lz(psi)) == 0) check(Lx(Ax(psi)) + Ly(Ay(psi)) + Lz(Az(psi)) == 0) check(H(Ax(psi)) == Ax(H(psi))) check(H(Ay(psi)) == Ay(H(psi))) check(H(Az(psi)) == Az(H(psi))) check(A2(psi) == 2 / m H(L2(psi) + hbar^2 psi) + Z^2 psi) "ok" Run