) and negative values (where ). Casimir invariants and the energy levels The Casimir invariants for negative energies are and have vanishing Poisson brackets with all components of D and L, C2 is trivially zero, since the two vectors are always perpendicular. However, the other invariant, C1, is non-trivial and depends only on m, k and E. Upon canonical quantization, this invariant allows the energy levels of hydrogen-like atoms to be derived using only quantum mechanical canonical commutation relations, instead of the conventional solution of the Schrödinger equation. This derivation is discussed in detail in the next section. Quantum mechanics of the hydrogen atom Poisson brackets provide a simple guide for quantizing most classical systems: the commutation relation of two quantum mechanical operators is specified by the Poisson bracket of the corresponding classical variables, multiplied by . By carrying out this quantization and calculating the eigenvalues of the 1 Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms (Figure 6) and, thus, their atomic emission spectrum. This elegant 1926 derivation was obtained before the development of the Schrödinger equation. A subtlety of the quantum mechanical operator for the LRL vector A is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of p and L must be defined carefully. Typically, the operators for the Cartesian components are defined using a symmetrized (Hermitian) product, Once this is done, one can show that the quantum LRL operators satisfy commutations relations exactly analogous to the Poisson bracket relations in the previous section—just replacing the Poisson bracket with times the commutator. From these operators, additional ladder operators for L can be defined, These further connect different eigenstates of L2, so different spin multiplets, among themselves. A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined, where is the inverse of the Hamiltonian energy operator, and is the identity operator. Applying these ladder operators to the eigenstates |ℓ〉 of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator, 1, are seen to be quantized, . Importantly, by dint of the vanishing of C2, they are independent of the ℓ and quantum numbers, making the energy levels degenerate. Hence, the energy levels are given by which coincides with the Rydberg formula for hydrogen-like atoms (Figure 6). The