page_title: Laplace–Runge–Lenz vector
section_title: Quantum mechanics of the hydrogen atom
hierarchical_section_title: Laplace–Runge–Lenz vector / Quantum mechanics of the hydrogen atom
caption_attribution_description: English: Diagram of the quantized energy levels of hydrogen. The fine structure level splitting (j=l +- ½) is also depicted, but invisible due to their nearness. Made by me using my own GPL software (written and run on 1 December 2006), visualized with Grace (also on 1 December 2006) and hereby released under the GFDL.
context_page_description: In classical mechanics, the Laplace–Runge–Lenz vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
  The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today.
  In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system.
context_section_description: 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  iħ.
  By carrying out this quantization and calculating the eigenvalues of the C₁ 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 Aₛ  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 H⁻¹ is the inverse of the Hamiltonian energy operator, and I is the identity operator.
  Applying these ladder operators to the eigenstates |ℓmn〉 of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator, C₁, are seen to be quantized, n² − 1.  Importantly, by dint of the vanishing of C₂, they are independent of the ℓ and m 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 additional symmetry operators A have connected the different ℓ multiplets among themselves, for a given energy (and C₁), dictating  n² states at each level. In effect, they have enlarged the angular momentum group SO(3) to SO(4)/ℤ₂ ~ SO(3) × SO(3).
caption: Figure 6: Energy levels of the hydrogen atom as predicted from the commutation relations of angular momentum and Laplace–Runge–Lenz vector operators; these energy levels have been verified experimentally.
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