Editing Laplace–Runge–Lenz vector (section)

== Poisson brackets ==
=== Unscaled functions ===
The algebraic structure of the problem is, as explained in later sections, {{math|1=SO(4)/'''Z'''<sub>2</sub> ~ SO(3) × SO(3)}}.<ref name="bargmann_1936" />
The three components ''L<sub>i</sub>'' of the angular momentum vector {{math|'''L'''}} have the Poisson brackets<ref name="goldstein_1980" />
<math display="block">
\{ L_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s,
</math>
where {{mvar|i}}=1,2,3 and {{math|''ε<sub>ijs</sub>''}} is the fully [[antisymmetric tensor]], i.e., the [[Levi-Civita symbol]]; the summation index {{mvar|s}} is used here to avoid confusion with the force parameter {{mvar|k}} defined [[#Mathematical definition|above]]. Then since the LRL vector {{math|'''A'''}} transforms like a vector, we have the following Poisson bracket relations between {{math|'''A'''}} and {{math|'''L'''}}:<ref>{{harvnb|Hall|2013}} Proposition 17.25.</ref>
<math display="block">\{A_i,L_j\}=\sum_{s=1}^3\varepsilon_{ijs}A_s.</math> 
Finally, the Poisson bracket relations between the different components of {{math|'''A'''}} are as follows:<ref>{{harvnb|Hall|2013}} Proposition 18.7; note that Hall uses a different normalization of the LRL vector.</ref>
<math display="block">\{A_i,A_j\}=-2mH\sum_{s=1}^3\varepsilon_{ijs}L_s,</math>
where <math>H</math> is the Hamiltonian. Note that the span of the components of {{math|'''A'''}} and the components of {{math|'''L'''}} is not closed under Poisson brackets, because of the factor of <math>H</math> on the right-hand side of this last relation.

Finally, since both {{math|'''L'''}} and {{math|'''A'''}} are constants of motion, we have
<math display="block">\{A_i, H\} = \{L_i, H\} = 0.</math>

The Poisson brackets will be extended to quantum mechanical [[canonical commutation relation|commutation relations]] in the [[#Quantum mechanics of the hydrogen atom|next section]] and to [[Lie algebra|Lie bracket]]s in a [[#Conservation and symmetry|following section]].

=== Scaled functions ===
As noted [[#Alternative scalings, symbols and formulations|below]], a scaled Laplace–Runge–Lenz vector {{math|'''D'''}} may be defined with the same units as angular momentum by dividing {{math|'''A'''}} by <math display="inline">p_0 = \sqrt{2m|H|}</math>. Since {{math|'''D'''}} still transforms like a vector, the Poisson brackets of {{math|'''D'''}} with the angular momentum vector {{math|'''L'''}} can then be written in a similar form<ref name="bargmann_1936" /><ref name="bohm_1993" />
<math display="block">
\{ D_i, L_j\} = \sum_{s=1}^3 \varepsilon_{ijs} D_s.
</math>

The Poisson brackets of {{math|'''D'''}} with ''itself'' depend on the [[sign (mathematics)|sign]] of {{mvar|H}}, i.e., on whether the energy is [[negative number|negative]] (producing closed, elliptical orbits under an inverse-square central force) or [[positive number|positive]] (producing open, hyperbolic orbits under an inverse-square central force). For ''negative'' energies—i.e., for bound systems—the Poisson brackets are<ref name="Hall 2013">{{harvnb|Hall|2013}} Theorem 18.9.</ref>
<math display="block">
\{ D_i, D_j\} = \sum_{s=1}^3 \varepsilon_{ijs} L_s.
</math>
We may now appreciate the motivation for the chosen scaling of {{math|'''D'''}}: With this scaling, the Hamiltonian no longer appears on the right-hand side of the preceding relation. Thus, the span of the three components of {{math|'''L'''}} and the three components of {{math|'''D'''}} forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to {{math|so(4)}}, the Lie algebra of the 4-dimensional rotation group {{math|SO(4)}}.<ref name="ReferenceA">{{harvnb|Hall|2013|at=Section 18.4.4.}}</ref>

By contrast, for ''positive'' energy, the Poisson brackets have the opposite sign,
<math display="block">
\{ D_i, D_j\} = -\sum_{s=1}^3 \varepsilon_{ijs} L_s.
</math>
In this case, the Lie algebra is isomorphic to {{math|so(3,1)}}.

The distinction between positive and negative energies arises because the desired scaling—the one that eliminates the Hamiltonian from the right-hand side of the Poisson bracket relations between the components of the scaled LRL vector—involves the ''square root'' of the Hamiltonian. To obtain real-valued functions, we must then take the absolute value of the Hamiltonian, which distinguishes between positive values (where <math>|H| = H</math>) and negative values (where <math>|H| = -H</math>).

=== Laplace-Runge-Lenz operator for the hydrogen atom in momentum space ===
Scaled Laplace-Runge-Lenz operator in the momentum space was found in 2022 .<ref>{{cite journal |last1=Efimov |first1=S.P. |title=Coordinate space modification of Fock's theory. Harmonic tensors in the quantum Coulomb problem |journal=Physics-Uspekhi |date=2022 |volume=65 |issue=9 |pages=952–967 |doi=10.3367/UFNe.2021.04.038966|bibcode=2022PhyU...65..952E |s2cid=234871720 |arxiv=2501.00010 }}</ref><ref>{{cite journal |last1=Efimov |first1=S.P. |title=Runge-Lenz Operator in the Momentum Space |journal=JETP Letters |date=2023 |volume=117 |issue=9 |pages=716–720 |doi=10.1134/S0021364023600635|bibcode=2023JETPL.117..716E |s2cid=259225778 |doi-access=free |arxiv=2411.14482 }}</ref> The formula for the operator is simpler than in position space:
: <math > \hat \mathbf{A}_{\mathbf p}=\imath(\hat l_{\mathbf p}+1 )\mathbf p -\frac{(p^2+1)}{2}\imath\mathbf\nabla_{\mathbf p  } ,   </math>
where the "degree operator" 
: <math > \hat l_{\mathbf p }=(\mathbf p \mathbf \nabla_{\mathbf p} )  </math>
multiplies a homogeneous polynomial by its degree.

=== Casimir invariants and the energy levels ===
The [[Casimir invariant]]s for negative energies are
<math display="block"> \begin{align}
C_1 &= \mathbf{D} \cdot \mathbf{D} + \mathbf{L} \cdot \mathbf{L} = \frac{mk^2}{2|E|}, \\
C_2 &= \mathbf{D} \cdot \mathbf{L} = 0,
\end{align}</math>
and have vanishing Poisson brackets with all components of {{math|'''D'''}} and {{math|'''L'''}},
<math display="block">
\{ C_1, L_i \} = \{ C_1, D_i\} = 
\{ C_2, L_i \} = \{ C_2, D_i \} = 0.
</math>
''C''<sub>2</sub> is trivially zero, since the two vectors are always perpendicular.

However, the other invariant, ''C''<sub>1</sub>, is non-trivial and depends only on {{mvar|m}}, {{mvar|k}} and {{mvar|E}}. Upon canonical quantization, this invariant allows the energy levels of [[hydrogen-like atom]]s to be derived using only quantum mechanical canonical commutation relations, instead of the conventional solution of the Schrödinger equation.<ref name="bohm_1993" /><ref name="ReferenceA" /> This derivation is discussed in detail in the next section.