The commutation relations involving the components of Runge-Lenz vector of the Hydrogen atom problem, ${\vec A}$ and the angular momentum ${\vec L}$ are given by $$ [L_i,L_j]=i\hslash\varepsilon_{ijk}L_k,\\ [A_i,A_j] = -i\hslash\varepsilon_{ijk} \frac{2H}{m} L_k,\\ [L_i,A_j]=i\hslash\varepsilon_{ijk}A_k $$ I am doubtful about the commutation relation in the middle the right-hand side of which contains the Hamiltonian operator. The righthand side does not seem to be a simple linear combination of $L_1,L_2$ and $L_3$. If $H$ were a constant, then this is the same as $SO(4)$ algebra. But how can this be $SO(4)$ where ${H}$ is an operator? Any help?
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asked Feb 2, 2022 at 16:08
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1$\begingroup$ Will "an operator commuting with all your generators behaves like a constant for all practical purposes" mean anything to you? $\endgroup$– Cosmas ZachosCommented Feb 2, 2022 at 16:23
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$\begingroup$ You should treat $H$ as a constant w.r.t. the $SO(4)$ generators - this is just the same thing as saying that $H$ is invariant under $SO(4)$ transformations. $\endgroup$– PraharCommented Feb 2, 2022 at 18:14
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Since both the $L_i$ and the $A_i$ are constants of motion, their Poisson bracket with the Hamiltonian vanishes and so it is "as good as a constant" for purposes of the algebraic structure - like a constant, it commutes with everything. Essentially you get one differently scaled $\mathfrak{so}(4)$ algebra for each subspace defined by constant energy $H(q,p) = E$.
answered Feb 2, 2022 at 16:33
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2$\begingroup$ Just to be really precise, for $E<0$. $\endgroup$ Commented Feb 2, 2022 at 16:37
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$\begingroup$ @ACuriousMind How about quantum mechanics? In that case, how do you scale the $A_i$'s to get the standard SO(4) algebra structure? Surely, we cannot divide by the square root of the hamiltonian operator. $\endgroup$ Commented Feb 2, 2022 at 16:40
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$\begingroup$ @Solidification I don't know what you mean - you still get one $\mathfrak{so}(4)$ on each eigenspace of $H$. $\endgroup$– ACuriousMind ♦Commented Feb 2, 2022 at 16:42
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1$\begingroup$ Ah! I think I got it. You are saying that this set of commutation relations when applied to a subspace of the Hilbert space corresponding to a fixed energy $E$, is a SO(4) algebra. Is that right? $\endgroup$ Commented Feb 2, 2022 at 16:46