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First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but vanishing for circular ones. The "lopsidedness" you are talking about amounts to the ellipse's eccentricity, a dimensionless quantity $|{\mathbf A} /me^2|$.

Quantizing, classical trajectories have a very uncomfortable connection to both Wigner functions and wavefucntions. In QM, this eccentricity quantity presents as $$ \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|H(L^2/\hbar^2+1)|} \qquad \leadsto \qquad \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|E|( l(l+1)+1)}~~. $$

You then see this eccentricity analog never vanishes in QM, even for s waves, spherical states! This angular momentum zero-point-shift is well known in deformation quantization, and informs the proper correspondence principle, however counter-intuitive. It has been thoroughly investigated in phase space by Dahl and collaborators, J P Dahl & M Springborg, Mol Phys 47 1001 (􏰑1982)􏰀.

So, formally, the classical eccentricity survives in QM and is crucial for its spectrum. Recall it is the basis of the original modern quantization of the hydrogen atom, W Pauli Jr, "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik" Zeitschrift für Physik 36 no. 5 (1926): 336-363.

For large n, and for the largest $l=n-1$ for that n, the square root tends to 1.

First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but vanishing for circular ones. The "lopsidedness" you are talking about amounts to the ellipse's eccentricity, a dimensionless quantity $|{\mathbf A} /me^2|$. Classically, the elliptical orbit is tantamount to angular momentum conservation in conjunction with LRL invariance.

Quantizing, classical trajectories have a very uncomfortable connection to both Wigner functions and wavefucntions. In QM, this eccentricity quantity presents as $$ \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|H(L^2/\hbar^2+1)|} \qquad \leadsto \qquad \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|E|( l(l+1)+1)}~~. $$

You then see this eccentricity analog never vanishes in QM, even for s waves, spherical states! This angular momentum zero-point-shift is well known in deformation quantization, and informs the proper correspondence principle, however counter-intuitive. It has been thoroughly investigated in phase space by Dahl and collaborators, J P Dahl & M Springborg, Mol Phys 47 1001 (􏰑1982)􏰀.

So, formally, the classical eccentricity survives in QM and is crucial for its spectrum. Recall it is the basis of the original modern quantization of the hydrogen atom, W Pauli Jr, "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik" Zeitschrift für Physik 36 no. 5 (1926): 336-363.

For large n, and for the largest $l=n-1$ for that n, the square root tends to 1.

First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but vanishing for circular ones.

That is because the "lopsidedness" you are talking about amounts to the ellipse's eccentricity, a dimensionless quantity $|{\mathbf A} /me^2|$, which, in QM, presents as $$ \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|H(L^2/\hbar^2+1)|} \qquad \leadsto \qquad \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|E|( l(l+1)+1)}. $$

You then see this eccentricity analog never vanishes in QM, even for s waves, spherical states! This angular momentum zero-point-shift is well known in deformation quantization, and informs the proper correspondence principle, however counter-intuitive. It has been thoroughly investigated in phase space by Dahl and collaborators.

So, formally, the classical eccentricity survives in QM and is crucial for its spectrum. Recall it is the basis of the original modern quantization of the hydrogen atom, Pauli Jr, W "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik" Zeitschrift für Physik 36 no. 5 (1926): 336-363.

For large n, and for the largest $l=n-1$ for that n, the square root tends to 1.

First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but vanishing for circular ones. The "lopsidedness" you are talking about amounts to the ellipse's eccentricity, a dimensionless quantity $|{\mathbf A} /me^2|$.

Quantizing, classical trajectories have a very uncomfortable connection to both Wigner functions and wavefucntions. In QM, this eccentricity quantity presents as $$ \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|H(L^2/\hbar^2+1)|} \qquad \leadsto \qquad \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|E|( l(l+1)+1)}~~. $$

You then see this eccentricity analog never vanishes in QM, even for s waves, spherical states! This angular momentum zero-point-shift is well known in deformation quantization, and informs the proper correspondence principle, however counter-intuitive. It has been thoroughly investigated in phase space by Dahl and collaborators, J P Dahl & M Springborg, Mol Phys 47 1001 (􏰑1982)􏰀.

So, formally, the classical eccentricity survives in QM and is crucial for its spectrum. Recall it is the basis of the original modern quantization of the hydrogen atom, W Pauli Jr, "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik" Zeitschrift für Physik 36 no. 5 (1926): 336-363.

For large n, and for the largest $l=n-1$ for that n, the square root tends to 1.

First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but vanishing for circular ones.

That is because the "lopsidedness" you are talking about amounts to the ellipse's eccentricity, a dimensionless quantity $|{\mathbf A} /me^2|$, which, in QM, presents as $$ \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|H(L^2/\hbar^2+1)|} \qquad \leadsto \qquad \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|E|( l(l+1)+1)}. $$

You then see this eccentricity analog never vanishes in QM, even for s waves, spherical states! This angular momentum zero-point-shift is well known in deformation quantization, and informs the proper correspondence principle, however counter-intuitive. It has been thoroughly investigated in phase space by Dahl and collaborators.

So, formally, the classical eccentricity survives in QM and is crucial for its spectrum. Recall it is the basis of the original modern quantization of the hydrogen atom, Pauli Jr, W "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik" Zeitschrift für Physik 36 no. 5 (1926): 336-363.

First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but vanishing for circular ones.

That is because the "lopsidedness" you are talking about amounts to the ellipse's eccentricity, a dimensionless quantity $|{\mathbf A} /me^2|$, which, in QM, presents as $$ \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|H(L^2/\hbar^2+1)|} \qquad \leadsto \qquad \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{|E|( l(l+1)+1)}. $$

You then see this eccentricity analog never vanishes in QM, even for s waves, spherical states! This angular momentum zero-point-shift is well known in deformation quantization, and informs the proper correspondence principle, however counter-intuitive. It has been thoroughly investigated in phase space by Dahl and collaborators.

So, formally, the classical eccentricity survives in QM and is crucial for its spectrum. Recall it is the basis of the original modern quantization of the hydrogen atom, Pauli Jr, W "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik" Zeitschrift für Physik 36 no. 5 (1926): 336-363.

For large n, and for the largest $l=n-1$ for that n, the square root tends to 1.