The crossing nothing couples

Sweeping the quantum defect of an n = 3 alkali shell in an electric field found that nothing dimensionless moves. The field at the shell’s one avoided crossing, divided by the zero-field gap between s and p, ran 0.04097, 0.04048 and so on down to 0.04000 as the defect fell by a factor of a hundred, and the essay named what that number probably was:

a ratio of two of the shell’s own angular integrals, most likely, in the same family as the four coincidence angles. Deriving it rather than measuring it would be the first exact statement made about the exact spectrum rather than about the estimates.

Deriving it turns out to be possible and to say three things the guess did not. The limit is not four hundredths. It is not a ratio of integrals. And the minimum it locates is not an avoided crossing: the two levels at it cannot interact.

The shell with the defect removed #

The screening in this model is a potential −λ/r², which replaces each angular momentum l by an effective one and gives a level of the shell the energy −1/2(3 − δ_l)², with the quantum defect δ_l ≈ λ/(l + ½) when λ is small. Every zero-field gap is therefore λ times a fixed number. The s–p gap is proportional to 2 − 2/3 = 4/3 and the p–d gap to 2/3 − 2/5 = 4/15, so the s–p gap is exactly five times the p–d gap in the limit — which is where the ratio 5.36 at the largest defect swept was heading as it fell to 5.003 at the smallest.

That makes the limit a problem with no λ in it at all. Divide the Hamiltonian by the s–p gap and measure the field in the same unit, so that f is the field over the gap. On the diagonal are 0 for the s level, 1 for the three p levels and 6/5 for the five d levels; off the diagonal is f times the shell’s own position matrix along the field. The only numbers left are the dipoles between the shell’s functions, whose squares are 54 between 3s and $3p_0$, 27 between $3p_0$ and $3d_0$, and 81/4 between the p and d functions with one unit of angular momentum about the field.

Every quantity the defect sweep watched settling is a property of that one nine-by-nine matrix. It can be diagonalised at any field, and its limits computed rather than read off the smallest defect anybody chose to sweep.

Four hundredths was on the way down #

The minimum of the zero-defect shell’s s–p gap sits at f = 0.0399865255.

The swept sequence was 0.040970, 0.040483, 0.040236, 0.040087, 0.040037, 0.040012 and 0.039997. It had not converged to 0.04000 at the smallest defect. It was passing through it — it crossed four hundredths between the two smallest defects swept — and it was still falling. A straight line through the two smallest points reaches 0.0399869 at zero defect, within four parts in ten million of the computed limit, so the limit and the sweep agree once the sweep is extrapolated rather than rounded.

The depth of the dip tells the same story. It ran down to 0.02356 at the smallest defect, and its limit is 0.0235249.

Five figures of agreement with a round number is what made four hundredths look like a closed form. The first five figures of 0.039997 do round to 0.04000, and a round number reached from above by a sequence that is slowing down invites the conclusion that it is the destination. That is the same trap a convergence checked against a value was designed to avoid, met from the other side: checking that a sequence approaches a value does not establish that the value is where it stops.

Two levels from two different blocks #

The more important result is not the digits. It is which two levels the minimum is between.

At f = 0.0399865 the lowest level of the shell is 92.5 per cent 3s, 7.3 per cent $3p_0$ and 0.2 per cent $3d_0$. The next level up is 74.3 per cent $3p_x$ and 25.7 per cent $3d_{xz}$. The minimum the sweeps counted is the separation between those two, 0.97648 of the zero-field gap.

Those two functions do not mix, for a reason that needs no computation. With the defect in, the zero-field energies depend only on l, so they do not change under any rotation. A field along z is unchanged by rotations about z. The Hamiltonian therefore commutes with the angular momentum about the field’s axis, and states with different values of it — m = 0 and |m| = 1 here — have no matrix element between them. It is the same kind of statement that makes a degeneracy a group theorem and that makes a selection rule a statement about an integrand, and it holds at every field.

The shell therefore falls into blocks the field cannot connect: 3s, $3p_0$ and $3d_0$ with m = 0; two identical pairs, $3p_x$ with $3d_{xz}$ and $3p_y$ with $3d_{yz}$, with |m| = 1; and $3d_{xy}$ and $3d_{x^2-y^2}$, with |m| = 2, which the field does not move at all. Found as the connected pieces of the position matrix, rather than typed in, those are exactly the blocks that come out, and the whole spectrum at every field is exactly their union. The largest element of the field between the m = 0 block and the |m| = 1 block is exactly zero, not small — the kind of zero a symmetry produces rather than the kind a calculation happens to land on.

Tilting the field changes nothing. The whole shell’s spectrum does not depend on the direction of the field, and along any direction the same blocks exist about that direction. A tilted basis makes the two levels look coupled because it describes them in functions quantised about a different axis; the levels themselves are the same two, with the same zero coupling.

Where two independent curves are parallel #

If the two levels do not interact, the minimum of their separation needs a different explanation from the one an avoided crossing supplies, and there is a short one.

The slope of a level with respect to the field is its induced dipole moment — the Hellmann–Feynman theorem, which says a level’s derivative with respect to a parameter is the expectation of the Hamiltonian’s derivative. A separation between two levels is stationary exactly where the two slopes are equal, so the minimum is where the two states carry the same induced dipole. At 0.0399865 both are −3.93340 bohr, to nine figures.

The two curves have different shapes, and the shapes are the reason there is a minimum at all. The |m| = 1 level has its d partner only a fifth of a gap above it, so it polarises fast at first: its dipole starts as 202.5 times the field, against 108 times for the s-like level. But the |m| = 1 block couples its level to only one partner, through a dipole of 9/2, so its induced dipole saturates at 4.5 bohr. The m = 0 block couples through two, and its largest induced dipole is $\sqrt{54 + 27} = 9$ bohr. So the |m| = 1 level falls faster at first and the m = 0 level faster later, and between the two regimes their slopes are equal once. The dipole of 9/2 against $3\sqrt{3}$ is the same pair of matrix elements whose ratio, $2/\sqrt{3}$, separated two of the shell’s crossover fields; here it decides where the two blocks stop polarising.

That is the whole mechanism of the minimum. It is a tangency of two independent curves, set by how differently two blocks of coupled levels polarise, and the depth of 2.35 per cent is how close two such curves happen to come.

What an avoided crossing needs #

The distinction is the one von Neumann and Wigner drew in 1929. Two levels that share a symmetry and depend on one parameter do not cross: as they approach, the coupling between them pushes them apart, and the smallest gap is set by that coupling. Two levels of different symmetry can cross freely, because there is nothing to push. An avoided crossing is a statement about an interaction, and a minimum between levels with no interaction is not one of them, however much it looks like one on a plot.

These two levels would have crossed if their curves had met. They do not meet — the gap bottoms out at 0.976 of its zero-field value and grows again — but the minimum that exists owes nothing to any coupling. The definition the tilted-field calculations settled on, “an interior minimum of the gap between adjacent levels, without presupposing which states are involved”, was chosen for being stable under refinement, and it is. What it does not do is check that the two states can interact, and that is the half of the definition that makes a minimum a crossing avoided.

It also changes what one earlier comparison was a comparison of. The nearest two-state estimate settled at 0.9067 of the exact field. Every two-state estimate is a field at which a coupled pair’s coupling matches its gap. The field it was divided by is where two uncoupled levels are closest. The ratio is real and it settles, and it is a ratio between properties of two different pairs of levels.

The minimum between levels that do interact #

A shell with coupled levels has to have somewhere they come closest, and it does.

Inside the m = 0 block the separation between the two lowest levels has a minimum at f = 0.0196001, where it is 0.99564 of the zero-field gap. The lower level is 98 per cent 3s; the upper one is 81.5 per cent $3p_0$ and 16.7 per cent $3d_0$, pushed down by the d level a fifth of a gap above it faster than the s level is pushed down by it. The coupling between them there is $\sqrt{54}$ times the field, 0.144 of the gap, and the dip it allows is shallow, 0.44 per cent.

This one was never seen, and the reason is the other block again. At f = 0.0196 both copies of the lowest |m| = 1 level sit at 0.96666 — between the two m = 0 levels. In the sorted spectrum those two are not neighbours, so a search over adjacent gaps compares each of them with an |m| = 1 level instead of with each other, and finds nothing. Run over the whole spectrum’s open gaps, the adjacent-gap search finds exactly one minimum, the uncoupled one at 0.0399865, and nothing within a factor of one and a half of 0.0196.

It is worth being clear about what this does and does not add. A shallow minimum of a separation between two coupled levels is not a dramatic avoided crossing either: at 0.44 per cent it is a gentle approach of two levels a whole gap apart. What it is, is the only feature of this spectrum that is about the field coupling levels at all — and it was invisible for a reason that would hide any such feature in any spectrum with more than one symmetry in it.

Nine levels, three kinds #

Drawn with their blocks coloured, the shell’s nine levels make the whole argument at once. The m = 0 block fans out into three levels heading for dipoles of −9, 0 and +9 bohr. The two |m| = 1 blocks fan into two levels each, heading for ±4.5. The |m| = 2 levels stay at 6/5, untouched. Levels of different colours start together at the zero-field p and d energies and separate with no sign of each other — nothing about one curve’s shape is decided by a curve of another colour, and had two of them met they would have passed straight through. The counted minimum is between the lowest curve and the lowest |m| = 1 curve above it, and the hidden one is between the two lowest m = 0 curves with an |m| = 1 curve in between.

A spectrum sorted by energy throws the colours away, which is why the two minima could be confused. Sorting is a convention about how to list numbers, and the physics is in which numbers can talk to which.

What kind of number 0.0399865 is #

The guess was that the limit would be a ratio of the shell’s own angular integrals, in the family of the four coincidence angles, each an arctangent of such a ratio. Those angles are properties of the two-state estimates, which are ratios of a gap to a dipole and are therefore built out of ratios. The exact spectrum’s features are roots.

Here the upper level has a closed form, since its block is two by two: E₁ = 11/10 − √(1/100 + 81f²/4). The lower level is the lowest root of the m = 0 block’s cubic, −E(1 − E)(6/5 − E) + 27f²E − 54f²(6/5 − E) = 0. The minimum is the field at which their derivatives with respect to f are equal. Eliminating the energy between those conditions leaves a polynomial equation in f whose coefficients are built from 54, 27, 81/4 and 6/5 — so 0.0399865255 is an algebraic number, a root of a polynomial with rational coefficients, and there is no reason to expect it to be a ratio of two integrals or anything simpler.

So the calculation named as “the first exact statement about the exact spectrum” does produce one, and it is not the one expected. The exact statement is structural rather than numerical: the counted feature is a tangency between the m = 0 and |m| = 1 blocks, the field at which two induced dipoles agree, and its value follows from the shell’s three dipoles and its gap ratio of five.

How the limit was computed #

The position matrix of the whole n = 3 shell is the one the whole-shell calculation uses, by quadrature, with its squared elements coming out at 54.00001, 27.00001 and 20.25000 against the exact 54, 27 and 81/4. The diagonal is 0, 1 and 6/5, derived from 1/(l + ½) rather than entered. Blocks are the connected components of the graph whose edges are the matrix’s non-zero elements, and each block is diagonalised separately.

The counted minimum is located on the whole nine-by-nine spectrum by a logarithmic scan and a golden-section refinement, exactly as the sweeps located it, and then again by bisection on the difference between the two levels’ induced dipoles; the two positions agree to a part in a million. Minima between levels of one block are searched for the same way, block by block.

The checks are these. At four fields the whole spectrum equals the union of the block spectra to 10⁻¹². The two levels at the counted minimum are in different blocks, with no element of the field between them. Their induced dipoles agree at the minimum to a part in a billion. The limit differs from 0.04 by more than 10⁻⁵, and the sweep’s own linear extrapolation lands within 2 × 10⁻⁶ of it. There is exactly one minimum between levels of one block, between the two lowest m = 0 levels, at a smaller field, with a level of another block between them. And the refusal: the adjacent-gap search over the whole spectrum must not find that one, and it does not.

What the zero-defect shell cannot say #

The limit is mathematical. A quantum defect of 0.0002 is already smaller than any real alkali’s, and zero is hydrogen, where the whole shell is degenerate and the problem changes character. The limit is used because the question was about a limit, and what it establishes — which levels the minimum is between — holds at every defect swept, since the blocking argument never used the defect’s size.

The screening model enters the energies only. Real screening changes the radial functions and therefore the dipoles, and the tangency’s position would move with them. Its nature would not: the blocks are a consequence of the symmetry, not of the model.

One shell. The next shell’s levels are far away at these fields and do not appear. Once they do, the m = 0 levels of two shells can approach each other with a coupling between them, and that is where avoided crossings in the full sense would first be expected.

And “avoided crossing” was a name, not a calculation. None of the numbers the earlier essays reported is wrong. The counts, the fields and the direction-independence all stand. What changes is what the one feature they counted is — a minimum between two independent levels — and the claims that relied on it being an interaction.

Name a feature by what it is between #

The transferable point is about defining a feature of a spectrum, and it is that a definition in terms of the shape of a curve is not enough.

“An interior minimum of an adjacent gap” is a shape. It is stable, it does not depend on the grid, and it does not presuppose which levels are involved — all of which is why it was chosen. What it cannot do is tell a minimum produced by an interaction from one produced by two curves that merely become parallel, because those have the same shape. The distinction lives in the symmetry of the two states, and the cheap check is to ask, before naming a minimum an avoided crossing, whether anything in the Hamiltonian connects the two levels at all. Here the answer was a block structure that took one line of reasoning about rotations and one connected-components search to find.

There is a second, smaller point about limits read from sweeps. A sequence that approaches a round number slowly and from one side looks like a sequence converging to it. The only way to know is to compute the limit or extrapolate honestly, and the extrapolation here, a straight line through two points, moved the answer past the round number in the fifth figure. How nearly a broken symmetry survives quoted crossover fields to a few figures for the same shell; nothing there depended on a fifth, but a closed form would have.

And a third about sorted lists. Ordering levels by energy is a convention that erases the one label that decides what can happen between them. A symmetry either holds or it does not, and while it holds the spectrum is several independent spectra printed on one axis.

Where the pieces come from #

The non-crossing rule is von Neumann and Wigner’s, from 1929, and the derivative of a level being the expectation of the derivative of the Hamiltonian is Hellmann’s and Feynman’s, from the late 1930s. The quantum-defect treatment of the alkali Stark effect is old, and so is the conservation of the angular momentum about a uniform field. The zero-defect limit of this shell, the identification of its counted minimum as a tangency between two blocks and the location of the hidden minimum are computed here.

The defect sweep deserves the credit for asking for a derivation rather than accepting five figures, and for saying that deriving it would be the first exact statement about the exact spectrum — which it was, though about the spectrum’s structure rather than its number. The degeneracy no group predicts is where this shell’s symmetry was first put to work; what was missing since is using the smaller symmetry that survives the field.

Still open: the next shell, and what the estimates were estimates of #

The obvious open question is the second shell, which the defect sweep already proposed for a different reason. With n = 4 added, levels of the same m from two shells approach each other as the field grows, with a dipole between them, and that is where a genuine avoided crossing — a minimum set by a coupling — should first appear. Whether it appears at fields comparable with the tangency found here or far beyond it, and whether the two-state estimates come closer to a real crossing than they came to a tangency, is the same diagonalisation on a larger matrix.

The nearer question is the estimates themselves. Each is a field at which a coupled pair’s coupling matches its gap, and the pairs they describe — s with $p_0$, p with d — are pairs within one block. The one feature of the exact spectrum that is about such a pair is the hidden minimum at 0.0196, not the counted one at 0.0400. Comparing the eight estimates with 0.0196, and asking which block each belongs to, would say whether the scatter of a factor of seven was a scatter around the wrong target all along.