None of the six was a crossing

Two events where there was one and the essays after it count events — fields at which a level in a shell and another level in the same shell come to interact strongly under an applied electric field. The count is a two-state estimate: for each coupled pair, the field at which the coupling matches half the zero-field gap between them, which is gap / 2d. It is the standard back-of-envelope for a Stark problem and it has served every count so far.

Four angles the shell chooses tilted the field out of the z axis and watched the count. It rose from three to six as the tilt mixed the two halves of the shell, then fell back at four particular angles — 45°, 49.107°, 60° and 63.435° — where two of the six estimates coincide, and those angles are arctangents of ratios of the shell’s own angular integrals: 1, 2/√3, √3 and 2. Forty-five degrees carries two collisions at once and gives four rather than five.

And it said, in its own closing section, that everything counted was a two-state estimate, that at each coincidence two such estimates land on one field so four levels are involved at once, and that the two-state picture is therefore at its least trustworthy exactly where the interesting thing happens. Diagonalising the full five-by-five at and around each coincidence would say whether the two crossings that meet produce one avoided crossing or two.

The answer is neither, and it is neither for a reason that reaches back through every one of those counts.

There are no crossings to avoid #

The tilted Hamiltonian is a real symmetric five-by-five: the zero-field energies down the diagonal, and F(cosθ·Z + sinθ·X) off it. Diagonalise it at a field, then at a slightly larger one, and follow the eigenvalues.

They fan apart, and none of them crosses another. That is not a fact about this shell; it is a fact about real symmetric matrices with no symmetry left in them. Two eigenvalues of such a matrix meet only on a set of codimension two, so a one-parameter sweep does not find one. Once the tilt has broken every symmetry the shell had, the five levels are five and they stay distinct.

That is worth pausing on, because the language of event counting quietly assumes otherwise. A crossing requires a degeneracy, and it matters exactly where a degeneracy comes from — it needs either a symmetry to enforce it or a coincidence to supply it, and the tilt destroys the first while a one-parameter sweep does not find the second. The exact spectrum of the tilted shell is five curves that never touch, at any field, at any angle in the interior of the range.

So “how many crossings” is not a question the exact spectrum answers by counting crossings. The nearest thing that is there is an avoided crossing: a field at which the gap between two adjacent levels has an interior minimum, which does not presuppose which two states are involved and does not need a two-state reduction to define.

Counting those gives one.

One, at every tilt #

At zero tilt there is one. At ten degrees, one. At twenty, thirty, forty, forty-five, forty-nine point one, fifty-five, sixty, sixty-three point four, seventy, eighty — one, one, one, one, one, one, one, one, one, one. Only at ninety degrees, with the field turned fully into x, does a second appear.

Meanwhile the two-state count does everything the tilt count said: three at each end, six in between, four at forty-five degrees, five at the other three coincidences. The two curves are not related.

The three at each end has its own explanation, and it is the one three events and a ratio of two dipoles gives: with the field along an axis exactly half the pairs have no dipole between them, so half the estimates are infinite and drop out. That remains true and remains a statement about which matrix elements vanish. It is simply not a statement about how many things happen to the spectrum.

And the one avoided crossing that does exist does not notice the coincidence angles. Its field is 1.6435 × 10⁻⁵ at forty degrees, 1.6808 at forty-five, 1.7270 at forty-nine point one, 1.8098 at fifty-five, 1.8903 at sixty, 1.9485 at sixty-three point four. That is a smooth monotone rise through two coincidences and past two more, and it varies by a factor of 1.34 across the whole ninety degrees of tilt.

There is nothing at 45°. There is nothing at 49.107°, or 60°, or 63.435°.

Why one estimate always looks right #

At sixty degrees the nearest two-state estimate to the exact crossing is 1.919 × 10⁻⁵ against a true 1.890 × 10⁻⁵ — agreement to one and a half per cent, on a quantity assembled from a matrix element and a gap. It is tempting to read that as the two-state picture working.

It is not, and the reason is in the same figure. At a general tilt the six estimates span a factor of four to twenty-one. At ten degrees they are 9.75 × 10⁻⁶, 1.13 × 10⁻⁵, 3.56 × 10⁻⁵, 6.38 × 10⁻⁵, 1.11 × 10⁻⁴ and 2.02 × 10⁻⁴, spread over a factor of twenty-one — and the exact crossing sits at 2.03 × 10⁻⁵, between the second and the third of them. At forty degrees they are packed into a factor of four and the crossing is again inside the pack. The crossing moves by 34 per cent across the entire sweep while the estimates move by orders. Scattering six numbers across a factor of ten and finding that one of them is near a fixed target is not evidence about the six.

The check is what happens at other angles, and it is the check that is made. At twenty degrees the nearest estimate is 37 per cent away. At ten degrees it is 76 per cent away. At sixty it is one and a half per cent away. The distribution of near-misses is what randomness produces, and no angle is systematically better than another — including, and especially, the coincidence angles, where the two-state picture was supposed to be at its most interesting.

And it is barely a crossing #

There is one more thing to say about the single avoided crossing, and it further weakens the vocabulary of events.

The gap at its minimum is 1.39 to 2.74 per cent below the gap at zero field. It does not close. It dips, shallowly, and reopens.

An avoided crossing in the sense the phrase usually carries — two levels approaching, repelling, and exchanging character — has a gap that closes to a small fraction of the level spacing. This one is a two per cent perturbation of a gap of 5.16 × 10⁻⁴, between the s level and the lower of the two p levels, at a field where the other three levels are also being pushed around. Calling it an avoided crossing is defensible because it satisfies the definition; calling the two states involved “a pair” is not, because at that field every level is mixed.

The depth also has a minimum near forty degrees and rises to either side, which is a real feature of the tilt and is one nothing in the two-state picture predicts. It is not at any coincidence angle.

That minimum is the only structure the exact spectrum has as a function of the tilt, and it is worth stating what it is not. It is not a symmetry point: forty degrees is not the arctangent of anything the shell computes, and the minimum’s position is not sharp — the depth changes by three hundredths of a per cent between thirty and forty-five degrees. Whether it sits at a definite angle at all, or whether the curve is simply flat there, is below what this sweep resolves, and the honest statement is that the exact spectrum’s dependence on the tilt is smooth and featureless to the precision measured.

The control: the search can find two #

A search that returns one everywhere is a search worth being suspicious of, and the way to check it is to point it at a case where the answer is known to be different.

With the field exactly along x, the search finds two. The second is between the two d levels, which are exactly degenerate at zero field, and its gap closes to 2.07 × 10⁻⁷ — three and a half orders of magnitude below the shallow dip that is present at every tilt, and unmistakably a real avoided crossing. At eighty degrees it is not there.

So the search can distinguish one from two, and it can see a deep crossing when there is one. Returning one at every interior tilt is a result rather than an insensitivity, and that is checked rather than assumed.

What was computed, and how #

The basis is the same five functions used for the tilt count: 3s, $3p_z$, $3d_{z^2}$, $3p_x$ and $3d_{xz}$, with hydrogenic radial functions and a quantum defect that lifts the l degeneracy. The zero-field energies come from that defect and depend only on l, so the p pair and the d pair are exactly degenerate before the field is turned on.

The field is F(cosθ ẑ + sinθ x̂), and the Hamiltonian is the zero-field diagonal plus F times the corresponding combination of the position matrices, both of which are already in hand. It is diagonalised by Jacobi rotation at three thousand fields spanning six decades, the four adjacent-level gaps are followed, every interior minimum is bracketed by the scan and refined by golden section.

The check requires eight things. That there is exactly one avoided crossing at every interior tilt. That the four coincidence angles are among those tilts and are no exception. That the crossing’s field at a coincidence sits inside the range spanned by its neighbours, so nothing marks it. That the field varies by less than sixty per cent across the whole tilt. That the two-state count exceeds the exact count at every tilt, and that the estimates span a factor of more than three while the crossing barely moves. That the dip is shallow, between nothing and ten per cent.

And two refusals. That the count is identical at 1,500, 3,000 and 6,000 scan steps, so it is not a property of the grid — a feature the scan could manufacture by scanning more finely is not a feature. And that the search finds the second crossing at ninety degrees, three orders deeper than the first, so it is known to be able to find two.

Where the model stops #

Five functions is not the shell. The n = 3 shell has nine orbitals and this basis holds the five that a field in the xz plane couples; the other four — $3p_y$, $3d_{xy}$, $3d_{yz}$, $3d_{x^2-y^2}$ — are excluded by the plane, exactly as they were for the tilt count. A field out of the plane makes the problem nine-dimensional and that open question is still there, unaffected by any of this.

The quantum defect is also the whole of the physics that lifts the l degeneracy, and it is a one-parameter caricature of a many-electron screening — the same caricature a broken symmetry was measured against, and it carries the same reservations here. The zero-field gaps — 5.16 × 10⁻⁴ between s and p, 9.97 × 10⁻⁵ between p and d — are set by that parameter, and the field at which the avoided crossing sits scales with them. What does not depend on the parameter is the count, since the count is a topological statement about a one-parameter family of real symmetric matrices.

And “avoided crossing” as an interior minimum of an adjacent gap is a definition, not the only one. A definition based on where an eigenvector’s character changes would be another, and it was tried first: it gives counts of four to seven that are unstable in the details and are counting the ordering of eigenvector components rather than anything in the spectrum. The gap-minimum definition is used because it is the one that is stable under refinement, and saying so is part of the result.

The generalisation #

The two-state estimate is not wrong. It is a good estimate of a quantity that is not the one being counted.

gap / 2d answers: at what field would the coupling between these two states, considered alone, become comparable to their separation? It is the same two-level reduction used throughout Stark problems and it is exact when there are two levels. That is a perfectly good question and its answer is a perfectly good number. What it is not is a prediction that anything happens in the spectrum at that field. Four successive arguments counted how many such fields there are and treated the count as a count of physical events, and it is not.

The failure is characteristic and it has a general form: a reduction that is exact for two levels is applied pairwise to five, and each application is fine while the collection of them is not. Six pairwise answers do not assemble into a five-level answer any more than three pairwise corrections assemble into a three-body one — the missing thing in both cases is everything that involves more than two objects at a time.

The right response is not to abandon the estimate. It is to know what it estimates, and to check it once against the full problem before building four arguments on the count.

There is a second habit here worth naming, and it concerns the definition rather than the physics. The first attempt here counted where an eigenvector’s dominant basis component changes, which is an intuitive reading of “the levels exchange character” and gives answers of four, five, six or seven depending on the angle in a pattern with no structure in it. It was discarded not because the numbers were inconvenient but because they were unstable in a specific way: the dominant component of an eigenvector flips whenever two of its coefficients are nearly equal, which happens for reasons that have nothing to do with the spectrum. A count that moves when the definition is nudged is measuring the definition, and the way to tell is to have two definitions.

Who found it, and when #

The Stark effect dates from 1913, the quantum-defect treatment of alkali spectra from the same era, and the fact that eigenvalues of a real symmetric matrix do not cross under a single parameter is von Neumann and Wigner, 1929. Everything computed here is original arithmetic on a five-level model, done to check a count that had been taken at face value.

The four arctangents remain exactly what they were said to be: angles at which two ratios of the shell’s angular integrals become equal. They are real, they are exact, and they are properties of the estimates.

Still open: nine levels, and where the two pictures part #

The obvious open question is the nine-dimensional problem, which is now a different question from the one first posed. That asked whether the coincidence curves on a sphere are the same four arctangents extended; the answer here makes that a question about the estimates, and the interesting version is whether the exact spectrum’s single shallow crossing stays single when four more levels are added. Nine levels give eight adjacent gaps to follow and the same calculation does it.

The nearer question is where the two pictures part company. The two-state estimate would be a good guide if the coupling were weak enough, and it is not obvious at what field or what defect it stops being one — the gaps here are set by the quantum defect and the couplings by the field, so there is a ratio that governs. Sweeping the defect from its present value towards zero, where the l degeneracy closes and the two-state gaps vanish, and watching whether the exact count rises to meet the estimated one, would say whether the two-state picture is a limit this problem is far from or a description of a different problem entirely. That is one parameter, and the calculation is already set up.