Two levels cannot make a minimum

The zero-defect shell of an n = 3 alkali atom in an electric field had one minimum everybody had counted and one nobody had. The counted one, at 0.0399865 of the zero-field s–p gap, turned out to be a tangency of two levels of different symmetry about the field, which no element of the field connects. The uncounted one sits at 0.0196001, inside the m = 0 block, between a lower level that is 98 per cent 3s and an upper one that is 81.5 per cent $3p_0$ and 16.7 per cent $3d_0$. It is the only place in the spectrum where two levels the field actually couples come closest.

That calculation closed by pointing at the two-state estimates. The defect sweep had counted eight of them, scattered over a factor of seven, and each is the field at which one pair of levels’ coupling matches half their gap — a statement about a pair within a block. The proposal was to compare them with 0.0196 instead of with the tangency, and to ask whether “the scatter of a factor of seven was a scatter around the wrong target all along”.

The comparison is quick, and it looks like a yes. Formed with the field along its own axis, the nearest estimate is 0.019245, 1.85 per cent below the minimum. This essay is about why that agreement is not an estimate estimating anything.

Eight estimates were eight projections #

A two-state estimate is gap / 2d, where d is the matrix element of position between two of the shell’s functions, taken along the field. The functions are the usual real orbitals, quantised along z. The exact spectrum does not care about that choice — the complete shell’s levels are the same for a field pointing anywhere, since a Hamiltonian that depends only on l is invariant under every rotation — but a matrix element between two fixed functions, taken along a direction, is a projection, and it does care.

With the field along z, fourteen coupled pairs collapse to three distinct estimates: 0.019245 for $3p_0$ with $3d_0$, 0.022222 for the two |m| = 1 pairs of p and d, and 0.068041 for 3s with $3p_0$. Along x the same shell gives three again, but a different three — 0.022222 from three pairs, 0.038490 from $3p_x$ with $3d_{z^2}$, and 0.068041 from 3s with $3p_x$. At the tilt of forty-five degrees from z and thirty round, which is where the defect sweep counted, it gives eight, from 0.027217 to 0.192450, and not one of the eight equals any of the axial three. The exact spectrum at the coupled minimum and at the tangency is the same along all three directions.

So the eight were a count of how many different lengths the shell’s dipoles have once they are resolved onto axes the problem does not have. The exact family proportions the defect sweep found and could not explain — 1, $2/\sqrt{3}$, 4/3, $4/\sqrt{3}$ and 4 among the p–d estimates, 1, $2/\sqrt{3}$ and 2 among the s–p — are those projections: every tilted matrix element is a full one times a direction cosine, so every tilted estimate is an axial one divided by it. The ratio the sweep watched settle towards 0.9067, the nearest tilted estimate over the tangency, has its limit here at 0.9075.

Only one of the three sets is a property of the problem rather than of a choice, and it is the one formed along the field’s own axis, where the functions are also the field’s symmetry functions and each estimate belongs to one block. The comparison worth making is between those three and the spectrum.

Two levels only ever separate #

Before any comparison, there is an arithmetic fact about what a two-state estimate can possibly be the location of.

Two levels a gap Δ apart, coupled by a field through a matrix element d, have energies that differ by $\sqrt{\Delta^2 + 4f^2d^2}$. That grows with the field from the first instant and never stops growing. Two coupled levels on their own have no minimum at all, at any field, whatever the gap and whatever the coupling.

The |m| = 1 block is exactly such a pair, $3p_1$ and $3d_1$, and its separation rises at every field. The s–$p_0$ pair, taken alone, does the same. Each estimate marked on its curve is a field at which something has happened, but not a closest approach: the coupling has reached half the gap, the two functions are mixed at twenty-two and a half degrees — each level carrying 14.6 per cent of the other function — and the separation is $\sqrt{2}$ times what it was at zero field, and still climbing.

So a two-state estimate is a statement about how mixed a pair is, never about where its levels are closest. The question of which estimate a minimum belongs to is malformed from the start. The only block in the shell with a minimum between coupled levels is the m = 0 block, and it is also the only one with three levels in it.

The third level makes the minimum #

The m = 0 block holds 3s, $3p_0$ and $3d_0$, at zero, one and six fifths of the s–p gap.

With $d_0$ in the block, the separation of the two lowest levels falls as the field is turned on, reaches a minimum 0.436 per cent below its zero-field value at 0.0196001, and rises after. With $d_0$ removed, the same two functions separate from the first instant, as two levels must.

What the third level does is push. The field couples $p_0$ to s below it and to $d_0$ above it. To second order the s level is pushed down and $p_0$ pushed up by the same amount, set by the square of their matrix element, 54; and $p_0$ is pushed down again by $d_0$, by the square of that matrix element, 27, divided by the d level’s height above p. When the push from above wins, the upper level of the pair is driven down faster than the lower one, and the pair closes before the growing field eventually separates everything.

Where it stops closing is a Hellmann–Feynman condition. The slope of a level with respect to the field is minus its induced dipole, so a separation is least where the two induced dipoles are equal — here −2.0671 bohr each. That is the condition that located the tangency as well. Both minima in this spectrum are points of equal dipole, and neither is a field at which a coupling matches a gap.

The p–d estimate, 0.019245, is drawn beside the minimum, 1.85 per cent to its left. It is the field at which $p_0$ and $d_0$ are mixed at twenty-two and a half degrees. The minimum is in the separation of s and $p_0$.

The minimum and the estimate move in opposite directions #

The d level’s height above p is fixed in this shell. As the quantum defect goes to zero it tends to λ/(l + ½), so the three levels stand at 0, 1 and 6/5 of the s–p gap with nothing to adjust. As a way of asking what places the minimum, though, the offset can be treated as a dial and moved.

The p–d estimate is the offset divided by $2\sqrt{27}$, so it is a straight line through the origin.

The minimum does the opposite. With $d_0$ almost on top of $p_0$, at an offset of 0.001, it sits at 0.0327 and is 8.29 per cent deep; at 0.1 it is at 0.0295 and 3.39 per cent deep; at 0.15, 0.0259; at the shell’s own 0.2, 0.0196 and 0.436 per cent; at 0.22, 0.0156; at 0.24, 0.0093 and two hundredths of a per cent deep. At 0.26 there is no minimum at any field.

The two curves cross once, at an offset of 0.2013. The shell’s own offset, one fifth, sits 0.65 per cent below the crossing, and that is the whole of the 1.85 per cent agreement. At an offset of 0.001 the same two quantities differ by a factor of 340. An estimate that tracked the minimum would move with it; this one moves against it and meets it once, at a place set by nothing either quantity knows about the other.

The s–p estimate, 0.068041, does not depend on the offset at all, and at the shell’s own offset it is three and a half times the minimum’s field.

A quarter, from second order #

The minimum’s disappearance is not gradual in the sense of fading out slowly: its field and its depth both go to zero together at one offset, and that offset can be written down.

The three pushes add up, to second order in the field, to a change in the separation of

$$(2 \times 54 - 27/\delta)\,f^2$$

with δ the offset in units of the s–p gap. The two pushes from s are a squared coupling over a gap that does not move; the push from d is a squared coupling over a gap that does, which is the difference between two ways of being second order and the whole reason an offset can decide anything. When the bracket is negative the separation starts by falling, and since at large fields the levels’ dipoles diverge and the separation must grow, it has to turn: a minimum exists. When the bracket is positive the separation starts by rising, and across every offset tried above the critical one it keeps rising. The bracket vanishes at

δ = 27/108 = 1/4,

which is half the ratio of the p–d squared matrix element to the s–p one. The numerical search, bisected on whether a minimum exists, stops finding one at 0.24999886. The exact separation’s curvature at a field of 10⁻³ reproduces the bracket at every offset from a tenth upward — −26.94 against −27 at one fifth, +4.17 against +4.15 at 0.26, +18.01 against +18 at 0.3.

The shell has its minimum because its own offset, one fifth, is below a quarter. The margin is twenty per cent, and it is decided by two squared matrix elements and a screening formula, none of which were chosen for it. It is a fact about matrix elements rather than about symmetry, the same kind of fact as a level that stays where it is with no symmetry protecting it.

That also answers, in part, a question the tangency left: what kind of number the shell’s features are. The tangency’s field is an algebraic number, the root of an equality of slopes, and the coupled minimum’s field is another. But whether the coupled minimum exists is a rational statement, a quarter against a fifth — the kind of ratio of the shell’s own integrals that the four coincidence angles are built from.

What each estimate was an estimate of #

Set out as a table, the correspondence the comparison was looking for does not exist in any row.

The m = 0 p–d estimate, 0.019245, is two per cent from the coupled minimum, but the minimum is in a different pair’s separation and moves the other way when the d level is moved. The |m| = 1 p–d estimate, 0.022222, is in a block with two levels and therefore no minimum. The s–p estimate, 0.068041, is in the right block for a minimum and three and a half times too large for the one there is. And the tangency, the minimum every earlier count was counting, has no estimate at all, because no coupling produces it.

Each estimate is exactly what the defect sweep said it was: a different quantity computed exactly. It is the field at which one pair is mixed by a fixed amount, and that is a perfectly good thing to know. The mistake was always to read a mixing field as the location of a feature of the spectrum, and there was no target for the estimates to scatter around. The same shape of error — an agreement that is a property of one setting of something nobody varied — turns up wherever a reduced account is checked at a single point, as a sign rule for a fitted susceptibility turned out to hold only between two poles.

What survives of the standing claims #

The eight estimates, and their factor of seven, were a property of the tilt. Formed along the field there are three, spanning a factor of 3.54, one per coupled pair within a block.

The proposal that the estimates were aimed at the hidden minimum does not survive. No two-state estimate can be the location of any minimum, and the nearest one’s agreement is two curves crossing at an offset of 0.2013.

The hidden minimum survives, with a condition attached. It is real, it is the only closest approach of coupled levels in the shell, and it exists because a fifth is less than a quarter. A shell whose levels stood differently would not have it.

And the tangency’s status is unchanged: the minimum every sweep counted is between levels nothing couples, and it is not the kind of thing a two-state picture could ever have predicted.

What was computed, and how #

The shell is the n = 3 shell with the quantum defect taken to zero: nine real functions, energies 0, 1 and 6/5 in units of the s–p gap, and the position operator between them by quadrature, with the squared matrix elements 54, 27 and 81/4 checked against their closed forms. The estimates are formed for every pair not degenerate at zero field with a non-vanishing matrix element along the field, at three directions, and grouped where they agree to a part in a million. The spectra at the three directions are compared at the coupled minimum and at the tangency.

A separation’s minimum is searched for on three thousand fields spaced logarithmically from 10⁻⁴ to 1 and refined by golden section. The offset is swept over twelve values and a further sixty for drawing. The critical offset is bisected on whether a minimum is found, and the crossing of minimum and estimate on which of the two is larger. The curvature is a finite-field difference at 10⁻³.

Ten results are checked. The estimate counts along the three directions, and that none of the tilted ones equals an axial one. That the spectrum is the same along all three. That the two-level pairs have no minimum. That the m = 0 minimum is the one already found, and a point of equal dipole. That the critical offset is a quarter in closed form and in the search. That a minimum exists at exactly the swept offsets below a quarter. That the curvature matches second order. That the minimum falls as the estimate rises, and that they meet within one per cent above a fifth. That at a small offset they differ by more than a factor of a hundred.

The refusal is the d level removed: without it the s–$p_0$ separation rises at every field drawn, so the minimum is the third level’s and not something the search manufactures.

Where the model stops #

The offset is not a free parameter of any real atom in this model. It is fixed at a fifth by the screening formula the defect is taken to zero through, and moving it is a way of asking what places the minimum rather than a description of a different atom. A real alkali atom’s defects do not stand in the ratio 1/(l + ½), and which side of a quarter its own shell falls on is a property of the atom.

One shell. The n = 4 levels are absent, and with them every coupling between shells.

One electron, as throughout, with the field entering only through the position operator.

Who found it, and when #

The two-level formula, second-order perturbation theory and the Hellmann–Feynman theorem are textbook material from the 1920s and 1930s. Their application to this shell — the estimates by direction, the offset sweep, the quarter and the crossing at 0.2013 — is new arithmetic.

Still open: the second shell, and a minimum set by a coupling #

The obvious open question is still the next shell. Nothing in the n = 3 shell is an avoided crossing in the sense a textbook means — a minimum whose depth is set by the coupling between the two levels at it — because the only coupled minimum here is set by a third level’s push and is less than half a per cent deep. With n = 4 added, levels of the same m from two shells approach each other as the field grows, with a matrix element between them, and that is where a minimum set by its own coupling could first appear. Whether a two-state estimate formed for that pair then lands on it, or misses it the way these missed everything, is the test the estimates have never yet been given a fair chance to pass.

The nearer question is the quarter in other shells. The condition $\delta < |\langle p|z|d\rangle|^2 / 2|\langle s|z|p\rangle|^2$ is general, and both sides change with n: the offset through the screening formula, the matrix elements through the radial functions. Whether n = 2, which has no d level, and n = 4, which has an f level pushing d the same way d pushes p, have their own coupled minima is a table of ratios that needs no diagonalisation at all.