Quantizing comets: semiclassical methods in action

One key aspect of theoretical physics is that the (rather few!) basic equations are not tied to a narrow field of applications, but that insights from celestial mechanics are equally relevant for the dynamics of quantum objects. I have written before how to interpret the Coulomb Green’s function of hydrogen or Rydberg molecules in terms of Lamberts theorem of cometary orbit determination.

Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.
Potential landscape around Jupiter (rotating frame) showing the unstable saddle points (Lagrange points), where comets and asteroids can enter or escape to join Jupiter for a while.

Equally interesting is the dynamics of small celestials bodies in the vicinity of a parent body (the “restricted three body problem“). Near the Lagrange points the attraction of the sun and the effective potential in a coordinate system moving with the parent body around the sun cancel and the small objects can be trapped.
The comet Shoemaker-Levy 9 (SL9) is a prime example: SL9 was captured by Jupiter, broke apart at a close approach, and finally the string of fragments collided with Jupiter in 1994.

What would have happened with SL9 if Jupiter was contracted to a point mass?

Since SL9 was once captured, it should also have been released again. Indeed, in 2014 SL9 would have left Jupiter, as shown in the numerically integrated JPL orbit. To illustrate and simplify the transient dynamics, I have assumed in a recent publication a circular orbit of Jupiter and that SL9, Jupiter, and the sun are located in a plane. In reality the changing distance from the sun can open and close the entry points and in conjunction with precise location of the comet an escape or trapping becomes feasible:

Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped).
Dynamics around Jupiter (located at (0,0)) for two slightly different initial kinetic energies. The shaded area indicates the energetically allowed regions. On the right: after encircling Jupiter many times, the object escapes (or if you reverse time, becomes trapped). (C) Tobias Kramer.

The Shoemaker-Levy 9 case has been studied extensively in the astronomical literature (see for instance the orbital analysis by Benner et al). So what new insights are there for quantum objects? The goal is not to claim that comets have to be treated as quantum mechanical objects, but to realize that exactly the same dynamics seen in celestial mechanics guides electrons in magnetic fields through wave guides. I refer you for the details to my article Transient capture of electrons in magnetic fields, or: comets in the restricted three-body problem, but want to close by showing the electronic eigenfunctions, which show a real quantum feature absent in the classical case: electrons can tunnel through the forbidden area and thus will always escape from the parent body:

Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field.
Spectrum and gallery of eigenstates for the quantized version of the celestial dynamics around Lagrange points. The quantum case describes the motion of an electron in a magnetic field. (C) Tobias Kramer.

Added on March 1, 2020: another case of a transient capture, this time around Erath: small body 2020 CD3:

Visualization of the orbit of small body 2020 CD3, ephemeris taken from the JPL Horizons system (the uncertainties of the 1 year backward integration are several hours), geocentric ecliptical coordinates. The inset shows the close encounters with Earth (blue disk) on April 4, 2019 and February 13, 2020.

Random scattering: the small scale structure of the universe

Branched flow
Emergence of branched electron flow from weak scattering across a random potential. (C) Tobias Kramer

Our universe displays various mass concentrations of matter and is not a homogeneous density soup of particles as assumed in the simplest cosmological models where isotropy is assumed (cosmological principle). Interestingly recent supernova data (see Colin et al Evidence of anisotropy of cosmic acceleration A&A 631 L13, also on the arxiv) shows deviations of the angular distribution of matter as seen from Earth. One possibility (also not taken into account in the standard model) is the topology of the cosmos, discussed in this blog by Peter Kramer before. But back to the “small structures” (< 100 Mega parsec): How do mass concentrations arise and how get small fluctuations amplified?

One universal mechanism at work across many domains of physics is structure formation and concentration into branches by random weak scattering. The key-point is: despite randomness this type of scattering does not smear out the density as expected by diffusion processes. This important mechanism has been independently discovered in various domains of physics, but has been rarely discussed and further explored within a consistent framework. Interestingly the fundamental branching behavior can be seen in both quantum mechanics and classical physics. Together with Rick Heller and Ragnar Fleischmann we have written a short overview  article (available on the arxiv: “Branched flow”) which provides some background information and serves as a starting point for further exploration. We also provide a short python script to generate intricate patterns out of random deformations (script available on github).

Sunlight creating a web of caustics at the bottom of a swimming poll. Picture (C) Tobias Kramer.

One example discussed in this blog is the formation of “dust concentrations” in the near nucleus coma of comets from a homogeneously emitting cometary surface. The structure formation of the universe (the cosmic web) is discussed by Y. Zeldovich and reviewed by P.J.E. Peebles in his monograph The Large-Scale Structure of the Universe (Princeton University Press, 1980) in the chapter “caustics and pancakes”. A more accessible incarnation of the caustics are the ripples of sunlight at a pool bottom.

Trilobites revived: fragile Rydberg molecules, Coulomb Green’s function, Lambert’s theorem

The trilobite state
The trilobite Rydberg molecule can be modeled by the Coulomb Green’s function, which represents the quantized version of Lambert’s orbit determination problem.

The recent experimental realization observation of giant Rydberg molecules by Bendkowsky, Butscher, Nipper, Shaffer, Löw, Pfau [theoretically studied by Greene and coworkers, see for example Phys. Rev. Lett. 85, 2458 (2000)] shows Coulombic forces at work at large atomic distances to form a fragile molecule. The simplest approach to Rydberg molecules employs the Fermi contact potential (also called zero range potential), where the Coulomb Green’s function plays a central role. The quantum mechanical expression for the Coulomb Green’s function was derived in position space by Hostler and in momentum space by Schwinger. The quantum mechanical expression does not provide immediate insights into the peculiar nodal structure shown on the left side and thus it is time again to look for a semiclassical interpretation, which requires to translate an astronomical theorem into the Schrödinger world, one of my favorite topics.

Johann Heinrich Lambert was a true “Universalgelehrter”, exchanging letters with Kant about philosophy, devising a new color pyramid, proving that π is an irrational number, and doing physics. His career did not proceed without difficulties since he had to educate himself after working hours in his father’s tailor shop. After a long journey Lambert ended up in Berlin at the academy (and Euler choose to “escape” to St. Petersburg).

Lambert followed Kepler’s footsteps and tackled one of the most challenging problems of the time: the determination of celestial orbits from observations. In 1761 Lambert did solve the problem of orbit determination from two positions measurements. Lambert’s Theorem is a cornerstone of astronavigation (see for example the determination of Sputnik’s orbit using radar range measurements and Lambert’s theorem). Orbit determination from angular information alone (without known distances) is another problem and requires more observations.

Lambert poses the following question [Insigniores orbitae cometarum proprietates (Augsburg, 1761), p. 120, Lemma XXV, Problema XL]: Data longitudine axis maioris & situ foci F nec non situ punctorum N, M, construere ellipsin [Given the length of the semi-major axis, the location of one focal point, the points N,M, construct the two possible elliptical orbits connecting both points.]

Lambert's construction of two ellipses.
Lambert’s construction to find all possible trajectories from N to M and to map them to a ficticious 1D motion from n to m.

Lambert finds the two elliptic orbits [Fig. XXI] with an ingenious construction: he maps the rather complicated two-dimensional problem to the fictitious motion along a degenerate linear ellipse. Some physicists may know how to relate the three-dimensional Kepler problem to a four-dimensional oscillator via the Kustaanheimo–Stiefel transformation [see for example The harmonic oscillator in modern physics by Moshinsky and Smirnov]. But Lambert’s quite different procedure has its advantages for constructing the semiclassical Coulomb Green’s function, as we will see in a moment.

Shown are two ellipses with the same lengths of the semimajor axes 1/2 A1B1=1/2 A2 B2 and a common focus located at F. The centers of the two ellipses are denoted by C1 and C2. Lambert’s lemma allows to relate the motion from N to M on both ellipses to a common collinear motion on the degenerate linear ellipse Fb, where the points n and m are chosen such that the time of flight (TOF) along nm equals the TOF
along the elliptical arc NM on the first ellipse. On the second ellipse the TOF along the arc NB2M equals the TOF along nbm. The points n and m are found by marking the point G halfway between N and M. Then the major axis Fb=A1 B1=A2 B2 of the linear ellipse is drawn starting at F and running through G. On this line the point g is placed at the distance Fg=1/2(FN+FM). Finally n and m are given by the intersection points of a circle around g with radius GN=GM. This construction shows that the sum of the lengths of the shaded triangle α±=FN + FM ± NM is equal to α±=fn+ fm ± nm. The travel time depends only on the distances entering α±, and all calculations of the travel times etc. are given by one-dimensional integrations along the ficticious linear ellipse.

Lambert did find all the four possible trajectories from N to M which have the same energy (=semimajor axis a), regardless of their eccentricity (=angular momentum). The elimination of the angular momentum from Kepler’s equation is a tremendous achievement and the expression for the action is converted from Kepler’s form

  • [Kepler] W(r,r‘;E)=√μ a Kc [ξ + ε sin(ξ) – ξ’ – ε sin(ξ’)], with eccentricity ε, eccentric anomaly ξ to
  • [Lambert] W(r,r‘;E)=√μ a Kc[γ + sin(γ) – δ – sin(δ)], with
    sin2(γ/2)=(r+r’+ |r‘-r|)/(4a) and sin2(δ/2)=(r+r’- |r‘-r|)/(4a).

The derivation is also discussed in detail in our paper [Kanellopoulos, Kleber, Kramer: Use of Lambert’s Theorem for the n-Dimensional Coulomb Problem Phys. Rev. A, 80, 012101 (2009), free arxiv version here]. The Coulomb problem of the hydrogen atom is equivalent to the gravitational Kepler problem, since both are subject to a 1/r potential. Some readers might have seen the equation for the action in Gutzwiller’s nice book Chaos in classical and quantum mechanics, eq. (1.14). It is worthwhile to point out that the series solution given by Lambert (and Gutzwiller) for the time of flight can be summed up easily and is denoted today by an inverse sine function (for hyperbolic motion a hyperbolic sine, a function later introduced by Riccati and Lambert). Again, the key-point is the introduction of the linear ficticious ellipse by Lambert which avoids integrating along elliptical arcs.

The surprising conclusion: the nodal pattern of the hydrogen atom can be viewed as resulting from a double-slit interference along two principal ellipses. The interference determines the eigenenergies and the eigenstates. Even the notorious difficult-to-calculate van Vleck-Pauli-Morette (VVPM) determinant can be expressed in short closed form with the help of Lambert’s theorem and our result works even in higher dimensions. The analytic form of the action and the VVPM determinant becomes essential for our continuation of the classical action into the forbidden region, which corresponds to a tunneling process, see the last part of our paper.

Lambert is definitely a very fascinating person. Wouldn’t it be nice to discuss with him about philosophy, life, and science?

Determining the affinities of electrons OR: seeing semiclassics in action

Electron trajectories for photodetachment in an electric field.

Negatively charged ions are an interesting species, having managed to bind one more electron than charge neutrality grants them [for a recent review see T. Andersen: Atomic negative ions: structure, dynamics and collisions, Physics Reports 394 p. 157-313 (2004)]. The precise determination of the usually small binding energy is best done by shining a laser beam of known wave length on the ions and detect at which laser frequency the electron gets detached from the atomic core.

For some ions (oxygen, sulfur, or hydrogen fluoride and many more) the most precise values given at NIST are obtained by Christophe Blondel and collaborators with an ingenious apparatus based on an idea by Demkov, Kondratovich, and Ostrovskii in Pis’ma Zh. Eksp. Teor. Fiz. 34, 425 (1981) [JETP Lett. 34, 403 (1981)]: the photodetachment microscope. Here, in addition to the laser energy, the energy of the released electron is measured via a virtual double-slit experiment. The ions are placed in an electric field, which makes the electronic wave running against the field direction turn back and interfere with the wave train emitted in the field direction. The electric-field induced double-slit leads to the build up of a circular interference pattern of millimeter size (!) on the detector shown in the left figure (the animation was kindly provided by C. Blondel, W. Chaibi, C. Delsart, C. Drag, F. Goldfarb & S. Kröger, see their orginal paper The electron affinities of O, Si, and S revisited with the photodetachment microscope, Eur. Phys. J. D 33 (2005) 335-342).

Observed time-dependent emergence of the interference pattern in an electric field. Video shown with kind permission of C. Blondel et al. (see text for full credit)

I view this experiment as one of the best illustrations of how quantum and classical mechanics are related via the classical actions along trajectories. The two possible parabolic trajectories underlying the quantum mechanical interference pattern were described by Galileo Galilei in his Discourses & Mathematical Demonstrations Concerning Two New Sciences Pertaining to Mechanics & Local Motions in proposition 8: Le ampiezze de i tiri cacciati con l’istesso impeto, e per angoli egualmente mancanti, o eccedenti l’angolo semiretto, sono eguali. Ironically the “old-fashioned” parabolic motion was removed from the latest Gymnasium curriculum in Baden-Württemberg to make space for modern quantum physics.

At the low energies of the electrons, their paths are easily deflected by the magnetic field of the Earth and thus require either excellent shielding of the field or an active compensation, which was achieved recently by
Chaibi, Peláez, Blondel, Drag, and Delsart in Eur. Phys. J. D 58, 29-37 (2010). The new paper demonstrates nicely the focusing effect of the combined electric an magnetic fields, which Christian Bracher, John Delos, Manfred Kleber, and I have analyzed in detail and where one encounters some of the seven elementary catastrophies since the magnetic field allows one to select the number of interfering paths.

We have predicted similar fringes for the case of matter waves in the gravitational field around us originating from trapped Bose-Einstein condensates (BEC), but we are not aware of an experimental observation of similar clarity as in the case of the photodetachment microscope.

Mathematically, the very same Green’s function describes both phenomena, photodetachment and atomlasers. For me this universality demonstrates nicely how mathematical physics allows us to understand phenomena within a language suitable for so many applications.