Solving the stationary Schrödinger (H-E)Ψ=0 equation can in principle be reduced to solving a matrix equation. This eigenvalue problem requires to calculate matrix elements of the Hamiltonian with respect to a set of basis functions and to diagonalize the resulting matrix. In practice this time consuming diagonalization step is replaced by a recursive method, which yields the eigenfunctions for a specific eigenvalue.
A very different approach is followed by wavepacket methods. It is possible to propagate a wavepacket without determining the eigenfunctions beforehand. For a given Hamiltonian, we solve the time-dependent Schrödinger equation (i ∂t-H) Ψ=0 for an almost arbitrary initial state Ψ(t=0) (initial value problem).
The reformulation of the determination of eigenstates as an initial value problem has a couple of computational advantages:
results can be obtained for the whole range of energies represented by the wavepacket, whereas a recursive scheme yields only one eigenenergy
the wavepacket motion yields direct insight into the pathways and allows us to develop an intuitive understanding of the transport choreography of a quantum system
solving the time-dependent Schrödinger equation can be efficiently implemented using Graphics Processing Units (GPU), resulting in a large (> 20 fold) speedup compared to CPU code
Recently, Christoph Kreisbeck has done a detailed calculations on the gate-voltage dependency of the conductance in Aharonov-Bohm nanodevices, taking full adventage of the simultaneous probing of a range of Fermi energies with one single wavepacket. A very clean experimental realization of the device was achieved by Sven Buchholz, Prof. Saskia Fischer, and Prof. Ulrich Kunze (RU Bochum), based on a semiconductor material grown by Dr. Dirk Reuter and Prof. Anreas Wieck (RU Bochum). The details, including a comparison of experimental and theoretical results shown in the left figure, are published in Physical Review B (arxiv version).
Einstein’s fundamental theory of Newton’s gravitation relates the interaction of masses to the curvature of space. Modern cosmology from the big bang to black holes results from Einstein’s field equations for this relation. These differential equations by themselves do not yet settle the large-scale structure and connection of the cosmos. Theoretical physicists in recent years tried to infer information on the large-scale cosmology from Cosmic microwave background radiation (CMBR), observed by satellite observations. In the frame of large-scale cosmology, the usual objects of astronomy from solar systems to galaxy clusters are smoothed out, and conditions imprinted in the early stage of the universe dominate.
In mathematical language one speaks of cosmic topology. Topology is often considered to be esoteric. Here we present topology from the familiar experience with the twisted Möbius strip. This strip on one hand can be seen as a rectangular crystallographic lattice cell whose copies tile the plane, see Fig. 2. The Möbius strip is represented as a rectangular cell, located between the two vertical arrows, of a planar crystal. A horizontal dashed line through the center indicates a glide-reflection line. A glide reflection is a translation along the dashed line by the horizontal length of the cell, followed by a reflection in this line. The crystallographic symbol for this planar crystal is cm. In three-dimensional space the planar Möbius crystal (top panel of Fig. 1) is twisted (middle panel of Fig. 1). The twist is a translation along the dashed line, combined with a rotation by 180 degrees around that line. A final bending (bottom panel of Fig. 1) of the dashed line and a smooth gluing of the arrowed edges yields the familiar Möbius strip.
Given this Möbius template in two dimension, we pass to manifolds of dimension three. We present in Fig. 3 a new cubic manifold named N3. Three cubes are twisted from an initial one. A twist here is a translation along one of the three perpendicular directions, combined with a right-hand rotation by 90 degrees around this direction. To follow the rotations, note the color on the faces. The three neighbor cubes can be glued to the initial one. If the cubes are replaced by their spherical counterparts on the three-sphere, the three new cubes can pairwise be glued with one another, with face gluings indicated by heavy lines. The complete tiling of the three-sphere comprises 8 cubes and is called the 8-cell. The gluings shown here generate the so-called fundamental group of a single spherical cube on the three-sphere with symbol N3. This spherical cube is a candidate for the cosmic topology inferred from the cosmic microwave background radiation. A second cubic twist with a different gluing and fundamental group is shown in Fig. 4. Here, the three twists combine translations along the three directions with different rotations.
The key idea in cosmic topology is to pass from a topological manifold to its eigen- or normal modes. For the Möbius strip, these eigenmodes are seen best in the planar crystal representation of Fig. 2. The eigenmodes can be taken as sine or cosine waves of wave length which repeat their values from edge to edge of the cell. It is clear that the horizontal wavelength of these modes has as upper bound the length L of the rectangle. The full Euclidean plane allows for infinite wavelength, and so the eigenmodes of the Möbius strip obey a selection rule that characterizes the topology. Moreover the eigenmodes of the Möbius strip must respect its twisted connection.
Similarly, the eigenmodes of the spherical cubes in Fig. 3 must repeat themselves when going from cube to neighbor cube. It is intuitively clear that the cubic eigenmodes must have a wavelength smaller than the edge length of the cubes. The wave length of the eigenmodes of the full three-sphere are bounded by the equator length of the three-sphere. Seen on a single cube, the different twists and gluings of the manifolds N2 and N3 shown in Figs. 3 and 4 form different boundary value problems for the cubic eigenmodes.
Since terrestrial observations measure the incoming radiation in terms of its spherical multipoles as functions of their incident direction, the eigenmodes must be transformed to a multipole expansion as done in my work. New and finer data on the CMB radiation are expected from the Planck spacecraft launched in 2009. These data, in conjunction with the theoretical models, will promote our understanding of cosmic space and possible twists in its topology.
An interesting and unfinished chapter of condensed mater theory concerns the quantum Hall effect. Especially the integer quantum Hall effect (IQHE) is actually not very well understood. The fancy cousin of the IQHE is the fractional quantum Hall effect (FQHE). The FQHE is easier to handle since there is agreement about the Hamiltonian which is to be solved (although the solutions are difficult to obtain): the quantum version of the very Hamiltonian used for the classical Hall effect, namely the one for interacting electrons in a magnetic field. The Hamiltonian is still lacking the specification of the boundary conditions, which can completely alter the results for open and current carrying systems (as in the classical Hall effect) compared to interacting electrons in a box.
Surprisingly no agreement about the Hamiltonian underlying the IQHE exists. It was once hoped that it is possible to completely neglect interactions and still to obtain a theoretical model describing the experiments. But if we throw out the interactions, we throw out the Hall effect itself. Thus we have to come up with the correct self-consistent solution of a mean field potential which incorporates the interactions and the Hall effect.
Is it possible to understand the integer quantum Hall effect without including interactions – and if yes, how does the effectively non-interacting Hamiltonian look like?
Starting from a microscopic theory we have constructed the self-consistent solution of the Hall potential in our previous post for the classical Hall effect. Two indispensable factors caused the emergence of the Hall potential:
repulsive electronic interactions and
equipotential boundary conditions at the contacts.
In graphene the dielectric constant is 12 times smaller than in GaAs and thus the Coulomb repulsion between electrons are stronger (which should help to generate the Hall potential). The observation of the FQHE in two-terminal devices has led the authors of the FQHE measurments to conjecture that hot-spots are also present in graphene devices [Du, Skachko, Duerr, Luican Andrei Nature 462, 192-195 (2009)].
These observations are extremely important, since the widely used theoretical model of edge-state transport of effectively non-interacting electrons is not readily compatible with these findings. In the edge-state model conductance quantization relies on the counter-propagation of two currents along the device borders, whereas the shown potential supports only a unidirectional current from source to drain diagonally across the device.
Moreover the construction of asymptotic scattering states is not possible, since no transverse lead-eigenbasis exists at the contacts. Electrons moving strictly along one side of the device from one contact to the other one would locally increase the electron density within the contact and violate the metallic boundary condition (see our recent paper on the Self-consistent calculation of electric potentials in Hall devices [Phys. Rev. B, 81, 205306 (2010)]).
Are there models which support a unidirectional current and at the same time support a quantized conductance in units of the conductivity quantum?
We put forward the injection model of the quantum Hall effect, where we take the Hall potential as being the self-consistent mean-field solution of the interacting and current carrying device. On this potential we construct the local density of states (LDOS) next to the injection hot spot and calculate the resulting current flow. In our model, the conductivity of the sample is completely determined by the injection processes at the source contact where the high electric field of the hot spots leads to a fast transport of electrons into the device. The LDOS is broadened due to the presence of the electric Hall field during the injection and not due to disorder. Our model is described in detail in our paper Theory of the quantum Hall effect in finite graphene devices [Phys. Rev. B, 81, 081410(R) (2010), free arxiv version] and the LDOS in a conventional semiconductor in electric and magnetic fields is given in a previous paper on electron propagation in crossed magnetic and electric fields. The tricky part is to prove the correct quantization, since the absence of any translational symmetries in the Hall potential obliterates the use of “Gedankenexperimente” relying on periodic boundary conditions or fancy loop topologies.
In order to propel the theoretical models forward, we need more experimental images of the Hall potential in a device, especially in the vicinity of the contacts. Experiments with graphene devices, where the Hall potential sits close to the surface, could help to establish the potential distribution and to settle the question which Hamiltonian is applicable for the quantum Hall effects. Is there anybody out to take up this challenge?
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 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?
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).
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.
For a while we have explored the usage of General Purpose Graphics Processing Units (GPGPU) for electronic transport calculations in nanodevices, where we want to include all electron-electron and electron-donor interactions. The GPU allows us to drastically (250 fold !!!) boost the performance of N-body codes and we manage to propagate 10,000 particles over several million time-steps within days. While GPU methods are now rather popular within the astrophysics crowd, we haven’t seen many GPU applications for electronic transport in a nanodevice. Besides the change from astronomical units to atomic ones, gravitational forces are always attractive, whereas electrons are affected by electron-donor charges (attractive) and electron-electron repulsion. Furthermore we have a magnetic field present, leading to deflections. Last, the space where electrons can spread out is limited by the device borders. In total the force on the kth electron is given by
Our recent paper in Physical Review B (also freely available on the arxiv) gives the first microscopic description of the classical Hall effect, where interactions are everything: without interactions no Hall field and no drift transport. The role and importance of the interactions is surprisingly sparsely mentioned in the literature, probably due to a lack of computational means to move beyond phenomenological models. A notable exception is the very first paper on the Hall effect by Edwin Hall, where he writes “the phenomena observed indicate that two currents, parallel and in the same direction, tend to repel each other”. Note that this repulsion works throughout the device and therefore electrons do not pile up at the upper edge, but rather a complete redistribution of the electronic density takes place, yielding the potential shown in the figure.
Another important part of our simulation of the classical Hall effect are the electron sources and sinks, the contacts at the left and right ends of the device. We have developed a feed-in and removal model of the contacts, which keeps the contact on the same (externally enforced) potential during the course of the simulation.
Mind-boggling is the fact that the very same “classical Hall potential” has also been observed in conjunction with a plateau of the integer quantum Hall effect (IQHE) [Knott et al 1995 Semicond. Sci. Technol.10 117 (1995)]. Despite these observations, many theoretical models of the integer quantum Hall effect do not consider the interactions between the electrons. In our classical model, the Hall potential for non-interacting electrons differs dramatically from the solution shown above and transport proceeds then (and only then) along the lower and upper edges. However the edge current solution is not compatible with the contact potential model described above where an external reservoir enforces equipotentials within each contact.