Two Unrelated Topics of Interest for the Purpose of Demonstration

April 5, 2001

This paper offers a demonstration of the appearance and general document features of the Physical Review style sheet as adapted from the American Physical Society specifications for journal submissions. Although the topics in this paper were written specifically for Publicon, it is hoped they will nonetheless be regarded with interest by the physics community. © 2004 Wolfram Research, Inc.

. HIGH-PRECISION VALUE FOR THE QUARTIC ANHARMONIC OSCILLATOR GROUND STATE

. Introduction

As is well known, only a very limited number of one–dimensional potentials allow for an exact solution of the Schrödinger equation. This means that for many model potentials one has to resort to numerical solution methods. For judging their accuracy, reliability, and speed it is important to have high–precision values of certain nonexactly solvable potentials. The most investigated of such potentials is the quartic anharmonic oscillator [1-19] described by

-(Subscript[ψ, k]^′)′(
            z) + (z^4) (Subscript[ψ, k])(
            z) = (Subscript[ε, k]) (Subscript[ψ, k])(
          z)

The eigenfunctions to the eigenvalues (Subscript[ε, k]) decay exponentially for z -> ±∞.

. The Hill Determinant Method

A classical method to solve Sturm–Liouville problems of type  is to calculate the eigenvalues of a truncated version of the corresponding Hill determinant. Using the harmonic oscillator basis (Subscript[ϕ, n])(z) we write (Subscript[ψ, 0])(
          z) = (∑_(k = 0)^∞) (Subscript[α, k]) (Subscript[ϕ, k])(
              z) where

-(Subscript[ϕ, k]^′)′(
            z) + (z^2) (Subscript[ϕ, k])(
            z) = (Subscript[ϵ, k]) (Subscript[ϕ, k])(
          z)

\(TraditionalForm\`\(ϕ\_n\)(
        z) = \(1\/\@\(\(\@π\) \(2\^n\) \(n!\)\)\) \(e\^\(\(-z\^2\)\/2\)\) \(\(H\_n\)(z)\)\)

Forming the matrix elements (Subscript[𝒽, m, n])=(∫_(-∞)^∞)(Subscript[ϕ, m])(
            z)(-(Subscript[ϕ, n]^″)(
                  z) + (z^4) (Subscript[ϕ, n])(
                  z))dz. For n ≥ m we obtain

(Subscript[𝒽, m, n])={({{0, n - m > 4}, {\(2\^\(\(m - n\)\/2 - 4\)\ \@\(\(m!\)\/\(n!\)\)\ \((32\ n\ δ\_\(m, n - 2\)\ \((n - 1)\)\^2 + 
                        16\ \((n - 3)\)\ \((n - 
                              2)\)\ n\ δ\_\(m, n - 4\)\ \((n - 
                              1)\) + 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4\ \((2\ n\ \((3\ n + 5)\) + 5)\)\ δ\_\(m, n\) + 
                        8\ \((n + 
                              1)\)\ δ\_\(m, n + 2\) + δ\_\(m, n + 4\))\)\), else}})

A rough estimation shows that one obtains about 0.2 digits per harmonic oscillator state. So by taking into account the first 500 eigenstates and carrying out the calculation with about five thousand digits one obtains about 120 reliable digits for (Subscript[ε, 0]). (This calculation takes about 20 minutes on a 2000 vintage workstation using Mathematica 4 [20].)

\(TraditionalForm\`ε\_0 = 
      1.0603620904841828996470460166926635455152087285289779332162452416959435  63044344421126896299134671703510546244358582525580982763829  …\)

The Hill determinant approach allows in addition to the calculation of the eigenvalues, the calculation of the eigenvectors. The following graphic visualizes the matrix of eigenvectors of (Subscript[((Subscript[𝒽, m, n])), 1 ≤ n, 
          m ≤ 100]). The graphic shows that the lowest eigenfunctions are quite similar to the harmonic oscillator eigenfunctions. Higher states are complicated mixtures of harmonic oscillator states. The overall “checkerboard”–like structure results from the fact that the contribution of the antisymmetric (symmetric) harmonic oscillator states to the symmetric (antisymmetric) anharmonic oscillator states is identical zero. The very high states are dominated by truncation effects and do not correctly mimic the anharmonic oscillator states.

. The New Algorithm

To get a very high–precision approximation of

-(ψ^″)(
            x) + (z^4) ψ(x) = λ ψ(
          x)

we start with the series expansion

ψ(
        x) = (Subscript[y, n])(
          x) = (∑_(k = 0)^n) (Subscript[a, k])(λ) 
            (x^k)

For the ground state we choose (ignoring normalization) ψ(0) = 1,  (ψ^′)(0) = 0. For “suitable chosen” \(TraditionalForm\`\(x\^*\)\) we then find high–precision approximations for the zeros of \(TraditionalForm\`\(y\_n\)(\(x\^*\))\) and \(TraditionalForm\`\(y\_n\%′\)(\(x\^*\))\). These zeros then bound (Subscript[λ, 0]) from below and above.

Using the differential equation one obtains the following recursion relation for the (Subscript[a, k])(λ):

\(TraditionalForm\`\(a\_k\)(λ) = \(\(a\_\(m - 6\)\)(λ) - λ\ \(\(a\_\(m - 2\)\)(λ)\)\)\/\(m\^2 - m\)\)

For large n (n -> ∞) we want the function (Subscript[y, n])(x) to vanish as x -> ∞. For a λ smaller than the smallest possible λ, the function \(TraditionalForm\`\(y\_n\)(\(x\^*\))\) will not have a zero, but the function \(TraditionalForm\`\(y\_n\%′\)(\(x\^*\))\) will have a zero for a certain \(TraditionalForm\`\(x\^*\)\). For a λ larger than the smallest possible λ, the function \(TraditionalForm\`\(y\_n\)(\(x\^*\))\) will have a zero, but the function \(TraditionalForm\`\(y\_n\%′\)(\(x\^*\))\) will not have a zero for a certain \(TraditionalForm\`\(x\^*\)\). This allows to find a bounding interval for (Subscript[λ, 0]). The next two graphics show (Subscript[y, 80])(x) and (Subscript[y, 80]^′)(x) for 10 equidistant values for λ from the interval [1.05, 1.08] to visualize this bounding process. (For more details, see [21].)

It is straightforward to implement the calculation of the bounding interval for (Subscript[λ, 0]) in Mathematica in a three–line program (see [22]). Using FindRoot we calculate high–precision values for the zeros of (Subscript[y, n])(ξ) and (Subscript[y, n]^′)(ξ).

λBounds[n_, ξ_, opts___] := 
   Function[f, λ /. FindRoot[f[n, λ, ξ] == 0, 
       {λ, 106/100, 107/100}, opts]] /@ {y, yPrime};

The calculation of (Subscript[y, n])(ξ) and (Subscript[y, n]^′)(ξ) is also straightforward based on a recursive calculations of the (Subscript[a, k])(λ).

y[n_, λ_Real, ξ_] := Module[{a6, a4, a2, ak, σ}, 
   {a6, a4, a2} = {1, -λ/2, λ^2/24}; 
    σ = a6 + a4*ξ^2 + a2*ξ^4; 
    Do[ak = a6 - λ*(a2/(k*(k - 1))); 
      {a6, a4, a2} = {a4, a2, ak}; 
      σ = σ + ak*ξ^k, {k, 6, n, 2}]; σ]

y[n_, λ_Real, ξ_] := Module[{a6, a4, a2, ak, σ}, 
   {a6, a4, a2} = {1, -λ/2, λ^2/24}; 
    σ = a6 + 2*ξ*a4 + 4*ξ^3*a2; 
    Do[ak = a6 - λ*(a2/(k*(k - 1))); 
      {a6, a4, a2} = {a4, a2, ak}; 
      σ = σ + k*ak*ξ^(k - 1), {k, 6, n, 2}]; σ]

Calculating now λBounds[16000, 16, startingValues, WorkingPrecision->6000, AccuracyGoal->600, MaxIterations->100] (where startingValues has been obtained from a call to λBounds recursively, one gets in a few minutes a 1184 digit approximation to the ground state energy of the quartic anharmonic oscillator.

\(TraditionalForm\`ε\_0 = 
      1.0603620904841828996470460166926635455152087285289779332162452416959435  630443444211268962991346717035105462443585825255808798082102931470131768363738  249357892262460047081754469601416374884172822569059357577908880617887902636015  493956902751961489009429348735844094426948979012139714642909519233545338283470  335057576151120257039888523720240221841103086573731091398915453658410311167940  
        583354860009227440069631126702388622971429699610592155832266713769355086736100  
        008318300275179262335739139061361807764985969618149941279280927284070795610604  
        407229468099491362757292738727913689027984247222617169444889547513704380684054  
        391877877295323424587437254317832319060381068741604403437453014684727813918612  
        940470431034013510716071103530089298232754276615189869505650471602527560895262  
        621910256882009644102878156400527052929324050763826502825911247736253847185471  
        440257228543848529745045857097884024906699957047684458770917620291243752732549  
        071164334402302947306923981908956853745359884460160023132919330593958693049166  
        442816339461633242870042614612377430099522342042085977356901535654168503089418  
        513487957341065854797194675964667966134676885864379526545195605682867159583388  84743467012042420714919290048732  …\)

Statistical analysis of the number does not show any regularity.

. Summary

A power series based approach to the high–precision calculation of the ground state of the anharmonic oscillator was presented. Mathematica code to carry out the calculation, as well as results were given. The method can straightforwardly be used to calculate ten thousands of digits of the quartic anharmonic, as well as other anharmonic oscillators. Work concerning the application of the method to higher states is in progress. 
 All calculations and visualizations have been carried out in Mathematica 4.

. Local Density of States for the Harmonic Oscillator

The next graphic shows the local density of states \(TraditionalForm\`\(𝒟\_E\)(
          x) = ⟨x | θ(E - H\&^) | 
            x⟩\) for the harmonic oscillator [21].

. Acknowledgments

The author would like to thank André Kuzniarek for making a prerelease version of the Publicon typesetting system. This work was supported by Wolfram Research, Inc.[23]

. ARE BRILLOUIN ZONES OF HIGH ORDER FRACTAL?

. Introduction

Brillouin zones are among the most popular objects a solid state physicists deals with [24-27]. Despite the fundamental importance for the explanation of most properties of crystalline solids, Brillouin zones as an own subject have rarely been investigated (the only ones we are aware of are [28-31]). For electronic properties, mostly the low order Brillouin zones matter, as an ownstanding subject the high order Brillouin zones are interesting. Mathematically, the (n + 1)th Brillouin zone is the set of points that a line to them crosses exactly n bisector planes. In computational geometry a nth order Brillouin zones is also called nth degree Voronoi region or nth nearest point Voronoi diagrams. The most important fact for high order Brillouin zones is that their shape approaches that of a thin spherical shell and their volume is a constant. Here, for the first time we report on some computational results of higher order Brillouin zones. All calculations and visualizations were done with Mathematica 4 [20].

Recursive definition of Brillouin zones: Given a lattice Λ in (ℝ^d)with lattice points (Subscript[h, i]) (i being a multiindex) the first Brillouin zone ℬ(Subscript[𝒵, 1]) is the closure of the set of all points x such that |x-0|≤|x-(Subscript[h, i])| for all (Subscript[h, i])≠0. The nth order Brillouin zones is the closure of the set of all points x such that |x-0|≤|x-(Subscript[h, i])| for all (Subscript[h, i])≠0 and x∉ℬ(Subscript[𝒵, n - 
                1]).

. 2D Hexagonal lattice

Figure  shows the first twenty Brillouin zones of a 2D hexagonal lattice. It is interesting to observe that the first, third, and fourth Brillouin zones have the shape of an hexagon. For higher orders the shape becomes much more complicated.

Figure  shows the 200th Brillouin zone in one symmetry unit (inside an angle of (30^°)). One sees many small and a few quite large Landsberg zones. The distribution p(A) of the area of the Landsberg zones in the limit n -> ∞ might be an interesting subject to study.

A good numerical fit to number of faces (line segments) (Subscript[♯, n]) of the nth Brillouin zone is (Subscript[♯, n]) ∝ (n^1.15). Figure  shows the circumference of the Brillouin zones normalized to the circumference of a circle with the same radius. Does the ratio approach a finite value in the limit n -> ∞?

. 3D Cubic Lattices

In [32] we gave a complete implementation for the effective calculation of higher order Brillouin zones of the three cubic lattice in (ℝ^3). Figure  shows the (outside of) 15th Brillouin zone for the simple cubic, Figure  shows the 18th for the face–centered cubic and Figure  shows the 10th for the body–centered cubic lattice. The higher order Brillouin zones show quite complicated behavior. Large faces alternate with small ones, relatively plane regions alternate with quite structured ones. The appearance of the n + 1th Brillouin zone is typically completely independent of the appearance of the nth Brillouin zone.

Using this implementation we analysed various properties of higher order Brillouin zones. Figure  shows the area of the Brillouin zones normalized to the area of a sphere of the same volume. The order of the three curves from the bottom is sc, fcc, bcc.

The last result to be given here is the number of faces of the Brillouin zones. By a face we mean any connected planar part of the outside facing side of a Brillouin zone (point contacts separate faces). Table  gives the results for the first 25 Brillouin zones.

. Summary

Preliminary results about some computational results about higher order Brillouin zones have been presented. Further work is in progress and will be published elsewhere.

All calculations and visualizations have been carried out in Mathematica 4.

. Acknowledgments

The author would like to take the opportunity to thank Amy Young for assistance in typesetting this paper.

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