Some Unrelated Topics of Interest

Michael Trott

April 5, 2001

High-precision Value for the Quartic Anharmonic Oscillator Ground State

Introduction

-(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

-(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].)

\(TraditionalForm\`ε\_0 = 
      1.060362090484182899647046016692663545515208728528977933216245241695943563044344421126896299134671703510546244358582525580982763829  …\)

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

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

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.

\(TraditionalForm\`ε\_0 = 
      1.060362090484182899647046016692663545515208728528977933216245241695943563044344421126896299134671703510546244358582525580879808210293147013176836373824935789226246004708175446960141637488417282256905935757790888061788790263601549395690275196148900942934873584409442694897901213971464290951923354533828347033505757615112025703988852372024022184110308657373109139891545365841031116794058335486000922744006963112670238862297142969961059215583226671376935508673610000831830027517926233573913906136180776498596961814994127928092728407079561060440722946809949136275729273872791368902798424722261716944488954751370438068405439187787729532342458743725431783231906038106874160440343745301468472781391861294047043103401351071607110353008929823275427661518986950565047160252756089526262191025688200964410287815640052705292932405076382650282591124773625384718547144025722854384852974504585709788402490669995704768445877091762029124375273254907116433440230294730692398190895685374535988446016002313291933059395869304916644281633946163324287004261461237743009952234204208597735690153565416850308941851348795734106585479719467596466796613467688586437952654519560568286715958338884743467012042420714919290048732  …\)

Statistical analysis of the number does not show any regularity.

Summary

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

Local Density of States for the Harmonic Oscillator

Acknowledgments

Are Brillouin Zones of High Order Fractal?

Introduction

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 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

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

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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