]> Two-Particle Schrödinger Equation and Correlation Function Animations of Wavepacket--Wavepacket Scattering

Two-Particle Schrödinger Equation and Correlation Function Animations of Wavepacket-Wavepacket Scattering

Bachelor of Science Thesis Requirement
David J. Vediner
Oregon State University
Department of Physics

Rubin H. Landau, Advisor

Abstract

Using several improvements developed over the previous few decades, an algorithm which numerically solves the two-particle, time-dependent Schrödinger equation is assembled and tested. Of particular interest is analysis of the correlation between the two particles. Plots and animations of the scattering of two wavepackets in both one and two dimensions can be found on the World Wide Web. Sections one through three were adapted from a previous paper by Maestri, Landau, and Paez.

1. Introduction

Examples in quantum mechanics textbooks are often solutions of the time-independent Schrödinger equation, presenting animations of a single particle interacting with an external potential. While this is a clear and easy to understand picture of what is happening in this process, a more realistic solution would involve the time-dependent Schrödinger equation with two (or more) wavepackets interacting in some external potential.

Figure 1: A time sequence of a Gaussian wavepacket scattering from a square barrier as taken from the textbook by Schiff [1]. The mean energy equals the barrier height.

In the classic quantum mechanics text by Schiff [1], examples of realistic quantum scattering are produced by computer simulations of wave packets colliding with square potential barriers and wells. While Fig. 1 is a good visualization of a simple quantum scattering process, the method developed in this paper presumably extends the realism of such quantum scattering animations. In addition, our extension goes beyond the treatment found in most computational physics texts which concentrate on one-particle wavepackets [2,3,4], or highly restricted forms of two-particle wavepackets [5].

The simulations shown in Fig. 1 were based on the 1967 finite-difference algorithms developed by Goldberg et al. [6]. Those simulations, while revealing, had problems with stability and probability conservation. A decade later, Cakmak and Askar [7] solved the stability problem by using a better approximation for the time derivative. After yet another decade, Visscher [8] solved the probability conservation problem by solving for the real and imaginary parts of the wave function at slightly different ("staggered") times.

In this paper we combine the advances of the last 20 years and extend them to the numerical solution of the two particle-in contrast to the one particle -time-dependent Schrödinger equation. Other than being independent of spin, no assumptions are made regarding the functional form of the interaction or initial conditions, and, in particular, there is no requirement of separation into relative and center-of-mass variables [5]. The method is simple, explicit, robust, easy to modify, memory preserving, and may have research applications. However, high precision does require small time and space steps, and, consequently, long running times. A similar approach for the time-dependent one-particle Schrödinger equation in a two-dimensional space has also been studied [4].

2. Two-Particle Schrödinger Equation

We begin with the two-particle time-dependent Schrödinger equation

\begin{matrix} i \frac{\partial}{\partial t} ψ (x_{1}, x_{2}, t) & = & H ψ (x_{1}, x_{2}, t), & (1) \\ H & = & - \frac{1}{2 m_{1}} \frac{\partial^{2}}{\partial x_{1}^{2}} - \frac{1}{2 m_{2}} \frac{\partial^{2}}{\partial x_{2}^{2}} + V (x_{1}, x_{2}) . & (2) \end{matrix}

where ℏ

= 1

and there are two spatial dimensions. The subscripts

i = 1, 2

denote each of two particles. Instead of

ψ (x_{1}, x_{2}, t)

, however, we will output the probability density, defined as

ρ (x_{1}, x_{2}, t) = {| ψ (x_{1}, x_{2}, t) |}^{2} . (3)

This function tells us the probability that particle 1 is located at

x_{1}

while particle 2 is located at

x_{2}

at some time

t

. We also have the additional constraint of normalization,

\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {dx}_{1} {dx}_{2} {| ψ (x_{1}, x_{2}, t) |}^{2} = 1, (4)

which tells us that particles 1 and 2 must be located somewhere in space at all times.

Visualizing $ρ (x_{1}, x_{2}, t)$ , however, can be very difficult; we do not have the necessary intuition to immediately grasp the meaning of a function over two spatial variables which do not correspond to two physically orthogonal dimensions. Therefore, we introduce two one-particle density functions and define them as

ρ_{i} (x_{i}, t) = \int_{- \infty}^{+ \infty} {dx}_{j} ρ (x_{1}, x_{2}, t), (i \neq j = 1, 2) . (5)

These one-particle density functions allow us to view the process as two separate wavepackets colliding. Of course, separation into two density functions introduces other issues, particularly and most interestingly particle correlation, which is discussed in detail in § 4.

If particles 1 and 2 are identical, then their total wave function should be symmetric or antisymmetric under interchange of the particles. We impose this condition on our numerical solution $ψ (x_{1}, x_{2})$ , by forming the combinations

\begin{matrix} ψ^{'} (x_{1}, x_{2}) & = & \frac{1}{\sqrt{2}} [ψ (x_{1}, x_{2}) \pm ψ (x_{2}, x_{1})] \Rightarrow & (6) \\ 2 ρ (x_{1}, x_{2}) & = & {| ψ (x_{1}, x_{2}) |}^{2} + {| ψ (x_{2}, x_{1}) |}^{2} \pm 2 Re [ψ^{*} (x_{1}, x_{2}) ψ (x_{2}, x_{1})] . & (7) \end{matrix}

The cross term in (7) places an additional correlation into the wavepackets.

3. Numerical Method

We solve the two-particle Schrödinger equation (1) using a finite difference method that converts the partial differential equation into a set of simultaneous, algebraic equations. A two-dimensional array must be set up so the function

ψ

can be evaluated on a grid of discrete values [6]:

ψ (x_{1}, x_{2}, t) = ψ (x_{1} = l Δ x_{1}, x_{2} = m Δ x_{2}, t = n Δ t) \equiv ψ_{l, m}^{n}, (8)

where

l

m

, and

n

are integers. The spatial part of the algorithm is based on Taylor expansions of

ψ (x_{1}, x_{2}, t)

in both the

x_{1}

and

x_{2}

variables up to

O (Δ x^{4})

; for example,

\frac{\partial^{2} ψ}{\partial x_{1}^{2}} \approx \frac{ψ (x_{1} + Δ x_{1}, x_{2}) - 2 ψ (x_{1}, x_{2}) + ψ (x_{1} - Δ x_{1}, x_{2})}{Δ x_{1}^{2}} + O (Δ x_{1}^{2}) . (9)

Using the discrete notation introduced in equation (8), the RHS of the Schrödinger equation (1) now becomes:

H ψ = - \frac{ψ_{l + 1, m} - 2 ψ_{l, m} + ψ_{l - 1, m}}{2 m_{1} Δ x_{1}^{2}} - \frac{ψ_{l, m + 1} - 2 ψ_{l, m} + ψ_{l, m - 1}}{2 m_{2} Δ x_{2}^{2}} + V_{lm} ψ_{l, m} . (10)

Using the formal solution to the time-dependent Schrödinger equation (1) and a forward-difference approximation for the time evolution operator we can express the time derivative in the Schrödinger equation in terms of finite time differences:

ψ_{l, m}^{n + 1} = e^{- i Δ tH} ψ_{l, m}^{n} \approx (1 - i Δ t H) ψ_{l, m}^{n} . (11)

This approximation scheme is unstable, however, since the term multiplying

ψ

has eigenvalue

(1 - iE Δ t)

and modulus

\sqrt{1 + E^{2} Δ t^{2}}

; this means that the modulus of the wave function increases with each time step [2]. Using a central difference algorithm also based on the formal solution (11) we can improve the stability [7]:

\begin{matrix} ψ_{l, m}^{n + 1} - ψ_{l, m}^{n - 1} & = & (e^{- i Δ t H} - e^{i Δ t H}) ψ_{l, m}^{n} \approx - 2 i Δ t H ψ_{l, m}^{n}, & (12) \\ \Rightarrow ψ_{l, m}^{n + 1} & \approx & ψ_{l, m}^{n - 1} - 2 i [{(\frac{1}{m_{1}} + \frac{1}{m_{2}}) 4 λ + Δ x V_{l, m}} ψ_{l, m}^{n} & (13) \\ - λ {\frac{1}{m_{1}} (ψ_{l + l, m}^{n} + ψ_{l - 1, m}^{n}) + \frac{1}{m_{2}} (ψ_{l, m + 1}^{n} + ψ_{l, m - 1}^{n})}], \end{matrix}

where we have assumed a "square" grid (

Δ x_{1} = Δ x_{2}

) and formed the ratio

λ = Δ t / Δ x^{2}

This algorithm has the advantage of increasing stability and only two previous time values must be stored at any time to determine all future solutions by iteration. An implicit solution would determine all future solutions in one step, but would require the inversion of exceedingly large matrices.

While the explicit method (13) produces a solution which is stable and second-order accurate in time it does not conserve probability well. Visscher [8] has deduced an improvement which takes advantage of the extra degree of freedom provided by the complexity of the wave function to preserve probability better. We can implement this extra accuracy by separating the wavefunction into real and imaginary parts,

ψ_{l, m}^{n + 1} = u_{l, m}^{n + 1} + i v_{l, m}^{n + 1}, (14)

and extending this separation to the previous algorithm (13) to arrive at a pair of coupled equations:

\begin{matrix} u_{l, m}^{n + 1} & = & u_{l, m}^{n - 1} + 2 [{(\frac{1}{m_{1}} + \frac{1}{m_{2}}) 4 λ + Δ {tV}_{l, m}} v_{l, m}^{n} & (15) \\ - λ {\frac{1}{m_{1}} (v_{l + 1, m}^{n} + v_{l - 1, m}^{n}) + \frac{1}{m_{2}} (v_{l, m + 1}^{n} + v_{l, m - 1}^{n})}], \\ v_{l, m}^{n + 1} & = & v_{l, m}^{n - 1} - 2 [{(\frac{1}{m_{1}} + \frac{1}{m_{2}}) 4 λ + Δ {tV}_{l, m}} u_{l, m}^{n} & (16) \\ - λ {\frac{1}{m_{1}} (u_{l + 1, m}^{n} + u_{l - 1, m}^{n}) \frac{1}{m_{2}} (u_{l, m + 1}^{n} + u_{l, m - 1}^{n})}] . \end{matrix}

Visscher's advance is in evaluating the real and imaginary parts of the wave function at slightly different (staggered) times,

[u_{l, m}^{n}, v_{l, m}^{n}] = [Re ψ (x, t), Im ψ (x, t + \frac{1}{2} Δ t)], (17)

and using a definition for probability density that differs for integer and half-integer time steps,

\begin{matrix} ρ (x, t) & = & {| Re ψ (x, t) |}^{2} + Im ψ (x, t + \frac{Δ t}{2}) Im ψ (x, t - \frac{Δ t}{2}), & (18) \\ ρ (x, t + \frac{Δ t}{2}) & = & Re ψ (x, t + Δ t) Re ψ (x, t) + {| Im ψ (x, t + \frac{Δ t}{2}) |}^{2} . & (19) \end{matrix}

These definitions reduce to the standard one for infinitesimal

Δ t

. The advantage comes from an algebraic cancellation of errors so that probability is conserved.

Output from the program using the parameters listed in table can be seen in Fig. 2. Notice that displaying the two one-particle density functions $ρ_{1} (x_{1}, t)$ and $ρ_{2} (x_{2}, t)$ , as in Fig. , separately results in a clear picture of what is happening. A two-dimensional plot of the two-particle density function $ρ (x_{1}, x_{2}, t)$ is less intuitive.

Frame 1 Frame 2

Frame 3 Frame 4

Frame 5 Frame 6

Figure 2: Six frames from an m-10m animation of the two one-particle density functions. In the frames at $t = 9000$ and $t = 16500$ the particles are interacting, while the other frames (mostly) have the particles relatively separated. It is quite clear the the particle which barely moves throughout the sequence is the heavier particle. Also notice the "edge" interaction effects in the last two frames.

$Parameter$ $Value$

$Δ x$ $0.002$

$Δ t$ $6.0 \times 10^{- 8}$

$k_{1}$ $+ 156$

$k_{2}$ $+ 156$

$m_{1}$ $0.5$

$m_{2}$ $5$

$σ$ $0.05$

$V_{0}$ $100, 000$

$α$ $0.062$

Table 1: Parameters for a m-10m collision in a repulsive square well potential.

4. Particle Correlation

4.1 The Correlation Function

An integral step in the derivation of § 2 is the separation of the (considering just two particles) two-particle density density function into two one-particle density functions to get:

\begin{matrix} ρ_{1} (x_{1}, t) & = & \int_{- \infty}^{+ \infty} {dx}_{2} ρ (x_{1}, x_{2}, t) & (20) \\ ρ_{2} (x_{2}, t) & = & \int_{- \infty}^{+ \infty} {dx}_{1} ρ (x_{1}, x_{2}, t) & (21) \end{matrix}

Once we have extracted both of the one particle density functions, we can combine them into a two-particle density function:

ρ (x_{1}, x_{2}, t) = ρ_{1} (x_{1}, t) ρ_{2} (x_{2}, t) (22)

Yet, unless the two particles are completely independent this equation leaves out all correlations between the particles. To complete this relation, we must introduce a function over both the spatial variables and time which accounts for the difference between the two one-particle density functions multiplied and the two-particle density function. We call this function the correlation function and use

C (x_{1}, x_{2}, t)

to represent it. We also define the correlation function so that it is equal to zero, instead of one, when the particles are completely uncorrelated:

ρ (x_{1}, x_{2}, t) = ρ_{1} (x_{1}, t) ρ_{2} (x_{2}, t) [1 - C (x_{1}, x_{2}, t)] (23)

Solving for

C (x_{1}, x_{2}, t)

we see that:

C (x_{1}, x_{2}, t) = 1 - \frac{ρ (x_{1}, x_{2}, t)}{ρ_{1} (x_{1}, t) ρ_{2} (x_{2}, t)} (24)

We notice that the correlation function, as defined, is a function of two spatial variables and time.

A good question to ask is "what is the expected behavior of this function?" Because the correlation function is defined so that it tells us the multiplicative difference (with a translation) between the two-particle density and the product of the one-particle densities, we expect $C$ to be large when the two particles are strongly correlated and zero when the two particles are uncorrelated. But when are the particles correlated and when are they uncorrelated?

The particles are initially far enough apart from one another that they can't "see" (interact with) each other, that is, their potentials do not significantly overlap. It makes sense to conclude that the particles, when in such a configuration, would be uncorrelated. Therefore, we would expect the correlation function to be initially zero, or very close to it. When the particles are interacting (colliding), however, each one is very much aware of the presence of the other, leading to a high level of correlation. We should expect the correlation function to reflect this. Ideally, after collision the particles are once again separate and distinct, and the correlation should again be zero.

Frame 1 Frame 2

Frame 3 Frame 4

Frame 5 Frame 6

Figure 3: Six frames from an animation of the correlation function using parameters from table 1. The frames at $t = 9000$ and $t = 16500$ are when the two particles are colliding. The frames at $t = 1500$ and $t = 24000$ are when the two particles should be uncorrelated. In the last two frames the particles are colliding with the wall of the potential barrier. Notice that none of the expected features of the correlation function can be seen in these frames.

Implementation of the correlation output is straightforward and only requires modification of the output routines. Six frames from an animation of the correlation function can be seen in Fig. 3. It is immediately clear that this function is not very illuminating in part because it does not match the expected characteristics of the correlation function. To begin with, the animation does not stay close to zero when the particles are expected to be non-interacting. Also, there is no noticeable change in the behavior of the function when the particles are interacting. Lastly, the function does not seem to go to zero after the particles have interacted and should once again be separate, although this could partially be an effect from collisions with the edge of the potential box.

Clearly, something is amiss with either our correlation function generating routines, or with our definition of the correlation function. Checking over our definition we see that it is probably reasonable, so we are left to examine our routines. What we discover is that while the problem is in our routines, the solution lies in changing our definition.

4.2 The Redistributed Correlation Function

When considering equation (24), one notices that the correlation function is really a measure (over both space and time) of the ratio between the two-particle density function and the product of the two one-particle density functions; indeed, that is exactly how we defined the function. However, in order for this function to give meaningful information, the precision of the algorithms must be significantly better than the precision of the computer hardware. Gaussian distributions, which the particle densities are modeled as, fall to zero very rapidly. Therefore, the numbers being entered into these algorithms (and which we divide by) are extremely small and presumably noisy for the vast majority of the space over which the correlation function is calculated. This yields mostly useless results, or, at the very least, introduces serious artifacts into the animations which hopelessly cloud the desired effects.

How do we get around this problem? What we need to do is multiply the already defined correlation function by another function which distributes the emphasis where it should be, that is, to where the gaussian distributions make the particle densities significant. It is obvious that such a redistribution function is the two-dimensional particle density. However, instead of this function being the two-particle density function, we use the product of the two one-particle density functions. We do this to make the operation of multiplying by the particle density precisely the same as redefining the correlation function to be not the ratio, but the difference between the two-particle density function and the product of the two one-particle density functions [compare equation (27) with equation (24)]. Using a difference automatically places the emphasis where we would like it.

To define this redistributed correlation function, we multiply both sides of equation (24) by $ρ_{1} (x_{1}, t) ρ_{2} (x_{2}, t)$ :

ρ_{1} (x_{1}, t) ρ (x_{2}, t) C (x_{1}, x_{2}, t) = ρ_{1} (x_{1}, t) ρ_{2} (x_{2}, t) - ρ (x_{1}, x_{2}, t) (25)

and then absorb this extra factor into our definition of the redistributed correlation function (

C_{R}

\begin{matrix} C_{R} (x_{1}, x_{2}, t) & = & ρ_{1} (x_{1}, t) ρ (x_{2}, t) C (x_{1}, x_{2}, t) & (26) \\ = & ρ_{1} (x_{1}, t) ρ_{2} (x_{2}, t) - ρ (x_{1}, x_{2}, t) & (27) \end{matrix}

Frame 1 Frame 2

Frame 3 Frame 4

Frame 5 Frame 6

Figure 4: Six frames from an animation of the redistributed correlation function using parameters from table 1. The frames at $t = 9000$ and $t = 16500$ are when the particles are interacting. The last two frames include boundary effects. The spatial axes are in $Δ x$ units.

Frames from an animation of the redistributed correlation can be seen in Fig. 4. While it is immediately obvious that this new function is more well-behaved (more like our original expectations) than our previous definition, our new definition still does not seem to go to zero (or close to it) after the particles have "finished" interacting. The function seems to demonstrate that there is a some kind of "correlation residue" left over from the interaction. While this "residue" is relatively small (it seems to be approximately twenty percent of the maximum value of the correlation function), it is easily large enough to warrant further investigation. However, it is difficult to draw any conclusions regarding this effect from Fig. 4 because shortly after the interaction the data becomes hopelessly tainted by interactions with the side of the potential box. The solution to this problem is readily apparent; make the box larger. This will provide more time and space over which the particles should be uncorrelated and not interacting with the wall.

Table 2 demonstrates the parameters used to make the potential well twice as large in both spatial dimensions. To achieve this, we make $Δ x$ larger by a factor of two, and, to maintain stability, we should therefore increase $Δ t$ by a factor of $\sqrt{2}$ [see our definition of $λ$ in equation (13)]. However, we actually decrease $Δ t$ by a factor of four. This is reflected in the change of parameters between Table 1 and Table . This introduces a concern of stability, but analysis of program output shows that it is stable. Because of this change in resolution, the time scales in Fig. 4 and Fig. are significantly different.

$Parameter$ $Value$

$Δ x$ $0.004$

$Δ t$ $1.5 \times 10^{- 8}$

$k_{1}$ $+ 156$

$k_{2}$ $+ 156$

$m_{1}$ $0.5$

$m_{2}$ $5$

$σ$ $0.05$

$V_{0}$ $100, 000$

$α$ $0.062$

Table 2: Parameters for a m-10m collision in a square well potential twice as large as table 1 in both spatial dimensions.

Fig. 5 shows six frames from an animation of the correlation function in a larger potential "box". The frames focus on the time between particle collision and interaction with the wall. Comparing Fig. 5 with Fig. 4, we see the same general behavior but with one important distinction. Given more time and space, it seems as though this correlation "residue" does actually get smaller. Over the interval of the animation, the function goes from approximately twenty percent of the maximum right after collision (the maximum being when the particles are most strongly interacting) to approximately three to five percent of the maximum just before potential wall interaction.

4.3 One-Dimensional Correlation and Conclusions

For two particles, the correlation function is clearly a function of the two spatial variables. However, are two variables really necessary to convey the necessary correlation information? We started out trying to leap directly into a one-dimensional correlation function, but could not get it to behave properly. Part of the problem was unrealized at that point, namely the problems inherent to the definition in equation (24) as discussed in § 4.2. The other part of the problem was with the issue of how to best reduce the two spatial variables into one. What subsection of the two dimensional space should be sampled over to define the one dimension? Perhaps

x = x_{1} - x_{2}

? Can it be as simple as just defining a line in the two dimensional space, or should we use something more sophisticated?

Frame 1 Frame 2

Frame 3 Frame 4

Frame 5 Frame 6

Figure 5: Six frames from an animation of the redistributed correlation function in a potential well twice as large in both spatial dimensions as the previous figures. This sequence begins at the time of collision (maximum interaction) and ends by hitting the potential wall. Notice that the spatial axes are in $Δ x$ units, so this box really is four times larger in area than the previous one, even though both are "50" units square.

Because there was little progress on the one dimensional front, we decided to move to a two dimensional correlation function, which we reasoned should be straightforward enough to extract. With the exception of the difficulties of § 4.2, it was, and we soon had reasonable correlation output. In hindsight, moving to two-dimensions was a good move because we can now build our insight into the one-dimensional correlation function on somewhat more solid ground, now being in possession of a better understanding of the two-dimensional version. Pursuing the one-dimensional correlation function is the next step.

Based on the output seen in the previous figures and the animations seen on the World Wide Web [9,10], we feel that the correlation output demonstrates that the two one-particle density functions contain useful information and allow us to visualize this problem as two distinct particles interacting. The correlation function shows us that the particles are not always in such a distinct state, but that is exactly as expected. We feel that analysis of the correlation function is a useful tool for gaining a deeper understanding of what is taking place in the collisions.

Acknowledgments

I would like to thank Prof. Rubin Landau for his support in this project. I would also like to thank both Prof. Landau and Jon J.V. Maestri for their development of the wavepacket-wavepacket interaction code and algorithms. Additionally, the previous papers written by Prof. Landau, Jon Maestri, and Manuel J. Páez [11] were very helpful in developing § 1 through § 3.

References

[1]: L.I. Schiff, Quantum Mechanics (third edition), McGraw-Hill, New York (1968), p 106.
[2]: S.E. Koonin, Computational Physics, Benjamin, Menlo Park, 176-178 (1986).
[3]: N.J. Giordano, Computational Physics, Prentice Hall, Upper Saddle River, 280-299 (1997).
[4]: R.H. Landau and M.J. Paez, Computational Physics, Problem Solving With Computers, Wiley, New York, 399-408 (1997).
[5]: S. Brandt and H.D. Dahmen, Quantum Mechanics on the Personal Computer, Chapt. 5, Springer-Verlag, Berlin, 82-92 (1990).
[6]: A. Goldberg, H. M. Schey, and J. L. Schwartz, Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena, Amer. J. Phys., 35, 177-186 (1967).
[7]: A. Askar and A.S. Cakmak, Explicit Integration Method for the Time-Dependent Schrödinger Equation for Collision Problems, J. Chem. Phys., 68, 2794-2798 (1978).
[8]: P.B. Visscher, A Fast Explicit Algorithm for the Time-Dependent Schrödinger Equation, Computers In Physics, 596-598 (Nov/Dec 1991).
[9]: Movies of Wavepacket-Wavepacket Quantum Scattering,
href://nacphy.physics.orst.edu/ComPhys/PACKETS/
[10]: Movies of Wavepacket-Wavepacket Correlation Output,
href://www.physics.orst.edu/~vedinerd/correlation/
[11]: J.J.V. Maestri, R.H. Landau, and Manuel J. Páez,
Two-Particle Schrödinger Equation Animations of Wavepacket-Wavepacket Scattering Accepted for publication, American Journal of Physics (Sept. 2000).
[12]: Our movies are animated gifs that can be viewed with any Web browser, or viewed and controlled with a movie player such as Apple's QuickTime. To create them, we have our C code packets.c output files of wavefunction data for each time. We plot each data file with the Unix program gnuplot to produce one frame, and then convert the plots to gif files. We then use gifmerge (Mark Podlipec and Rene K. Mueller 1996, http://www.the-labs.com/GIFMerge/) and gifsicle (Eddie Kohler 1997, http://www.lcdf.org/gifsicle/) to merge the frames into an animation. Further information and instructions for making movies using different operating systems and formats can be found on the Visualization Of Scientific Data section of the Landau Research Group Web pages, http://nacphy.physics.orst.edu/DATAVIS/datavis.html.