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Web-Schrödinger is a program for the interactive solution of the stationary (time independent) and time dependent two dimensional (2D) Schrödinger equation. The program itself runs on our server and can be used through the Internet with a simple Web browser (Internet Explorer, Mozilla, Opera, Chrome was tested). Nothing is installed on the user's computer. The user can load, run, and modify ready-made example files, or prepare her/his own configuration(s), which can be saved on her/his own computer for later use.
The time evolution of the quantum mechanical wave function ψ(r;t) is governed by the time dependent Schrödinger equation:
where r = (x,y) is the position coordinate, t is the time and H = K + V is the Hamilton operator, K is the operator of the kinetic energy, and V = V(x,y) is the operator of the potential energy.
When the potential function V(x,y) and the initial wave function ψ(x,y,t0) = ψ0(x,y) is known, the time dependent Schrödinger equation determines the wave function ψ(x,y,t) for any time value. We can calculate all observables from the wave function, for example the rho(x,y,t) probability density and the j(x,y,t) probability current density.
rho(x,y,t) gives the probability of finding the quantum
mechanical particle around the point (x,y)
at time t. We call those ψ(x,y,t)=ψ(x,y)
states, where ψ(x,y) is independent of time, stationary
states. The stationary (time independent) states are given by the
stationary Schrödinger equation:
Hψ(r) = Eψ(r)
where E is the energy of the state.
All functions of the program are available through a menu system. Upon starting the program a default configuration is loaded, the user can immediatelly run this through the Calculation menu, or load another configuration with the Load Example, or Load menu points. All parameters can be modified in the Edit menu and the current setup can be saved anytime with the help of the Save function.
We have prepared several characteristic examples, illustrating the most important phenomena of quantum mechanics, including the spreading of the wave packet, tunneling, bound states, etc. The current list of the examples is given in Appendix A. The example library is continuously expanding, see Appendix A for the up to date status. After loading an example setup the user can study and modify the parameters through the Edit menu or go straight to Calculation to calculate the time development and/or the stationary states.
This function makes it possible to load the user's own configuration files, from her/his own computer. Such parameter files can be created either by saving a (possibly modified) example configuration (or the default configuration) or writing a configuration file from the scratch with a text editor or any other program.
The current state of the parameters can be saved anytime to the user's own computer.
The wave function and the potential is represented on a 2D mesh. Here you can specify the number of mesh points (Nx , Ny) in the x and y direction and the size of the calculation region in Angström (sx, sy). For typical applications the Δx = sx/Nx, Δy = sy/Ny values should be between 0.1 - 1 Å. The origin of the coordinate system is in the middle of the calculation region.
The numerical algorith uses a periodic boundary condition, i.e. what goes out of the calculation region at the right side, comes in at the left side. It is like if the whole plane were "tiled" with the calculation region. As a consequence when the wave packet approaches the boundary of the calculation box, it "meets" its copy at the neighboring box and this causes unphysical interference effects to appear in the probability density. The parameters of the calculation (spatial- and temporal mesh, potential, and initial state) should be carefully chosen to avoid this effect.
V0 gives the default value of the potential in
eV (Electronvolt).
Note: due to the difference of the algorithms used for the solution
of the time dependent and stationary Schrödinger equations,
generally a finer mesh is necessary for the time dependent calculation.
E.g. a Nx=256 is typical
value for the time dependent, and Nx=64
for
the
stationary
calculation
The potential V(x,y) can be interactively assembled from objects of several types: circle, rectangle, and plane. Any number of these objects can be given. For each object the user can specify its geometrical parameters and its potential value. For pixels where several objects overlap, the object given most recently determines the pixel potential value. The program shows the potential function generated from the current set of objects as a grayscale image.
Here the user can specify the initial wave function ψ0(x,y), which is the input of the time dependent calculation (it is not used at the stationary calculation). Its general form is a so called truncated plane wave [7] wave packet, i.e. a Gaussian wave packet convolved with a 2D square window function. The program displays the chosen initial state together with the potential function, as a composite color image. In order to ensure that the wave packet has its ideal form (minimal size and flat envelope) when it hits the potential, a time retardation procedure is included into the initial state preparation. The user can specify the retardation time by giving the the bx, by distance values, which mean that after proceeding such distances in x, and y the wave packet should have its "ideal" form.
ax, ay give the spatial
width of the wave packet. The initial state should be specified such a
way, that its overlap with the potential objects is negligible.
The user can place horizontal or vertical line segments (detectors)
into the calculation window. The program calculates the probability
current I(t) passing through
each line segment during the time evolution of the wave packet and also
its time integral T for the
whole calculation time. T is
called transmission, because it gives the probability that the quantum
particle crosses the given line segment (detector).
Here we can specify the parameters of the time dependent and the
stationary
calculation.
Parameters used for the time evolution calculation: The number of time points is Nt and Δt gives calculation time step. Δt has to be given in atomic time units, 1 au time = 0.0242 fs (femtosecond).
The numerical algorithm imposes a condition on the maximal Δt value that can be used: Δt < 4/π (Δx)2 / D, where D is the number of dimensions, D=2 in 2D. (This formula is valid in atomic units, i.e. one has to insert Δx in Bohr, 1 Bohr = 0.529 Å. For the default Δx = 0.3 Å, Δt = 0.2 au is suitable and this is the default time step.)
It is not necessary, however, to display the results in such a fine
time scale. Therefore the user can input the "display timestep", i.e.
the number of calculation time steps, when the wave function is
displayed.
Parameters used for the stationary
calculation: Nstat
gives the number of states calculated.
When the user hits the "RUN" button, the time development calculation starts on the server. The progress of the calculation is shown by small thumbnail images. For typical parameters the time development calculation takes 1-2 minutes. (If there are more concurrent jobs on the server – either from this user or from others – the calculation may be somewhat slower. The program writes out the number of concurrent jobs – if there is any – after hitting the "RUN" button.)
After the time development calculation is completed on the server,
the time development of the probability density is displayed in
composite color images. The program first calculates the global maximum
of the probability and normalizes each frame using this value. A
nonlinear color scale (γ=2.5) is used in order to facilitate
presentation.
If the user placed detectors into the calculation window before the
start of the calculation, the program also displays the I(t) probability current functions
and T transmission values for
each of the detectors.
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