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Please read this writeup for a description of the theory associated with this applet, and an accompanying set of exercises. Documentation of the applet itself is listed below.

This simulation demonstrates the scattering of an incoming wave on a one-dimensional crystal. The one-dimensional crystal is represented as a series of evenly-spaced atoms, each atom of which is modelled as a Dirac Delta Function. The simulation shows two windows, with a set of controls at the right.

The left window demonstrates graphically the parameters of the system. The x-axis represents position, and the y axis represents energy. In reality, y-axis represents momentum, but for a user not in the quantum-mechanics class, this can be thought of as energy. There are several objects in the left window, most of which can be adjusted. To adjust a parameter, one can drag the little red boxes attached to the adjustable objects.

• Drawing of the Crystal
  The crystal is drawn in WHITE in the left window. Each atom of the crystal is represented as a white spike.
• Representation of Incoming Wave
  The momentum of the incident wave is represented in yellow. The more energy the wave has, the higher up this line will be.
• Adjustable parameters
  • The yellow momentum line can be adjusted.
  • The strength of the atoms can be adjusted by changing the height of the white atoms profile. Notice, in general, that the higher the energy, the harder for the wave to traverse through the crystal.
  • Spacing of the atoms can be adjusted. The number of atoms in the right-hand controls must be set to a number greater than one for this control to be accessible.

The right window demonstrates the transmission of the wave through the crystal. There are no axes on the graphs yet, but they will be added soon. Pure reflection (hence no transmission) is represented by a point at the left of the graph, perfect transmission at the right side of the graph. The vertical axis of the graph representes momentum (ie, energy), and is calibrated to be equivalent to the vertical scale on the left window.

• Transmission Curve The dark blue curve plots the transmission through the crystal for the energy range represented by the graph.
• Current Transmission The bright cyan dot on the transmission graph corresponds to the transmission represented by the current level of wave momentum (represented by the yellow line in the left window).

• The top slidebar controls the number of atoms in the crystal. Note that more atoms are added to the left window as this is adjusted. Also, only a finite number can be displayed in the window at one time.
• The next 4 outputs represent the current status of the system, showing the values of the relevent parameters, such that a specific value can be recorded or compared to the theory.
• The "Random Spacing" button will randomize the spacing of the delta-functions within the range (a * n).
• The "Random Height" button randomizes the 'height' of the delta-functions individually, between zero and 2*kappa.
• The "Initialize" button resets the system back to it's default starting setup.
• The "Toggle Redraw" turns on and off a dynamic redraw of the system when various parameters are adjusted. For example, if one is adjusting the number of atoms or strength of the atoms, the transfer curve needs to be continually recalculated. For crystals with many atoms, this can take while to calculate, and is difficult to update. Thus, one can click this toggle switch, so the program won't update the transfer curve until the user is done dragging the control. Notice that if this is so, the transfer curve is grayed out.

• Value Entry
  Components will be added that will empower the user to manually input desired parameters instead of manually adjusting everything from the screen. This will allow the user to compare almost exact values they have calculated from the theory with the simulation, and then let them tweak things later on.
• Anti-aliasing
  Currently, the output plots (in the y axis, or 'k' momentum axis) are calculated as pixel rows. Anti-aliasing could be employed to give the user better visualization of the range of transmissions within a pixel-width's uncertainty.