QTime Applet Help

Click to find out:

• what this applet does.
• what units are used.
• how to see the data entry area.
• how to see the display.
• what the display shows.
• how to get quantitative wave function information.
• how to start/stop the time development.
• how to reset the applet.
• how to change wave functions and/or potentials.

Description. The QTime Applet displays the time evolution of a one dimensional, single particle quantum mechanical system. The complex valued (spatial) initial wave function and the potential function are user defined. The evolution is done by iterating the time independent Schroedinger Equation. The iteration time step is preset.

Units. In this applet, the following units are used
[NOTE: h=Planck's constant, h-bar=h/(2*pi), c=speed of light, m=particle mass. The units described below are "natural" units in which h-bar=c=m=1.]

• position (x) is measured in units of the Compton wavelength of the particle divided by 2*pi, sometimes called the angular Compton wavelength ((h-bar)/mc)
  
• energy (U) is measured in units of the particle rest energy (mc²)
  
• time (t) is measured in units of time it takes light to travel the angular Compton wavelength (h-bar/mc²).

Tabs (located at top of applet )

• Wavefunction. Click to show wave function display
  
• QM. Click to show area for input of wave function and potential
  
• Help. Click to show credits

The "Wave function" display.

• To view the display. Click the "wave function" tab at the top of the applet.
  
• Display elements:
  
  • Graph axes: The scale and range of the horizontal axis (x) is preset. The scale and range of the vertical axis (U) autoscales with the potential energy function
  • Potential energy: The potential energy is shown on the graph with a single black line.
  • The wave function: The wave function, PSI, is complex valued, with modulus |PSI| and phase PHASE, PSI = |PSI|*exp(i(PHASE)). Both the modulus and phase vary in space and time. |PSI| is proportional to the width of the wave function envelop. This means that the probability density is proportional to the square of the width of the wave function envelope. PHASE is indicated by color coding (red=0 ^o, blue=180 ^o). The average energy of the wave function (a constant of the motion) can be read as the vertical coordinate of the center of the wave function envelope.

• To obtain quantitative wave function information, click and hold the mouse button on the graph.
  
  • X is the location of the cursor along the x axis
    
  • |Psi| is the modulus of the normalized wave function.
    
  • < is the phase of the wave function, -180^o< PHASE<180^o.

• Buttons (located below display)
  
  • Stop/Run button toggles the time evolution on/off.
    
  • Reset button restores the initial wave function and sets the time display to zero.

The "QM" Input

• To view the input entry. Click the "QM" tab at the top of the applet.
  
• Input elements. Standard calculator functions and notation are accepted. The step(x) function is implemented. NOTE: decimal values must be entered with zero to left of decimal (i.e. 1/sqrt(2)=0.707).
  
  • Real. Enter the real part of the wave function into the box. The function does not have to be normalized. The vertical size of the wave function in the display is proportional to the value of the function entered.
    
  • Imaginary. Enter the imaginary part of the wave function into the box. This should be a real valued function. Do not enter an "i". The function does not have to be normalized. The vertical scale of the wave function in the display is proportional to the value of the function entered.
    
  • Potential. Enter a function into the box.
    
  • New System. Click the Submit Functions button to enter changed input for display. If input is unrecognizable, the problem input will be highlighted in red