|
Time
Development of a Quantum Wave Packet
Click here to go back to the
Physlets page.
This QTime physlet displays the probability density of the wave packet along with the
quantum phase. The vertical thickness of the wave packet is proportional to the probability.
Time Development Examples Using QTime
Be sure the simulation has finished loading before you begin.
Instructions
Click on an example below to initialize the simulation at the top of the page. You
may start and stop the simulations using the buttons on the bottom of the animation. Click
and drag inside the animation to read the coordinates and probability.
QTime Applet Help
Contents
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, -180o< PHASE<180o.
- 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
|
