Time Evolution Exploration #1

Please answer the  following 2 questions regarding the Physlet:

Pick n =   ; then

check, then click the set state button to see the alternative visualization!

Description

Shown is the time-dependent wave function for a particle in an infinite square well of length a=2.   The wave function evolves with time according to the TDSE.  You may change the state by choosing an n.   Restart

Note: In the student version we do not initially show the color strip.

Questions

  1. What does the "height" of the wave function correspond to? Why?
  2. For n = 1, what does a time of t = 1 correspond to?  Why?
  3. For n = 1, what then does the color of the wave function at t=0, 0.25, 0.5, 0.75, and 1 correspond to?

Student Answers:

1) The "height" of the wave function changes with x and with n. So, depending on where you are on the x-axis and what n you have, you will have a certain height. The max height is always the same, however, it does not depend on anything. What does the "height" mean- well, the height times the height integrated over x from a starting point to an ending point is the probability that the particle will be in that region. So, it also in a way corresponds to probability.

2)For n=1, t=1 corresponds to the color when t=n where n= 0,1,2,. . . . When t=n, this color is a royal blue. Only the color of the graph changes with time, so color or time, correspond only to each other.

3)For n=1, when t=n (t=0 or 1)the color is royal blue (see #2) for t=.25+n (t=.25) the color is light green; for t=.5+n (t=.5) the color is a brownish green; and for t=.75+n (t=.75)the color is a nice pink. As you can see, the colors themselves form a kind of curve dependent on time where the "wavelength" is 1 unit of time. So, each color corresponds to a certain time + n. Meaning, it will return to that color every time you add an integer to that first time.


1. It looks like the height is just the wave amplitude ... but it seems like that's too simple 2. t=1 is the period of the wave function b/c the color returns to original 3. Is the color a phase shift of some sort?


1. The height of the wave function corresponds to the center of the well. This is what a normal wave function with N=1 would look like besides the color. The height corresponds to the area within which the wave would be most likely found.

2. I would guess that due to the color, that t=1 is the same as t=0. The same wave pattern returned back to the original starting pattern.

3. The color of the wave function goes from blue, to green, to puke green, to bright pink, and back to blue. I'm guessing that these colors correspond to the intensity of the wave.


First of all, this is really confusing. The only thing that I can come up with is that the changing colors have to do with the phase of the wave equation. If this is a superposition of two waves with different phases w/r/t time, then maybe the colors change based off of how the two waves are lining up. t=0... in phase; t=.25... 1/4 out of phase; t=.5... out of phase; t=.75... 3/4 out of phase; t=1... in phase again; The t=1 would then correspond to a complete 'beat" of the two waves. That still doesn't explain what the height of the wave function corresponds to. Maybe it is the energy of the particle in at that particular x?? I really don't understand it.

 

 

Copyright 2003 Mario Belloni and Wolfgang Christian.