A few suggestions for playing with the Particle-in-a-Box Applet
Preparations:
•
click on link or go to URL http://www.falstad.com/qm1d/
•
check “Stopped” box
•
hit “Clear” button
•
pulldown “View”, deactivate “State Phasors” and “Momentum”; in the “View” sub
menu “Wave Function”, activate “Probability”
•
there is a grey line at about the height of the “Stopped” box; this is a delimiter
separating the top and bottom parts of the window; you can pull it down a bit to get a
better view of the energy levels in the upper part of the window
•
pull the “Well Width” sliding bar to the center
Points of interest and food for thought:
•
Play with the “Well Width” sliding bar. As the well width changes, the energy levels
of the PIB (grey lines in the top part of the window) move up and down. Recall that
the energy of the PIB levels goes as
En =
h2 2
n
8ma 2
where h is Planck’s constant, m is the mass of the particle and a is the well width.
Follow the behavior of the En as you decrease and increase the well width a, and see
that it makes sense according to the equation above.
•
Play with the “Particle Mass” sliding bar. Follow the behavior of the En as you
decrease and increase the mass, and see that it makes sense according to the equation
above.
•
Center the “Well Width” sliding bar and reduce the particle mass until you can clearly
resolve the lowest few energy levels. Move the mouse pointer over the line
corresponding to the ground state. The yellow line appearing in the lower part of the
window is the wave function of the state your pointer is touching. The white filled
area is the probability density ψ (x) of the state you last clicked. Click on the
2
various states to see how wave function and probability density behave. Count the
nodes in the wave function.
•
As n increases, the wavelength of the particle becomes shorter. You can rationalize
this the following way: All energy the particle has inside the well is kinetic energy
(since V(x) = 0 inside). This means that if n grows, En = Ekin grows, too. Decreasing
the deBroglie wavelength λ = h / p implies that the momentum p increases, which is
consistent with increasing kinetic energy.
•
Increase the particle mass and look at higher and higher excited states. See how the
number of nodes grows. In the limit of really high quantum numbers n, the maxima in
ψ (x) seem to “smear” into one another. In classical physics, the probability of
2
finding the particle anywhere in the box would be equal at each point. As the maxima
come closer and closer to one another and the deBroglie wavelength becomes shorter
and shorter, the maxima will eventually be spaced by less than the size of the particle
itself, and the probability density averaged over the particle size will approach a
constant. This shows that the QM description converges to the classical result in the
limit of large quantum numbers n.
....
There is a lot in this applet to play with and I encourage you to do so. We will get back to
this and other applets as the semester progresses. Have fun playing!