This is a collaborative problem*. Please share your questions and results here. We want to get to the answers and an understanding of this problem through our collective effort. Posting comments here is my idea about how that collective effort can take place. This is a somewhat difficult problem. Your efforts and comments here will be very much appreciated!
4. Consider a well of the width we found for problem 1, and with a depth of 100 eV:
a) Is the energy of the ground state above or below -90 eV?
b) How many bound states does it have and what are their energies?
c) Sketch the ground state. What is the length scale associated with the exponentially decreasing part of the wave-function which extends outside the well? (We can call this the evanescent part of the wf.)
d) Sketch the 1st-excited state. What is the length scale associated with the
exponentially decreasing part of the wave-function which extends
outside the well for this state?
e) Sketch the highest energy bound state. What is the length scale associated with the
exponentially decreasing part of the wave-function which extends
outside the well for this state?
f) extra credit: Create normalized bound state wave-functions for any or all of these bound states. Do a nice graph and maybe compare them to the corresponding bound state from a infinite square well of the same width.)
*The way I envision this is that we work on this problem collaboratively here, freely sharing and comparing our ideas, questions and results. And, in addition, you hand in a nicely organized written solution. Not one or the other, but both. Does that make sense?
The energies I got after graphing the function are -99eV and 57 eV. They're close to your answers but I feel are too large. Did anyone else get anything similar?
ReplyDeleteSince we have a finite well I said K was sqrt(2m(E-U))/hbar and when i plugged it in to the graphing program I let E=100eV and U=x and of course make sure to graph both sides separately or else it won't graph it. I'm not 100% sure that is right, but I do know I'm on the right track.
DeleteBetter late than never but here is what I got for the allowed energy states: -3.146,-40.544,-72.78,-93.103.
ReplyDeleteFor how far the "tails" extend beyond the well I used x=sqrt(-hbar^2/s*m*E) this will give a decay of 1/e for every x distance outside the well.
The last excited energy state I got a decay of 1/e at an x = .11nm! Meaning it extends rather far outside our .194nm well.
Let me know what you guys got or if I am doing this right.