1-D Free Particle

In the scenario below, a source of particles of mass M at minus infinity streams towards plus infinity at a constant total energy E. You can change M and E as well as the width, a, height (V > 0), or depth (V < 0) of a potential energy discontinuity. Pay attention to the way the particles' wavelength changes in each region, and look for non-classical behavior: reflections, tunneling, and resonances. These effects show up in the second graph, which plots T, the transmission probability (see Figure 12.5 on page 399), as a function of total energy. (Note that this text figure uses T to symbolize kinetic energy. Away from the potential step or well, where V = 0, the kinetic energy is the total energy.) Since the wavefunction has both a real and an imaginary part, the final graph below shows three curves. From top to bottom, these are the square of the total wavefunction (proportional to the probability), the real part of the wavefunction, and its imaginary part. The gray area spans the region of the potential discontinuity. (The mass is in amu units so that M = 1 is H, M = 4 is He, etc. The unit of length is the Ångström, 10^-10 m, and the unit of energy is 10^-20 J, a chemically reasonable amount.) Click here for a color pdf version of the class handout based on this topic.

1-D Trapped Particle

The particle-in-a-box model system is very important (see Figure 12.3 on page 395). Note how the wavefunction changes with quantum number n, and explore the probability distribution. The wavefunction is in black, its square is in red, and the Probability integration range, from xmin to xmax, is the green bar. Change the quantum number n, the box length L, and the integration range variables xmin and xmax. The probability of being in the right half of the box = the probability of being in the left half = 0.5, but does the probability of being in the middle half from 0.25L to 0.75L also equal 0.5?

Back to Mixing wavefunctions. Ahead to 2-D systems. Back to the Index.   Copyright 1999,2000 John S. Winn. All rights reserved. Last updated September 8, 2000.