Visual Quantum Mechanics
Prepared for Contemporary Physics by
Dean Zollman, Wally Axmann, Bob Grabhorn,
Carol Regehr, and Paul Donovan
Spring, 1994
From Kansas State University:http://bluegiant.phys.ksu.edu/dvi/vqm/vqm.html
Visual Quantum Mechanics: Table of Contents
8. Decreasing Wavefunctions: The Classically Forbidden Region
In regions where the total energy is less than the potential energy,
a region where in classical physics there would be no probability of
finding a particle, the amplitude of the wave function decreases.
Further away from the
boundary where the potential energy changes, the probability of the
object being located there decreases. However, unlike classical physics,
in the region where the
total energy is less than the potential energy, the probability of
finding the object is not always zero.
We would also expect that, for a given total energy, the rate of
decrease should depend on the size of the potential energy. If the
potential energy is large, the object should be more restricted than
if it is small. The results show that this does occur as illustrated
here.
To put some numbers on the rate of decrease of the wave function in regions where the total energy is
less than the potential energy, we introduce some approximations based
on wave motion. First, we introduce a characteristic length, l, with
which we describe the decrease:
When we use the electron-volt and nanometer units, we obtain an equation similar to the
one we used for wavelength:
We use this length as a unit to measure the decrease of the wave
function's amplitude. The basic (approximate) rule
is that when we move six times l in a region where PE > TE the
amplitude of the wave function decreases to one-third of its value:
We can apply this rule to determine how much the wave function
decreases in any region where the total energy is less than the
potential energy.
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Visual Quantum Mechanics: Table of Contents