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
6. Determining the Wavefunction from the Potential Energy
Potential energy diagrams will be used to describe the forces which
act on small objects. They will form the basis of our solutions to
Schrödinger's Equation. Thus, the first
rule for obtaining wave functions is:
Draw a diagram of the potential energy versus position for the object.
As an example consider an electron which is moving through empty space
and then enters a metal. When it enters the metal, it interacts with
the electrons in lots of atoms. These interactions slow
the electron, so its kinetic energy decreases. Because the total
energy does not change, the potential energy of the electron must
increase while the kinetic energy is decreasing. The electron
experiences a sudden change in kinetic and potential energies when
it enters the metal. Thus, the potential energy
will reflect a rapid change at that boundary:
Because the wavelength of an object is
associated with its momentum, we can find a
relation between wavelength and kinetic energy. The
first step is to establish a relation between kinetic energy and momentum.
Both kinetic energy and momentum are related to the speed
of the object. Thus, we can relate the kinetic energy to the momentum:
Then, we can use these equations with the wavelength-momentum relation to write the
wavelength in terms of the kinetic energy.
We can use this equation for any object. To describe an electron
using units of
electron volts (eV) for energy and nanometers (nm) for wavelength,
the equation above becomes
We can use this result to find the wavelength of the wavefunction in each region of potential energy. In general we will not calculate the
exact number for the wavelength. Instead, we will want to know
how the wavelength in one region compares to that in another. For that
comparison we can use the equation above to note that:
as the kinetic energy goes up the wavelength goes down
and
as the kinetic energy goes down the wavelength goes up.
The equations used are those of classical physics, and
looking at the equation, we can also see that the kinetic energy must
be positive for the wavelength to have any meaning. If the kinetic
energy is negative, we get the square root of a negative number. That
gives us a meaningless wavelength. Thus, we will use this result only
for positive kinetic energies, that is, when the total energy is
greater than the potential energy.
This discussion leads to our second rule for obtaining solutions to
Schrödinger's
Equation:
When the total energy is greater than the potential energy, we can determine the
wavelength of the wavefunction by using the relation between the wavelength
and the kinetic energy.
As an example, use the potential energy for an electron entering a metal:
Next we can redraw this diagram and add
a dotted line to indicate the total energy of
the electron:
To determine the kinetic energy of the electron we
look at the difference between the total energy and the potential
energy. As shown in the figure below, the kinetic
energy on the left side is about four times greater than that on the
right side of the edge of the metal.
So, we conclude that the wavelength on the left
side will be about one-half of that on the right side. The
approximate wavefunction looks like the one in
this figure
Next:Constraints on the Wave Function
Visual Quantum Mechanics: Table of Contents