401
40
OneDimensional Quantum Mechanics
Recommended class days:
3
Background Information
At this point, students have seen that:
• Atomic particles exhibit wavelike behavior.
• Standing de Broglie waves lead to quantization of energy.
• The Bohr model suggests that real atoms have quantized energy levels.
•Waveparticle duality can be understood with a probabilistic wave function.
Now we need to assemble these bits and pieces into an actual
theory
of matter on the atomic scale.
The goals of this chapter are quite limited. Physics majors and those engineers who might use
quantum mechanics extensively will go on to take a modern physics course and perhaps a full
course in quantum mechanics. For them, this chapter is an introduction. A larger group of students
needs to understand the
significance
of energy levels and wave functions, particularly as they apply
to issues such as molecular bonds or tunneling, but they’ll never have to solve the Schrödinger
equation for themselves. Hence the focus of this chapter is on:
•Modeling quantum systems.
• Interpreting and using solutions of the Schrödinger equation.
• Understanding quantum phenomena such as tunneling.
These goals can be met with onedimensional quantum mechanics as applied to time
independent, boundstate problems with realvalued solutions. Students are not expected or asked to
solve differential equations, but they should learn to recognize and interpret the major features of a
solution of the Schrödinger equation.
Classical mechanics analyzes phenomena by developing a
model
of the forces acting on a
particle. Some forces may be accurately known, others are reasonable approximations. Once the
forces of the model are identified, the particle’s trajectory can be found by solving Newton’s second
law. We also use models in quantum mechanics, but now the models are expressed in terms of a
potential energy function. This is much harder for students to deal with. Although potential energy
diagrams were emphasized in Chapter 10, most students will need to review how potential energy
diagrams are interpreted classically to yield turning points, kinetic energies, forces, and so on. You
will also need to be very explicit with your explanations of how a particle confined to a certain
region can be
modeled
as being in a potential energy well.
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Instructor’s Guide
Unpublished research at Kansas State University has found
that students often interpret graphs and diagrams very differently
than we intended. For example, we like to show wave functions
“in” the potential well, each oscillating about its energy level.
Students find these graphs to be confusing because we’re graphing
both energy
and
wave functions on the vertical axis. Conse
quently, some students interpret this graph as saying that
ψ
3
is
always larger in value than
ψ
1
because it is “above,” and hence
more positive, than
ψ
1
. They don’t recognize, and textbooks rarely
state, that each wave function is drawn such that the energylevel
line about which it oscillates is the “zero” for that wave function.
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 Spring '10
 kant
 Potential Energy

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