|
This Mathcad worksheet uses two Gaussian
functions to guide students as they explore
the uncertainty principle. The standard deviations
of Gaussian functions provide convenient
measures of position operator uncertainty.
Students are asked to build a linear combination
of orthogonal particle-in-a-box eigenfunctions
to represent Gaussian functions. The document
permits students to discover that a Gaussian
function with smaller standard deviation
requires more eigenfunctions in its expansion
than a Gaussian function with a larger standard
deviation. The magnitude of momentum uncertainty
is related to both the number of eigenfunctions
(required in the expansion of the Gaussian
function) and the quantum numbers of these
eigenfunctions. Through the many document
exercises, students are guided to discover
the relationship between position and momentum
uncertainty. The document can also be used
as a classroom instruction tool. Answers
are embedded in the document. These can be
deleted in copies of the document provided
to students for in-class work or homework.
Plots
of f(x) as a function
of x for
two Gaussian functions where the solid
line is for the function with σ2 =
0.25 and the dotted line is for the function
with σ2 = 2.5.
|
Citation
