Question:
How do I introduce the radial probability function and
related graphs to 11th grade students?
Replies:
I assume this question is related to the solution of the Schroedinger
equation for the hydrogen-like wave functions. Without some background in
calculus, the answer requires a lot of "faith" on the part of the students
because the math is a bit "messy" in its details. First, the student must
accept that the Schroedinger equation describes the behavior of the atom.
Second, the solutions to the Schroedinger equation depend upon a radial
variable and two angular variables. This is "convenient" because this make
the problem have a center of symmetry, and various "small" interactions
involving the coupling of the spin of the nucleus and the electron are
neglected, the difficulties associated with taking relativity into account
are also ignored. Third, the solution of Schroedinger's equation involves
three variables: one radial and two angular variables. Fourth, this
"separation of variables" allows the equations to be expressed as a
function of the proton-electron distance, and a recurrent set of solutions
that appear in many physics problems called "spheric harmonics". Fifth, it
is the "square" of the radial wave function that defines the radial
probability of the position of the electron with respect to the proton.
Having accepted all of the above as true it is possible to draw graphs of
the square of the radial part of the solution. These are the 'orbitals'
one finds in elementary texts.
The five steps are way beyond what can be done in detail here. A good
resource that "walks the student" through the various mathematical steps is
the "classic" text: "Introduction to Quantum Mechanics" by Linus Pauling and
E. Bright Wilson.
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