The angular solution of hydrogen activitiy is designed for use in
an undergraduate level course in physics, chemistry, or mathematics.
In an upper level undergraduate mathematics course,
this activity could be used to illustrate eigenvalue problems
in applied mathematics.
In a lower level undergraduate physics course, this activity
could be used to introduce Schrodinger's equation and
particle wave duality.
In an upper level undergraduate course in modern physics,
quantum mechanics, or chemistry, this activity could be used to introduce
numerical solutions to Schrodinger's equation. In this case,
it might be appropriate to require students to build their own
model rather than using the one supplied.
Azimuthal Term
As with the radial and polar terms, the question here is symmetry
and boundary conditions. For a wave solution to make physical
sense it must be periodic over 2 pi.
Students should find that integer values of the coefficent
in the sine function leads to a function that is periodic
over a range of 2 pi. They should check both the sine
and cosince functions, as the two can look somewhat different,
particularly with regards to coefficients in the sine and
cosine functions that are equivalent to half-integers.
Some half-integer coefficients will look fine over a range
of 0 to 2 pi, but will not look periodic if the students look
at them over a range from -pi to pi.
Polar Term
If students numerically solve Legendre's equation, they should
find that solutions that meet the boundary condition of being
finite at x = -1 and x = 1, as well as having a function which is
either odd or even occur when lambda = 0, 2, 6, 12, 20, ...,
or when lambda = l (l+1) for l = 0,1,2,3,4....
In addition, students should see that as m increases, the lower
eigenvalues of lambda no longer have meaningful solutions,
and only values of l(l+1) for l>=|m| will work.
Exercises
Students might visualize these in a variety of ways.
The suggested solution is to multiply the spherical harmonic
equation by exp(-r) and visualize it using a 3-D density plotting
tool. This most resembles the effect of an electron cloud.
However, students could also simply plot these as surface plots,
or as 3-D surfaces where r is a function of theta and phi.
l=0, m=0 term
The l=0, m=0 term should appear spherically symmetric.
l=1, m=0?
The harmonic appears as two lobes along the z axis.
l=1, m=1 (real, imaginary, and magnitude (Y*Y)?)
The real harmonic appears as two lobes along the x axis.
The imaginary harmonic appears as two lobes along the y axis.
The magnitude of the harmonic appears as a toroid in the
x-y plane, centered along the z axis.
Higher order terms?
Higher order terms will increase in complexity and degrees
of summetry.