Lesson Plan - Angular Solution of Hydrogen Atom

Audience

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.