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===== Visualizing Spherical Harmonics Using a Balloon: Instructor's Guide =====



==== Main Ideas ====
Spherical harmonics are continuous functions on the surface of a sphere.

The $\ell$ and $m$ values tell us how the function oscillates across the surface.

Spherical harmonics are complex valued functions.


==== Students' Task ====
//Estimated Time: 30 min//

Students set up a spherical coordinate system on a balloon, draw a spherical harmonic, and use the balloon as a prop to describe the main features of their spherical harmonic to the class.


==== Prerequisite Knowledge ====
Students should be familiar with the spherical coordinate system, and the equations describing spherical harmonics.

==== Props/Equipment ====

  * [[http://www.amazon.com/gp/product/B0046EAWBI/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=B001F5P0OK&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=03BY3KJNZ88H4P5SNYKC|Punch ball balloons]] (almost spherical), or standard party balloons (less perfect).
  * White board markers




==== Activity: Introduction ====
The instructor points out that visualizing complex numbers on a spherical surface is challenging, and then describes a way to visually represent a field of complex numbers. The visual representation is based on stylized Argand diagrams (examples below).

{{activities:guides:visualize_complex_numbers.jpg?300|}}

Note that the size of each circle represents magnitude, and the direction of the radial spoke represents phase.

Each small group is assigned a spherical harmonic from the list below:
  * $(\ell = 1, m = 1)$
  * $(\ell = 1, m = 0)$
  * $(\ell = 1, m = -1)$
  * $(\ell = 2, m = 1)$
  * $(\ell = 2, m = 0)$
  * $(\ell = 2, m = -1)$


There is room on the balloon to draw 8 complex numbers around the equator $(\theta = π/2)$. Similarly, complex numbers can be drawn around the balloon at $\theta = π/6$, $2π/6$, $4π/6,$ $5π/6$. Example below.

{{activities:guides:balloon_for_spherical_harmonic_bright.jpg?300|}}

Groups work independently on their spherical harmonic.









==== Activity: Student Conversations ====
  * During this time the Instructor and TA ask questions such as
    * "Show me $\phi = 0$ on your sphere"
    * "Show me $\theta = \pi/2$ on your sphere"
    *" Is your function well behaved (no discontinuities) everywhere on the surface?"

  * If students need help with their coordinate system, ask them to draw a dot representing $\theta = 0$ and line to represent all points with $\phi = 0$. 
  * Not all students fully understand the stylized Argand diagrams, but most can still use them appropriately.






==== Activity: Wrap-up ====
After students have made their plots, each group shows the main features of their spherical harmonic. For example, 
  *Are there nodes at the poles?
  *Does the phase wind up 360 degrees as we move around the equator?
  *How is $m = 1$ different from $m = -1$?

Useful points to review are:
  *Unless $m = 0$, the spherical harmonics must have nodes (zeros) at the $\theta = 0$ and $\theta = \pi$ to ensure a continuous function.
  *The index $m$ is equivalent to the number of times the phase gets wound as you move around the equator.

What happens if we add two spherical harmonic functions? For example ($/ell = 1$, $m = 1$) and ($\ell = 1$ $m = -1$). Discuss the constructive and destructive interference of complex numbers.







==== Extensions ====

  * What linear combination of the $\ell = 0$ and $\ell=1$ spherical harmonics would describe a probability density that is big in the northern hemisphere and small in the southern hemisphere? 
  *What linear combination of the $\ell = 0$ and $\ell = 1$ spherical harmonics would describe a probability density that is big in the $x$direction but and small in $-x$ direction?

An alternative activity only asks students to show how the phase (not the magnitude) changes about the equator. Have one group at each table show $e^{i\phi}$ and one group show $e^{-i\phi}$ around the equator. By asking them to then add their functions together, you can use this as a way of introducing the superposition of states and to talk about how physicists' counting of states ($p_1$, $p_0$, $p_{-1}$) differs from chemists' counting of states ($p_x$, $p_y$, $p_z$).