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Finding the Square of the Norm in Polar Form

The Prompt

“Compute the square of the norm of $z$, where,

$$ z=a e^{i\beta}\, \text{.”} $$


This SWBQ designed to have students think about how to take the complex conjugate of a complex number in polar form and discover what the result is after taking the square of the norm of a complex number.
This SWBQ can be used as an extension from (or prequel to) any activity that features complex numbers, especially those in “polar” form. In particular, lectures featuring relative phases in quantum states benefit greatly from this prompt.

Wrap Up

Students should see that the exponential factor disappears. Some students may forget to conjugate and instead just square the entire quantity. Reminding students that the square of the norm, the “magnitude”, must be a real number may alleviate some confusion.
Be sure to indicate to students, if they have not already discovered so, that the “polar” form of a complex number allows the magnitude to be calculated by inspection, and is therefore the simplest form for this operation. This SWBQ can be exchanged with, or followed/preceeded by, Taking the Square of the Norm, which contains a complex number expressed in rectangular form instead.

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