Editing 849: Complex Conjugate

{{comic
| number    = 849
| date      = January 19th, 2011
| title     = Complex Conjugate
| image     = complex_conjugate.png
| titletext = Fun fact: if you say this every time a professor does something to a complex-number equation that drops the imaginary part, they'll eventually move the class to another room and tell everyone else except you.
}}

==Explanation==
This was the second comic with one of [[Randall|Randall's]] [[:Category:Fun fact|fun facts]].

This comic is a joke on the phrase "Shit just got real", which means "something has suddenly increased in difficulty and become genuinely challenging or dangerous".

[[Cueball]] is standing in front of a board delivering a lesson, and is about to multiply a wavefunction by its complex conjugate. A {{w|wave function}} is a mathematical description of a quantum system which uses {{w|complex number|complex}} values - numbers that have both a {{w|real number|real}} part and an {{w|imaginary number|imaginary}} part. Multiplying a wavefunction by its complex conjugate is a common thing to do, as it yields the probability density of where a particle is likely to be found, which is a real-valued function.

Complex numbers can be written in the form a + bi, where a is the real part and bi is the imaginary part. i is the imaginary unit, defined so that i²=-1. The {{w|complex conjugate}} of a complex number simply reverses the sign on the imaginary part - so for the number above, the complex conjugate is a - bi.

Multiplying the complex number by its own complex conjugate therefore yields (a + bi)(a - bi). If you multiply out the brackets, you get a² + abi - abi - b²i². The abi cancel each other out, and i² can be replaced by -1. Thus, the result is a² + b², a real number, so "shit just got real" as Cueball promised.

The title text notes that you can make this joke in class every time a calculation is performed that drops the imaginary part from a complex number, but warns that it would be so annoying that the professor will eventually find a way to have the class without you in it. Because nonreal numbers are often considered to have no physical significance, turning them into real numbers to produce a final answer is so common that this joke would quickly become tedious.

The wave function shown on Cueball's board is the {{w|Schrödinger_equation|time-dependent Schrödinger equation}} <math>i\hbar\dfrac{\partial}{\partial t} = \hat{H}\Psi</math>, a {{w|Differential equation|differential equation}} that the {{w|Wavefunction|wavefunction}} Ψ, which determines the possible positions of a quantum particle over time, always satisfies. The {{w|Derivative#Partial_derivatives|derivative}} should be written with ∂Ψ on the top; the omission of the Ψ may be a mistake.

==Transcript==
:[Cueball, holding a marker pen down in one hand, is standing at a whiteboard with two equations, one of which is the time-dependent Schrodinger equation and the other shows what the wavefunction equals, but from there it becomes unreadable. Below is a graph with a bell-shaped curve. There are also other unreadable markings on the board below the second equation and next to the graph.]
:Cueball: Okay, anyone who's feeling like they can't handle the physics here should probably just leave now.

:[In a frame-less panel Cueball is seen writing on the whiteboard. This is seen from the side of the board, so it is just a thin line with a shelf at the bottom for putting the pen on.]
:Cueball: Because I'm multiplying the wavefunction by its complex conjugate.
:Cueball: That's right.

:[Dramatic zoom in on Cueball appears to be writing the final part of his next equation.]
:Cueball: Shit just got ''real''.

{{comic discussion}}

[[Category:Fun fact]]
[[Category:Comics featuring Cueball]]
[[Category:Math]]
[[Category:Physics]]
[[Category:Puns]]