849: Complex Conjugate
Title text: 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.
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 wave function is a mathematical description of a quantum system which uses complex values - numbers that have both a real part and an 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 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 has 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 time-dependent Schrödinger equation , a differential equation that the wavefunction Ψ, which determines the possible positions of a quantum particle over time, always satisfies. The derivative should be written with ∂Ψ on the top; the omission of the Ψ may be a mistake.
- [Cueball is standing at a whiteboard covered in equations, one of which is the time-dependent Schrodinger equation.]
- Cueball: Okay, anyone who's feeling like they can't handle the physics here should probably just leave now.
- Cueball: Because I'm multiplying the wavefunction by its complex conjugate.
- Cueball: That's right.
- [Dramatic zoom. It appears Cueball is writing on the side of the panel.]
- Cueball: Shit just got real.
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