system
Given the following post, respond as if you're commenting on Reddit:
human
What does it mean if a wave function equals 1, independent of position (x)?

I have been looking at wave functions on MIT's YouTube channel (I'm not studying a course, or anything) and there's one thing that seems so weird. If you take the rootnorm of a wave function e^ikx you get the probability which is in this case 1. I've gotten that through my own calculations and they confirm it in the MIT lecture. However, my interpretation is that, whatever position (x-value) you enter you get 1, and therefore all the positions in the universe have probability of finding the particle with 100% success rate. It doesn't matter where you look, because the probability is suddenly independent of the position, and even stranger, is always 1. But that doesn't make sense!! In the lecture they just say that there's an even probability everywhere and therefore no position has a favor, but they're not mentioning that the probability is 1 everywhere and that the particle should therefore certainly be everywhere (for real, not like the uncertainty principle). 

I'm guessing it's safe to assume that the professor in the lecture is right and my amateur interpretation is incorrect, but could anyone explain how I have misunderstood this 100% probability of finding the particle everywhere?
gpt
You are correct. The problem is that the distribution is not normalized. If you place this particle in an imaginary box of volume V, the normalized wavefunction is:

e^(ikx)/sqrt(V).

Now the probability density is 1/V. This is a properly normalized uniform distribution. It means that you're equally likely to find the particle at any position inside the box.