Question

@empty QM

Answer 1

why must it go to 0

Answer 2

pls dont say to normalize

Answer 3

\[\Psi ^* \frac{ \partial \Psi }{ \partial x } =0 ?\]

Answer 4

Ah ok, so you remember the divergence test from calculus 2?

Answer 5

It's not really directly important here, I'm just wondering if you remember the reasoning for why it works.

Answer 6

Yeah I think so, diverges if limit does not go to 0 as it approaches infinity or something

Answer 7

Yeah, like if the "final" term of your infinite series isn't 0, then you'll be stuck at infinity adding some non zero number, which makes it diverge

Answer 8

Right

Answer 9

Similarly, we can consider either the \(\Psi\) or \(\Psi'\), we know that if \(\Psi \to 0\) at x=0, that's the same as the divergence test. But being zero at infinity really isn't enough.

What you really need is for it to be flat there as well. So we also require that the derivative be zero at infinity as well too, otherwise it's just flat out not a normalizeable wave function.

Answer 10

Yeah, cause like remember, the Harmonic series passes the divergence test but doesn't converge. That's why we need the extra condition of the derivative being zero too.

Answer 11

Actually I might be thinking of something else, I don't know if the restrictions are quite so intense on wave functions as I've said. It's really weird cause like momentum eigenstates for example aren't even normalizeable so it's almost like sorta mathematical nonsense junk.

Actually I found something you might like, watch for about a minute or so where he says it's completely nonsense https://youtu.be/TWpyhsPAK14?list=PLUl4u3cNGP61-9PEhRognw5vryrSEVLPr&t=4134

Answer 12

Ah gotcha, ok I was thinking it had to be 1, so to be clear this is the result for \[\frac{ d }{ dt } \int\limits_{- \infty}^{\infty} |\Psi(x,t)|^2dx = 0\] so if it were anything else than it's non normalizable right as it would not cause the flat line? (I guess I should just think of this as the divergent test)
How does this effect the probability of a particle though, or does it at all?

Answer 13

Yeahhhh lol

Answer 14

Yeah cause the total probability is a constant, 1. This doesn't change over time, which is a very sane assumption. If over time the probability of finding your particle at some point in space became less than (or more than?!) 100% then that would mean conservation of matter is violated and stuff just started disappearing from reality. There is no change, which is the exact content of this derivative being zero, not an exaggeration.

Answer 15

Remember, this integral represents the probability of finding the particle if you look everywhere it can possibly exist. You have to find your car keys if you look everywhere in existence (assuming your keys can't melt in the center of the earth or something weird like that, they're magic keys in the same sense that this is an ideal model like the ideal gas law is a model).

Answer 16

Yup, this is insane I think, so if I were to find the velocity of something, essentially I'd be integrating over all of space as it can be anywhere...err I'm not quite there yet but this is crazy talk

Answer 17

Nah, that's just cause you have a misconception about what a wave function represents.

Answer 18

Yeah I think so, it's hard for me to grasp the wave function being spread out in space, where classically a particle is localized haha.

Answer 19

Let's say your entire space of reality is really tiny... Just three possible points on a line. You have 25% probability of being at point 0, 50% probability of being at point 1, and 25% probability of being at point 2.

Now what does your wave function look like?