Question

What's an eigenfunction (QM)

Answer 1

Eigenfunction- each of a set of independent functions that are the solutions to a given differential equation.

Answer 2

Haha thanks :)

@Kainui

Answer 3

Lel, you didn't want the definition? XD

Answer 4

Hey ok, so at its very core, every operator has eigenfunctions with associated eigenvalues, so they all come together.

So for this example, I picked the operator:

\[\hat O = \frac{d^2}{dx^2}\] and now I personally know that \(f = 6 \sin (3x)\) is an eigenfunction of this operator. What's it's eigenvalue?

Well, eigenvalue equations are always of this form with \(\lambda\) as our eigenvalue:

\[\hat O f = \lambda f\]

So if you plug everything in:

\[\frac{d^2}{dx^2}(6 \sin (3x)) = -54 \sin(3x) \]

Now it's tempting to say the eigenvalue is -54 -- but it's NOT! Look at the right side of this equation, \(\lambda \) and \(f\) are distinct!
\[\hat O f = \lambda f\]
So if we use parenthesis that will help us see it more clearly I think:

\[\frac{d^2}{dx^2}(6 \sin (3x)) = -9(6 \sin(3x)) \]

What is the eigenvalue now? It should hopefully be clear to see that it's really \(-9\)! I hope this made sense.

Answer 5

Wow, thats hot, thanks!

Answer 6

Haha yeah, so normalization constants and eigenvalues are two totally different things. I can give you a quick question to test your understanding:

What's the momentum of this wave function:

\[\psi(x) = \frac{1}{2} e^{-i24x /\hbar }\]

:O

Answer 7

\[\hat p \psi(x) = \lambda \psi (x)\] I'm going to start with this

Answer 8

Perfectomundo

Answer 9

You could have written an unhatted p as the eigenvalue, cause that's what it is, but it's just letters haha. \(p = \lambda\)

Answer 10

it's just easy to mix up \(\hat p\) with \(p\) if you're not careful. One is an operator the other is just a number :P

Answer 11

Lol yeah I realized that after, thought you wouldn't notice but you are kainui...ok so the operator is \[\hat p = \frac{ \hbar }{ i } \frac{ d }{ dx }\] so we get \[\hat p \psi (x) = -12 \psi (x)\]

Answer 12

I simplified in my head

Answer 13

I suggest simplifying on paper

Answer 14

:P

Answer 15

Loll

Answer 16

Oh I missed the h bar squared

Answer 17

No, nothing like that, it's almost completely right... Except you made the same mistake I explained in the first post I am teaching you to fix.

Answer 18

nvm - 24 i cant math

Answer 19

Yeah there we go haha. phew

Answer 20

OK I NEED TO WRITE THIS DOWN

Answer 21

Haha yeah I get the idea though

Answer 22

THAT'S WHAT MATTERS

Answer 23

Yeah, this is pretty important. All observables are real eigenvalues. Pretty important to have observables if you wanna do physics imo :P

Answer 24

Haha yes I also want to say the reason the momentum operator even worked is because it's a hermitian operator and observables are represented by hermitian operators!

Answer 25

Trying to use these terms

Answer 26

Hell yeah. Also Hermitian operators only have real eigenvalues, I think that they prove that in a few lines with linear algebra in the second semester of QM at MITOCW, so something fun to look forward to heh

Answer 27

ooh they have lectures for the second course to? Both the courses are just going through griffiths book

Answer 28

In a sense

Answer 29

Yeah, the second semester is quite different cause it's more linear algebra but they don't really like expect you to know any linear algebra which is nice I think, but I am not sure how true to their word they mean this since I was pretty comfortable with linear algebra before watching it.

He does suggest a good linear algebra book at some point too during that second semester, it's like Linear Algebra Done Right or something like that. Anywho yeah pretty fun, linear algebra is really the backbone of QM, calculus is not really that important imo.

Answer 30

Haha well I'm taking linear algebra this sem to where you learn eigenvalues, subspace, etc but it's hard to find a connection since I find linear algebra a bit harder to understand than like calculus.

Answer 31

The theory the math is ok I think XD

Answer 32

I should quit procrastinating and just watch the lectures on mit rofl

Answer 33

Yeah no way around it, linear algebra is kinda hard and weird but after a while you'll get kinda comfortable with it and I think like for me I didn't really get eigenvalues and eigenvectors too much until I started proving things with them in linear algebra for specific applications in physics and then I understood why they mattered.

Plus, diagonalizing matrices is just fun imo.

Answer 34

Haha yeah I haven't found much use for it, till now, since I haven't applied it to any applications or done anything useful with them but I'll get there weee