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Eigenvalues come from linear algebra. I think a difficult think in general with understanding them is often the failure of most middle/high school math teachers to teach matrix operations at all. (I’m guessing because matrix multiplication never shows up on SAT/ACT). Here’s a good explanation for the math on finding eigenvalues and eigenvectors.

But basically eigenvalues are going to be associated with certain matrixes/vectors. You take a “Hamiltonian” of a system, which is a way of describing possible energy values in the system, and it’ll give you a set of possible answers - pairs of eigenvalues and eigenvectors that describe the system.

In effect - you get things like the quantum numbers. That the 1st energy level has 1 subshell can hold 2 electrons, both with opposing spins. That the 2nd energy level has a 2s subshell that holds two, that 2p holds six. You get your n (1st energy level, 2nd so on as you go down periods of the periodicity table), l (subshell - don’t get a SPeeDy F), m (which breaks down where in the subshell they are) and the need for opposing spins.

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someacnt@sh.itjust.works ⁨1⁩ ⁨week⁩ ago
Thank you for in-depth explanation! Though I already know the eigenvalues and eigenvectors, as a math major. What I am curious of is: why can’t we only observe e.g. energy values? I heard that one can only observe commutative operators or something, but honestly why is quite unclear.

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- andros_rex@lemmy.world ⁨1⁩ ⁨week⁩ ago
  I’ll try to dig out Griffith for a better explanation but has to do with the fact that when you do a partial derivative you kinda lose information I guess?
  
  (Idk, this is heady trying to make math into reality shit and I got a “c” in the class (for reasons partially related to other things) - also, there might be a way to do latex in markdown but I’m a bit too stoned to figure out)
  
  So go back how often we do implicit differential because it’s just an opportunity to look at how sexy the chain rule is. d(xy)/dx = xy’+x’y god fucking dammit that gorgeous
  
  But okay. Think about position and velocity. Velocity is the derivative of position right (and also connected to energy - KE = 1/2mv^2 and E = mc^2 lol)
  
  But since velocity is a derivative of position, it loses information. d(mx+b)/dx turns into m, no way to ever get b back with an initial value condition.
  
  Then - omigod, when you take a partial - you have to ignore dependence. curlyd(xy+by)/curlydx turns into y and then things is really fucked if there was any dependence on y.
  
  There are some operators that are just exclusionary. Once you chose to look for one, you’ve discounted the chance of finding the other.
  
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