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 periodic 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.
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.
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 periodic 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.
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.