>>11438983Basically: treat particles as waves, look at the energies and shapes that a given potential forces them into. For simple potentials like infinite square well the waves sit in sinosoidal stationary states much as a string fixed at two points vibrating. Because the well has a fixed width and the wave is totally confined within it, you get only certain allowed energies.
Because the SE is linear, more general states for a given potential can be broken down into a sum of stationary states with different time-dependent phase factors.
Fundamentally, the time-independent SE says: Here are the states of fixed Total Energies. Hopefully, these fixed-energy states form a well-behaved basis for vector space of all possible states on that potential (with some boundedness and normalization reqrs).
We can then define operators on these states. If an operator acts on a state and returns that state times a scalar, then the state is an eigenstate of that operator with the scalar its eigenvalue. All these eigenstates in turn form another basis of the vector space for that potential and BC's; I like to think of this as some infinite dimensional rotation of the basis states much as you could rotate your basis vectors in a normal 3D vector space.
>>11439709okayish, but remember that for example a free particle has no such quantization; any quantization in elementary quantum mechanics comes from the potential and is not inherently present in the particle. In QFT you quantize the fields themselves and this becomes more fundamental.
