A linear transformation T that stays in the same vector space could have some points that stay relatively fixed. A fixed vector v would have T(v)=v. Many transformations only have the 0 vector as their only fixed point.

But, what happens in a lot is that there are certain vectors that get scaled by the transformation. That is, there exists a scalar λ such that T(v)= λv. Since all scalars are unaffected by T, this same λ works for all scalings of v. If v has this property, it is an eigenvector for T, and λ is an eigenvalue.

A transformation can have multiple independent eigenvectors.

Another cool property is that if λ is less than 1, then T repels vectors from v, and if λ is greater than 1, then it attracts T.

The universe is a song, singing itself.

No, really. The solutions of the Schrödinger Equation are harmonics, just like musical notes.

Quantum state of an electron orbiting a hydrogen atom where n=6, l=4, and m=1 (spin doesn’t matter):

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This is an equal superposition of the |3,2,1> and |3,1,-1> eigenstates:

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This is an equal superposition of the |3,2,2> and |3,1,-1> eigenstates:

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This is an equal superposition of the |4,3,3> and |4,1,0> eigenstates:

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What is an eigenstate? It’s a convenient state to use as a basis. We get to decide which quantum states are “pure” and which “mixed”. There’s an easy way and a hard way; the easy way is to use eigenstates as the pure states.

More mathematically: the Schrödinger equation tells us what’s going on in an atom. The answers to the Schrödinger equation are complex and hard to compare. But phrasing the answers as combinations of eigenfunctions makes them comparable. One atom is 30% state A, 15% state B, 22% state C … and another atom is different percentages of the same states.

Just like vectors in 3-D space, where you can orient the axes differently – you can pick different directions for x, y, and z to point in. But now the vectors are abstract, representing states. Still addable so still vectors. Convex or linear combinations of those “pure” states describe the “mixed” states.


SOURCE: Atom in a Box