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An eigenvector is a vector that doesn't change direction when a transformation is applied. So in the case of, say, an energy (Hamiltonian) eigenstate, it's a state that doesn't change 'direction' (not exactly sure what to map that to). But why is that more special/important/useful than any of the other states?

Is it because you can potentially 'construct' any of the other states with the eigenstates, so that's all you need to worry about?

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    $\begingroup$ It's just that solving a linear algebra equation is really easy once you write the inputs in terms of eigenvectors. Is that what you're asking? $\endgroup$ – DanielSank Aug 10 '16 at 17:17
  • $\begingroup$ We were taught, in marine engineering, that the eigenfrequencies of the hull limit the speed of the vessel. That is, the maximum vibrational speed of the engine must be less than the lowest eigenfrequency of the hull. This sets the flank speed; higher speeds will cause the hull to resonate, leading to catastrophic failure. Only in StarTrek can this fate be ignored! The eigenvectors are the directions of maximum resonance. $\endgroup$ – Peter Diehr Aug 10 '16 at 18:30
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You can write down quantum states as $| \psi \rangle = \sum_k c_k | \phi_k \rangle$ for any basis $| \phi_k \rangle$. However, if you choose basis which is not an eigenbasis of Hamiltonian the coefficients will change with time and it would be extremely difficult to keep track of them. Therefore such expansion is not very useful. To sum up: every basis is fundamentally as good as any other. We usually use particular bases, such as eigenbasis of Hamiltonian because then it's easier to solve real physical problems.

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    $\begingroup$ It's important to understand that the eigen-problem is not limited to quantum things. It also appears in coupled oscillation problems and general rotations to name two other subject which come up in a undergraduate education as well as in other subject you are likely to meet only if you dig in so some subject very deeply. And in all the places it appears identifying the eigenvectors (for a generalized understanding of "vector") simplifies the solution of the general problem. $\endgroup$ – dmckee Aug 10 '16 at 17:55
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    $\begingroup$ I was taught that there are three important rules in linear algebra: First, never choose a basis. Second, never even think about choosing a basis. Third, when you choose a basis, choose it wisely. $\endgroup$ – WillO Aug 10 '16 at 18:22

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