# Why are eigen-things better/more useful than non-eigen-things?

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?

• 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? – DanielSank Aug 10 '16 at 17:17
• 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. – Peter Diehr Aug 10 '16 at 18:30

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.