Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun One of a set of quantummechanical descriptions of the possible state of a physical system, using the mathematics of eigenvectors and generally existing in a state of superposition until the moment of observation.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun physics A
dynamic quantum mechanical state whosewave function is aneigenvector that corresponds to aphysical quantity
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
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Examples

In the Copenhagen formalism, one could calculate the probabilities of various outcomes assuming a measurement was taken at a particular time; during the interval between measurements, those “probability amplitudes” changed in definite, calculatable ways, while at the instant of measurement, another process took over and caused the system being measured to settle into a single eigenstate.

For ten points, calculate in the Heisenberg picture the time evolution of a harmonic oscillator state produced by acting on the lowestenergy eigenstate with a spatial translation operator.

I suspect that DM consists largely of neutralinos, which is a mixed eigenstate of the supersymmetric pairs of the photon, Higgs and Z particle.
Dark Matter Detector Heading to the ISS This Summer  Universe Today 2010

Provided we maintain the eigenvalueeigenstate link, the quantum description by means of that state function will yield neither conclusion, and hence the quantum description is incomplete.
The EinsteinPodolskyRosen Argument in Quantum Theory Fine, Arthur 2009

In this formulation of the argument it is clear that localityseparability conflicts with the eigenvalueeigenstate link, which holds that a quantity of a system has an eigenvalue if and only if the state of the system is an eigenstate of that quantity with that eigenvalue (or a mixture of such eigenstates).
The EinsteinPodolskyRosen Argument in Quantum Theory Fine, Arthur 2009

Probably the likeliest candidate is the neutralino, which is a mixed eigenstate or condensate of the supersymmetric pairs of the photon, higgs and Z_0 particle.
New CMB Measurements Support Standard Model  Universe Today 2009

Of course it may also be possible to break the EPR argument for the dilemma plausibly by questioning some of its other assumptions (e.g., separability, the reduction postulate, or the eigenvalueeigenstate link).
The EinsteinPodolskyRosen Argument in Quantum Theory Fine, Arthur 2009

B.when measured, unless the quantum state is an eigenstate of the measured observable A, the system does not possess any categorical property corresponding to A's having a specific value in the set B. Putnam seems to assume that a realist interpretation of (*) should consist in assigning to A some unknown value within B. for which quantum mechanics yields a nontrivial probability.
Puppet X: 1 2009

That eigenstate applies to the reality there and that eigenstate enables us to predict a determinate position for Niels 'system with probability one.
The EinsteinPodolskyRosen Argument in Quantum Theory Fine, Arthur 2009

It certifies that the predicted position value, corresponding to the position eigenstate, is an element of the reality that pertains to Niels 'system.
The EinsteinPodolskyRosen Argument in Quantum Theory Fine, Arthur 2009
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