The concept of an eigenstate is key to the mathematical formulation of quantum mechanics. Each particle in a quantum mechanical system is described by a mathematical object called a state vector or state for short. Each observable quantity, such as position, momentum, and spin, is associated with a set of states that are called its eigenstates. If a particle is in an eigenstate of an observable, it has a definite value for that quantity (i.e. there is no quantum uncertainty in that quantity). Each possible value (called an eigenvalue) for the observable quantity corresponds to at least one eigenstate
Conceptual Utility of Eigenstates
A set of eigenstates for an observable quantity forms a complete basis for the system. This means that any possible state of the particle can be written as a superposition of eigenstates for each observable. This superposition is key when a measurement of the system is made. A measurement connects the quantum system to a classical system which must obey the deteministic laws of classical mechanics. Hence, the measurement must find a definite value of the observable quantity, which only occurs when the particle is in an eigenstate of that observable. So measurements end the superposition and force the system into an eigenstate corresponding to the quantity measured.
It is possible that two different observable quantities can have the same set of eigenstates. In this case the two observables are said to commute. (Yes, I know that's weird terminology, bear with me) Physically, this means that the two quantities can have definite values at the same time, whereas two observables that do not commute obey some kind of uncertainty relation and cannot both have simultaneously definite values.
Eigenstates come in both discrete and continuous sets. The set of position eigenstates and the set of momentum eigenstates are both continuous, since there is an eigenstate corresponding to each of the possible values of these quantities. Discrete sets of eigenstates are often infinite, but can also be finite in size, particularly for observables related to angular momentum. Discrete sets of eigenstates are where many of the interesting features of quantum mechanics occur, as values of the observable quantity between the eigenvalues are simply not possible, in contrast to the continuous observables of classical mechanics.
In Dirac's powerful formalism of quantum mechanics, states are represented as 'kets' (|ψ>), and observable quantities correspond to mathematical operators (notated in boldface) that act upon the states. The eigenstates, |ψ>, of an arbitrary operator Q are determined by the equation:
Q|ψ> = q|ψ>
The q is the eigenvalue. Note that the operator leaves the actual state unchanged, rather than transforming it into a different state. This is the defining property of being an eigenstate.
Operators do not obey the same sort of algebra as numbers, but rather they obey the same algebra as matrices. The knowledgeable reader will notice that this implies that operators do not obey the commutative law and so for operators A and B, AB may not be the same as BA. Some operators, however, do obey AB = BA; these operators commute. It can be proven that two operators that commute must have the same set of eigenstates; this is a general theorem in linear algebra.
The most important set of eigenstates is that associated with the Hamiltonian operator, which measures total energy. The time-independent Schrodinger equation is just the eigenstate equation for the Hamiltonian:
H|ψ> = E|ψ>
Since energy is conserved over time, we would expect a time-independent state to have a definite, constant amount of energy. So any such 'stationary state' must be an eigenstate of the Hamiltonian, whatever it might be.
However, there are often a number of states with the same energy eigenvalue (referred to as degenerate), so the equation as written may not uniquely describe every state. It is, however, usually possible to describe all states using other operators that commute with the Hamiltonian. In the case of the hydrogen atom, for example, the electron states are described by eigenstates of the Hamiltonian, the total angular momentum operator, and the z-axis angular momentum operator.
Eigenstates are the building blocks used by physicists to build full quantum mechanical descriptions of physical systems. They are, by definition, those states that we can have definite knowledge about, making them the key we can use to unlock the quantum puzzle of small-scale matter.
This writeup is copyright 2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .