In quantum mechanics, a quantum state refers to the complete description of a physical system. It encompasses all the relevant properties and characteristics of the system that can be measured or observed. Mathematically, a quantum state is represented by a vector in a complex vector space called a Hilbert space.
On the other hand, an eigenstate, also known as an eigenvector or eigenfunction, is a specific type of quantum state that possesses a distinct property. When a physical observable (such as position, momentum, or energy) is measured on a system described by an eigenstate, the result of the measurement is always a specific value corresponding to that eigenstate. In other words, eigenstates are associated with eigenvalues, which represent the possible outcomes of a measurement.
To clarify further, let's consider the example of an electron in an atom. The quantum state of the electron describes its position, momentum, and other properties. However, when we measure the energy of the electron, we find that it can only have certain discrete values, represented by eigenstates. Each eigenstate corresponds to a particular energy value, and the probabilities of measuring different energy values are given by the coefficients of the quantum state when expressed in terms of the eigenstates.
In summary, a quantum state is a general term encompassing the complete description of a physical system, whereas an eigenstate is a specific type of quantum state associated with a measurable property of the system, yielding definite outcomes when measured.