Question

What are the quantum states in a system?

Answer 1

For any given attribute of a quantum system (e.g., position), said `hat X` the variable (i.e., observable) identifying such attribute, a quantum state is a "formula" describing all "possible states" of the observable, that is all possible "sharp values" of the variable, that have been identified through measurements.

So, a quantum state is NEVER an objective physical state, as it ALWAYS depends from the experimenter's information collected from the observation (i.e., measurement) of the system.

Thus, said `hat x_1 … hat x_n` the "basis states" that form the observable's spectrum of a quantum system, a quantum state can be written as a wave function:

`vec psi = c_1 hat x_1 + c_2 hat x_2 + … + c_n hat x_n`

Therefore, in the case of an observed (i.e., measured) quantum system, a quantum state "embodies" the results of a series of experiments conducted on it, where for each possible state of the system (the basis states `hat x_1, … hat x_n`) it gives the probability to occur at a given time from the start of the experiment (the complex coefficients `c_1, … c_n`).

As a definition, the term "quantum state" can be used to refer to both either any of the basis states of a system (`hat x_1, … hat x_n`) or their superposition (`vec psi`).