A quantum state is a mathematical description used to calculate probabilities for measurement outcomes. It is not simply a list of properties that an object must possess with definite values before every possible measurement.
For a particle moving in one dimension, the state may be represented by a wave function. At a given time, the squared magnitude of the wave function gives a probability density for finding the particle at a position. To obtain the probability of finding it within an interval, we add this density across the interval by integration.
Probability is not the whole story
Two quantum states can give the same position probabilities and still behave differently in a later experiment. The reason is that a quantum amplitude has phase as well as magnitude. Relative phases affect interference and therefore affect future measurement probabilities.
In a two-path experiment, the amplitudes associated with the alternatives combine before we calculate the probability. If information that distinguishes the paths becomes available through interaction with the environment or a detector, the interference can be reduced or lost.
This is why a quantum superposition cannot generally be replaced by an ordinary statement that the system was secretly following one alternative and we simply did not know which.
Measurement must be specified
The same state can predict a definite result for one measurement and a range of possible results for another. A state with a sharply defined momentum, for example, is not also a state with a sharply defined position.
The uncertainty relation concerns the spreads of outcome distributions for an ensemble of identically prepared systems. It is not merely a statement about poor instruments or an observer being careless.
Here Δx and Δp are the standard deviations of position and momentum. The relation sets a constraint on the quantum state itself.
What should we imagine?
It is useful to distinguish the operational use of a state from its interpretation. Operationally, we prepare a system, allow it to evolve, choose a measurement, and compare the observed statistics with the theory.
Whether the wave function is a physical object, a description of information, or part of a deeper account is a question on which interpretations differ. Those differences should not obscure the experimentally testable predictions that make quantum theory so successful.
A good starting habit is to ask three questions: How was the system prepared? What is being measured? What probabilities does the theory assign? These questions keep the explanation connected to an actual experiment.
