I haven't studied quantum mechanics in any depth at all. The meaning I, as a layman, derive from this statement is: in the formal QM system a particle has no property labelled "position". There is perhaps an emergent property called position, but it is not fundamental and is not always well defined, just like there are no ice-cream atoms. Is this wrong?
Yes, it's wrong. In the QM formalism position is a fundamental property. However, the way physical properties work is very different from classical mechanics (CM). In CM, a property is basically a function that maps physical states to real numbers. So the x-component of momentum, for instance, is a function that takes a state as input and spits out a number as output, and that number is the value of the property for that state. Same state, same number, always. This is what it means for a property to have a well-defined value for every state.
In QM, physical...
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