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 properties are more complicated—they’re linear operators, if you want a mathematically exact treatment. But here’s an attempt at an intuitive explanation: There are some special quantum states (called eigenstates) for which physical properties behave pretty much like they do in CM. If the particle is in one of those states, then the property takes the state as input and basically just spits out a number. Whenever the particle is in that state, you get the same number. For those states, the property does have a well-defined value.
But the problem in QM is that those are not the only states there are. There are other states as well. These states are linear combinations of the eigenstates, i.e. they correspond to sums of eigenstates (states in QM are basically just vectors, so you can sum them together). These linear combinations are not themselves eigenstates. When you input them into the property, it spits out multiple numbers, not just one. In fact it spits out all the numbers corresponding to each of the eigenstates that are summed together to form our linear combination state. So if A and B are eigenstates for which the property in question spits out numbers a and b respectively, then for the combined state A + B, the property will spit out both a and b—two numbers, not just one.
So the property isn’t just a simple function from states to numbers; for some states you end up with more than one number. And which of those numbers do you see when you make a measurement? Well, that depends on your interpretation. In collapse theories, for instance, you see one of the numbers chosen at random. In MWI, the world branches and each one of those numbers is seen on a separate branch. So there’s the sense in which properties aren’t well-defined in QM—properties don’t associate a unique number with every physical state. This is all pretty hand-wavey, I realize, but Griffiths is right. If you really want an understanding of what’s going on, then you need to study QM in some depth.
Also, I should say that in MWI there is something to your claim that the position of a particle is emergent and not fundamental, but this is not so much because of the nature of the property. It’s because particles themselves are emergent and non-fundamental in MWI. The universal wavefunction is fundamental.
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 properties are more complicated—they’re linear operators, if you want a mathematically exact treatment. But here’s an attempt at an intuitive explanation: There are some special quantum states (called eigenstates) for which physical properties behave pretty much like they do in CM. If the particle is in one of those states, then the property takes the state as input and basically just spits out a number. Whenever the particle is in that state, you get the same number. For those states, the property does have a well-defined value.
But the problem in QM is that those are not the only states there are. There are other states as well. These states are linear combinations of the eigenstates, i.e. they correspond to sums of eigenstates (states in QM are basically just vectors, so you can sum them together). These linear combinations are not themselves eigenstates. When you input them into the property, it spits out multiple numbers, not just one. In fact it spits out all the numbers corresponding to each of the eigenstates that are summed together to form our linear combination state. So if A and B are eigenstates for which the property in question spits out numbers a and b respectively, then for the combined state A + B, the property will spit out both a and b—two numbers, not just one.
So the property isn’t just a simple function from states to numbers; for some states you end up with more than one number. And which of those numbers do you see when you make a measurement? Well, that depends on your interpretation. In collapse theories, for instance, you see one of the numbers chosen at random. In MWI, the world branches and each one of those numbers is seen on a separate branch. So there’s the sense in which properties aren’t well-defined in QM—properties don’t associate a unique number with every physical state. This is all pretty hand-wavey, I realize, but Griffiths is right. If you really want an understanding of what’s going on, then you need to study QM in some depth.
Also, I should say that in MWI there is something to your claim that the position of a particle is emergent and not fundamental, but this is not so much because of the nature of the property. It’s because particles themselves are emergent and non-fundamental in MWI. The universal wavefunction is fundamental.
Thanks for the detailed explanation! Now I have more fun words to remember without actually understanding :-)
Seriously, thanks for taking the time to explain that.