Quantum states after real world measurements

Question

Regarding measurements of an observable in a quantum system. My understanding, from the postulates of quantum mechanics, is that when we measure an observable quantity, the state of the system collapses to an eigenfunction of the linear Hermitian operator which corresponds to the observable: $$\hat{A}|\psi \rangle = y|y \rangle$$ where $y$ is the eigenvalue and $|y \rangle$ is the eigenstate. Then if we project onto the basis of the observable we get the dirac delta function. Let's consider the position operator for example, then:

$$\langle x| \hat{A}|\psi \rangle = y \langle x| y \rangle = y \delta(x-y).$$

From what I understand, in real world measurements the state of the system is not exactly a dirac delta function but rather some wave packet. What is the nature of this wave packet and what determines the shape and corresponding function thereof? Why can't the function be a dirac delta function in real world measurements?

Thanks.

Answer 1

A Dirac delta function has a vanishing width. To "collapse" the wavefunction to a delta function, one's measuring apparatus would need to have infinite precision, i.e. zero uncertainty. Since no measuring apparatus is perfect, no measurement can force the wavefunction to have zero uncertainty, i.e. zero width. Therefore, measurement will collapse the wavefunction to a width related in some way to the uncertainty of the measuring apparatus.

The shape of the wavefunction following measurement depends on the nature of the measurement act. This would be impossible to model without knowledge about the measuring apparatus.