Undergrad Wave function collapse and measurement rule

[emdezla]

Hi everyone,

I'm kind of new in the QM world and I'm having difficulties understanding the superposition and the measurement principles together with the have function collapse. This is how I understand these principles:

Superposition: While not measuring, the particle is in a superpsotion of all possible eigenstates,
Measurement: While measuring, the particle can only be in one precise state. (in other words, the wave function associated to the particle collapses)

And here's my question: when we stop doing the measurement, will the wave function be "rebuilt" so that the particle will be in a superposition of states or, in contrast, will the wave function continue collapsed until the end of the days?

Thank you very much for your anwers!

 

[Gene Naden]

Good question! To me, the measurement is an event rather than a process. So the wave function cannot be rebuilt after the measurement stops. The wave function may change as other things come along and influence it, though.

Prior to measurement, it is not really a superposition of all possible eigenstates but it may be a superposition of a combination of states.

 

[Mentz114]

[..]
Superposition: While not measuring, the particle is in a superpsotion of all possible eigenstates,
Measurement: While measuring, the particle can only be in one precise state. (in other words, the wave function associated to the particle collapses)

From those assumptions there appears to be no unitary evolution that can give single outcomes with frequencies corresponding to the eigenvalues of the operator.
Luckily the first assumption is very rarely true and often impossible so the problem is more in the mind that in actuality.

And here's my question: when we stop doing the measurement, will the wave function be "rebuilt" so that the particle will be in a superposition of states or, in contrast, will the wave function continue collapsed until the end of the days?

Thank you very much for your anwers!

With some operators (POVMs) the state after measurement is the eigenstate which was registered.

 

[jambaugh]

Keep in mind that by "superposition" we mean a given state vector can be resolved in a specific basis as a linear combination of basis state vectors. It is not that some states are and some states are not "in superposition". This qualifier rather relates a given state with a particular choice of basis states. For example, a vertically polarized photon is "in a single state" when you are using a Vertical vs Horizontal polarization basis but it is also in a superposition of left circular and right circular polarization states.

The "wave function collapse" is simply the fact that by measuring the system with respect to some basis, you are resolving it as one of those basis states when previously it might have been in a superposition of this basis. You likewise might have know what the earlier state was because you had earlier measured it to be in that state from among a different set of measurable states. For example you observe a photon to be L-circularly polarized then measure its V vs H polarization.

 

[Nugatory]

And here's my question: when we stop doing the measurement, will the wave function be "rebuilt" so that the particle will be in a superposition of states or, in contrast, will the wave function continue collapsed until the end of the days?

It depends on whether whatever you're measuring commutes with the Hamiltonian. If it does, then once the system is an eigenstate of that observable, it will stay there. If it does not, then it will evolve away from that state.