# Wave function collapse and measurement rule

• I
• emdezla
In summary, the conversation discusses the principles of superposition and measurement in quantum mechanics. It is explained that while not measuring, a particle is in a superposition of all possible eigenstates, but when measuring, it can only be in one precise state. The concept of wave function collapse is also discussed, with the conclusion that it depends on whether the observable being measured commutes with the Hamiltonian as to whether the particle will remain in that state or evolve away from it.

#### 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!

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.

emdezla
emdezla said:
[..]
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.

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emdezla
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.

emdezla
emdezla said:
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.

emdezla and Mentz114

## 1. What is wave function collapse?

Wave function collapse is a fundamental concept in quantum mechanics. It refers to the phenomenon where a quantum system in a superposition of multiple possible states "collapses" into a single definite state upon measurement or observation.

## 2. What is the measurement rule in quantum mechanics?

The measurement rule, also known as the Born rule, is a mathematical formula that describes the probability of obtaining a certain measurement result for a quantum system in a superposition of states. It states that the probability of a particular measurement outcome is equal to the square of the amplitude of the corresponding state in the superposition.

## 3. How does wave function collapse relate to the measurement rule?

Wave function collapse occurs when a quantum system is measured, causing it to "choose" a single state from its superposition. This chosen state is then used to calculate the probability of obtaining a specific measurement outcome using the measurement rule.

## 4. Is wave function collapse a real physical process?

There is ongoing debate among physicists about the nature of wave function collapse and whether it is a physical process or just a mathematical tool for calculating probabilities. Some theories, such as the Many-Worlds interpretation, suggest that collapse is not a real process, while others, such as the Copenhagen interpretation, view it as a fundamental aspect of quantum mechanics.

## 5. Can wave function collapse be observed or measured?

No, wave function collapse itself cannot be directly observed or measured. It is a theoretical concept used to explain the behavior of quantum systems. However, the effects of wave function collapse can be observed and measured through experiments and observations of quantum systems.