How is Zig-zag Motion Observable in Quantum Mechanics Given Wave Function Collapse?

Question

I'm puzzled by a concept I read about in a physics text concerning quantum measurement. The text describes the potential to observe a "zig-zag" motion if one could capture images of an object at intervals around Planck time seconds. My confusion arises from the quantum mechanics principle where the wave function of an object collapses to a specific location upon measurement, thus fixing its position at that precise point.

How is it possible to observe such zig-zag motion when theoretically, subsequent measurements should find the object in the same position unless a significant time has elapsed allowing the wave function to re-expand? This seems to contradict the expected behavior where the object, post-collapse, should remain in a defined state until enough time has passed for the wave function to spread out again. Could this zig-zag observation be an error from the book or they mean we would observe the zig zag movement if we wait some time for the spread of the wave function between each measurement.

Answer 1

This seems to contradict the expected behavior where the object, post-collapse, should remain in a defined state until enough time has passed for the wave function to spread out again.

This i an incorrect statement. Assuming that the system is described by Hamiltonian H, and we measured its energy, thus collapsing the system to an eigenstate of H - then the system would remain in this state.

Now, if we talk about a particle in potential field, with Hamiltonian $H=\frac{\mathbf{p}^2}{2m}+V(\mathbf{x})$ (or simply a free particle $H=\frac{\mathbf{p}^2}{2m}$), then the position operator does not commute with the Hamiltonian. That is, if we measure the particle at position $\mathbf{x}$, localizing it in this point (by the wave function collapse), its wave function will start spreading (e.g., as a spherical wave emanating from this point, in case of a free particle), and the next measurement may find the particle elsewhere - with equal probability in any direction. And so on.

Another way to see this is from the point of view of the Heisenberg uncertainty principle: localizing a particle at a point, means that its momentum is uncertain - in magnitude and direction. Thus, the next measurement may find the particle having moved in any direction - hence the zigzag (rather than a smooth trajectory that we expect in classical mechanics, where the momentum predicts where the particle move, and is changing only gradually under influence of a measurement.)

Answer 2

I believe the problem in your reasoning is assuming that the position/wave function of the object evolves in discrete intervals.

You state

subsequent measurements should find the object in the same position unless a significant time has elapsed allowing the wave function to re-expand?

This seems to imply that either a threshold of time elapsed needs to be reached for the wave function to evolve (and significantly, that threshold of time is larger than the Planck Time, so the theory still allows us to speak about changes within that time frame) or a threshold of wave function spread needs to be crossed for the position to be able to measured elsewhere.

However, QM gives us no reason to think so. The solutions of the Schroedinger equation (or any of the alternative QM equations we choose to use) evolve continuously over time. That is to say immediately after we measure an object in a definite position, the wave function will have already started spreading around that position. How much it spreads depends on how much time you wait, but it's not going to be zero no matter how small the timespan.

Given a wave function distributed over space, even if that space is very small, we can derive from the Born rule that there is a probability for the object to be measured in any position within that space. There's no rule that says the distribution needs to be large before the Born rule applies or before we can make a measurement.

Therefore, if we make a measurement every Planck time seconds, on the first measurement we will find the object on a definite position; on the second measurement, the wave function will have spread into a small volume around that position and we can measure the object anywhere within that volume. On the third measurement, the wave function will have spread a little in a volume around the second position and we can measure the object anywhere within that volume. There's no reason to assume that the three measured positions so far would align in a line, so it's likely they'll form an angle. Continuing like this, the repeated angles will form a zig-zag pattern.

Generally speaking we can't actually observe this zig-zag pattern (at least to my knowledge/on that scale), because our measurement equipment does not have the extreme accuracy required to measure such small distances. This is why it might look like the object isn't moving unless a significant amount of time does not pass (assuming an object with no momentum). However, the question hypothesised the possibility of being able to make a measurement every Planck time seconds, so I engaged with this hypothetical scenario where we have measurement equipment of perfect accuracy.

Answer 3

The equations of motion of quantum theory such as the Schrodinger equation don't include wave function collapse. Including collapse would require modifying quantum theory, e.g. - spontaneous collapse theories

https://arxiv.org/abs/2310.14969

Such theories haven't reproduced the predictions of relativistic quantum field theories, i.e.- almost all predictions of quantum theory in the real world:

https://arxiv.org/abs/2205.00568

A measurement is an interaction that produces a record that can be copied of the measured quantity. Such interactions suppress quantum inteference, an effect called decoherence:

https://arxiv.org/abs/quant-ph/0306072

This leads to a set of relative states that are highly peaked in position and momentum on the scales of everyday life

https://arxiv.org/abs/0903.1802

https://arxiv.org/abs/1111.2189

All of the states continue to exist but they evolve independently to a good approximation. This is often called the many worlds interpretation but it is just an implication of quantum theory without collapse. Decoherence absolutely does not and cannot lead to a particle being localised at a single point under any set of circumstances that is physically realistic.

Repeated measurements and continuous measurements can be treated easily in quantum theory without collapse by modelling the measured system, the measurement device, the environment and their interactions with the appropriate Hamiltonian. A repeated measurement just involves turning the coupling between the measurement device on when doing a measurement and off the rest of the time. A continuous measurement can be modelled by keeping the relevant measurement interaction turned on. When the measurement interaction happens at a rate comparable to that of the measured system this will suppress evolution of the measured system: the quantum Zeno effect. If the measurement interaction happens at a rate that is slow compared the the evolution of the measured system the Zeno effect is reduced, see Sections 3 and 4 of:

https://arxiv.org/abs/1604.05973

and Home and Whitaker's paper on the quantum Zeno effect:

https://www.sciencedirect.com/science/article/abs/pii/S0003491697956992