Here are 2 papers that discuss quantum measurement issues. Note that they both assume, that measurement is an unsolved problem. On reflection, perhaps this is not surprising. Quantum systems, in our case, consist of one electron and a potential; a measurement involves a macroscopic apparatus. An idealized problem we could envision might involve a simple quantum system, like the 1D-HO or inf. sq well, for t<0, t="0">0). Then to correctly represent the measurement in the context of quantum theory, we would have to solve for the energy eigenstates of the measurement apparatus and then look at the evolution of the state function for the system. Since the measurement apparatus involves trillions of electrons and protons, this is, at the very least, a very challenging problem.
The paper by Philip Ball discusses issues and experiments related to measurement. Decoherence is an important concept here, and I would not suggest that is easy to understand or that there are simple ways to model it.
The earlier paper by P.W. Anderson uses the word "emergent" in the title. The concept of emergence is generally associated with extremely challenging and perhaps unsolvable matters that appear in science. It is related, i think, to the issue of "what is knowable", and used in the context of understanding and facing limits to reductionism in science. For example, people might say that the ineffectiveness of quantum theory in predicting the behavior of a mouse, a human (or a many-electron Hubbard model system) is not associated with any problem or incompleteness in our knowledge of the Schrodinger equation, or any question that it, in principle, governs the behavior of every electron in that mouse, but rather that there are "levels of emergence" separating phenomena at the mouse-level of complexity from the one-electron (or several electron) regime, and that these borders cannot be crossed.
As far as I noticed, neither paper mentions "wave-function collapse". Wave-function collapse, as I understand it, is something that got "invented" when people were very confused about how to understand what the Schrodinger equation was telling us. From a historical perspective, the Schrodinger equation was predicting and explaining things in a stunning level of detail. Most of these were related to energy, e.g., the H-atom energy levels, the details of their fine-structure, and their dependencies on magnetic and electric field.
My understanding, and this may not be everyone's view, is that wave-function collapse is something that got concocted in a hurry when people were desperate to "explain" the meaning of the new "quantum theory". Unlike the Pauli exclusion principle, the Heisenberg commutation relations and the normalization condition, which along with the Schrodinger wave equation form the basis of quantum theory, it has drifted along in a nether world in which it is not part of the core of quantum theory, but it has not gone away either. (Perhaps because measurement theory remains an unsolved problem?) Wave-function collapse, as I understand it, postulates a time-dependence which is different from, and exists in addition to, the time dependence which comes from the Schrodinger equation (via separation of variables, using energy as a separation parameter, solving for the energy eigenstates, and obtaining a exp{-iEt/hbar} time dependencies. I am not aware of there being any particular mathematical expression associated with wave-function collapse time dependence, though probably that is just my lack of familiarity with that area, and perhaps a number of possibilites have been proposed?
So when we measure a particle it doesn't instantaneously collapse but it goes through a process of quantum decoherence? In this final state however, are we saying it is now in an eigenstate? A SINGLE state and not a superposition of these eigenstates? If so, wouldn't there still be an associated probability of the particle being located here or there and so did not even actually "collapse" into a single reality as I originally took the perception of "wave function collapse" to mean (that it exhibits PARTICLE behavior and not undulatory behavior)? Man I am confused
ReplyDeleteI am not sure if you are thinking about an eigenstate of the small "quantum" system, or if you are referring at that point to the macroscopic measurement system? In any case, it might be helpful to recall that quantum decoherence is not a completed theory that replaces wave-function collapse. In a sense they are both essentially speculations regarding the nature of the outcome of an unsolved problem. Perhaps in the next 50 years or so, both the formulation and the solution (or partial solution) to the measurement problem will emerge. Hopefully you will live to see and understand that. In the meantime, it probably makes sense that it is confusing.
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