There is an implication appearing in a lot of works on the measurement problem in QM:
(1) |S> |M0> = ( Σj αj |sj> ) |M0> → Σj αj |sj> |Mj>
where |S> = Σj αj |sj> is the state of the quantum system before the interaction with the apparatus, { |sj> } is the set of eigenstates of the measured operator, |M0> is the initial state of the macroscopic apparatus and |Mj> is the apparatus state which records the |sj> microscopic state.
The questions are:
1) What is |M0>? A macroscopic state of the apparatus, or a microscopic state of a microscopic part of the apparatus?
2) What are |Mj>? Macroscopic states of the apparatus, or microscopic state of the same microscopic part of the apparatus?
I opened this discussion because some people argue ad absurdum that |Mj> are macroscopic states of the apparatus, although the QM formalism does not support such a thing - Schrodinger's cat cannot be in a superposition of alive and dead.
|M0 > is the state of the system the quantum system of interest is coupled to. If the former system can be assumed to be in one state, while the quantum system evolves, the combined system will be found in the superposition shown.
If the backreaction of the quantum system of interest on the former system can't be neglected, then the former system, too, is found in a superposition of its states and the combined system is, then, found in the second superposition shown.
Whether the |Mj > are the states of a ``macroscopic'' system or not isn't relevant-these are, just, words. What matters is, whether the interaction between the two systems can change the state of one of them, or not.
Dear Stam,
For being specific, let's take as an example the ion chamber. |M0> cannot be the state of the gas in the chamber. The gas is classical, no quantum state describes it. On the other hand, the studied quantum system |S> interacts with the internal state of one or more atoms/molecules in the gas, by ionizing them. This internal state yes has a quantum description. Don't you think so?
"Whether the |Mj > are the states of a ``macroscopic'' system or not isn't relevant-these are, just, words. "
Ah, my Stam, you have a way of declaring irrelevant exactly what's the most important in my question. Whether |Mj> are the microscopic or macroscopic states is the core of a debate.
Let me explain. It seems to me that the interaction between |S> and the gas has two ways of evolving. One way is that |S> interacts with one or more molecules in the unperturbed state |M0>, and evolves into the superposition Σj αj|sj> |Mj>. However, at this step something happens, some decision. Either the interaction continues, i.e. one of the |sj> |Mj> is chosen and instead of |Mj> a whole avalanche grows - which is already a current, a macroscopic phenomenon. In this case the detector clicks. Or, the process is stopped, i.e. no further ionizations occur, and the detector remains silent.
But, you see, there are people who insist that |M0> and |Mj> are macroscopic states. We know that a macroscopic object doesn't have a state of superposition, i.e. ( |M1> + |M2> ) with |M1> and |M2> being macroscopic states (see Schrodinger's cat). But those people claim that when the microscopic object is involved, the superposition Σj αj|sj> |Mj> does exist, with |Mj> being macroscopic states. Please look at this justification:
" . . . this entangled "Schrodinger's cat state" between a superposed quantum object S and a detector D . . . is merely a superposition of correlations BETWEEN S and D, . . . . This is not a "macro superposition." It is a superposition of correlations between a micro object and a macro object."
Do you understand what he means by "superposition of correlations"? It seems to me a very high philosophy on spiritual level, having nothing to do with the quantum formalism. In what consists such a correlation from the said superposition? Not in a product of two states, one of one object and the other of the other object? Then if |Mj> is macroscopic state, how Σj αj|sj> |Mj> is not a macro superposition?
With kind regards
I think |M0> and |Mj> are macroscopic state of the system.
in order to study the interaction between the system with its environment theoretically, we model the environment as a set of independent harmonic oscillators that coupled to the system. then we examine the coupling of each environments oscillator to the system and write the equation of motion for each oscillator. then we can determine the state of each of the environments oscillator theoretically.
but there is problem; because the environment is a system of many degrees of freedom, we can't determine its microstate (for example consider a chamber contains a gas of 10^23 atoms). so we consider it's macrostate (for example consider an environment as a macroscopic system in thermal equilibrium) and describe the system with its density matrix and we apply many assumptions in an investigation of the dynamics of the environment (for example Markov approximation). I think |M0> and |Mj> are the superposition of microstates that corresponds to a specific macrostate of the environment.
Dear Hossein Allahverdi ,
"I think |M0> and |Mj> are the superposition of microstates that corresponds to a specific macrostate of the environment."
Your answer is beautiful and interesting. However, as we know, the superposition Σj αj|sj> |Mj> founders at a certain step of complexity, i.e. of the number of the particles engaged in the entanglement. For the experiment in entirety, i.e. along all the trials, of course we can represent the result of the interaction of our particle with the bath of oscillators, as a mixture. But in each trial of the experiment, one member of the mixture, one of the |Mj> prevails. But this problem is beyond the scope of the question.
With thanks and best regards,
Sofia
Dear Sofia D. Wechsler ,
I'll think about your question.
thank you very much for your attention.
Hossein
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