From: <[email protected] <mailto:[email protected]>>
On Tuesday, June 5, 2018 at 7:03:28 AM UTC, Bruce wrote: From: <[email protected]>On Tuesday, June 5, 2018 at 1:18:29 AM UTC, Bruce wrote: From: <[email protected]>Remember that the analysis I have given above is schematic, representing the general progression of unitary evolution. It is not specific to any particular case, or any particular number of possible outcomes for the experiment. Bruce *OK. For economy we can write, **(|+>|e+> + |->|e->), where e stands for the entire universe other than the particle whose spin is being measured. What is the status of the interference between the terms in this superposition? For a quantum superposition to make sense, there must be interference between the terms in the sum. At least that's my understanding of the quantum principle of superposition. But the universe excluding the particle being measured seems to have no definable wave length; hence, I don't see that this superposition makes any sense in how superposition is applied. Would appreciate your input on this issue. TIA, AG*A superposition is just a sum of vectors in Hilbert space. If these vectors are orthogonal there is no interference between them. *As a graduate student, in one of those standard problems, I seem to recall solving for the wf of some system using the SWE, and then expanding the solution using an orthonormal set of eigenfunctions as the basis (or maybe it was claimed there exists such an expansion). Are you saying there is no interference between the basis eigenvectors? TIA, AG*Basis vectors need not be orthogonal. But if you choose and orthonormal set then the individual basis vectors do not interfere, though the superposition made up of such a set may be thought of as an interference between them. But unless you take the product of this superposition with something else, you do not have interference cross terms. This is similar to the basis problem I referred to earlier. If we have a basis of |dead> and |alive>, being mutually orthogonal, then the basis vectors (|dead>+|alive>) and (|dead>-|alive>) are also orthonormal. But this basis is not stable under decoherence, so the environmental states corresponding to this basis do interfere to produce the stable |e_dead> and |e_alive> states. Thus, if the orthogonal basis you choose is such that the interaction of each basis vector with the environment leads to orthogonal environment vectors, then there is no interference between these environmental states -- this is how classical states emerge. Bruce *So in the case of the S Cat, the superposition, ( |alive> |undecayed> + |dead>|decayed> ) , does NOT imply the cat is simultaneously alive and dead because the states in this superposition are orthogonal? If that's your conclusion, and if I am not misreading the discussions of this problem incorrectly, most, if not all of the texts which discuss this problem are completely misleading. Is that the situation? AG*
I am not sure exactly what you are wanting to discover in this discussion. From my perspective, interference is due to cross terms between states, so if there are no cross terms, such as if the states are orthogonal, then there is no interference. In the cat example, the cat is not both dead and alive simultaneously. If someone wants to claim that, then they are saying that the live and dead states are not actually orthogonal, so the original state can be re-established by the interference between live and dead cats. However, classical states do not interfere because the classical dead state is orthogonal to the alive state. In MWI, the term FAPP occurs here, because MWIers claim that the interference terms never vanish completely. So probably the issue comes down to how you react to FAPP orthogonality. If that means that the states can still interfere, then they are not actually orthogonal. If they can't interfere to reconstruct the original state, then decoherence has led to genuinely orthogonal states.
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