From: <[email protected] <mailto:[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.
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