On 6/5/2018 2:05 AM, Bruce Kellett wrote:
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.
The MWI view is that the cat is in FAPP orthogonal states which include
orthogonal environments, and obeservers, and hence constitute FAPP
orthogonal "worlds". So the cat exists in both these worlds, one
alive and one dead, but with different probabilities. This over
simplifies the situation though. The radioactive decay could happen at
any time, and so there is a continuum of worlds, corresponding to the
different times at which the cat died.
Brent
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