On Thursday, January 17, 2019 at 1:48:57 PM UTC, [email protected] wrote: > > > > On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal wrote: >> >> >> On 17 Jan 2019, at 09:33, [email protected] wrote: >> >> >> >> On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote: >>> >>> >>> >>> On 1/16/2019 7:25 PM, [email protected] wrote: >>> >>> >>> >>> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote: >>>> >>>> >>>> >>>> On 1/13/2019 9:51 PM, [email protected] wrote: >>>> >>>> This means, to me, that the arbitrary phase angles have absolutely no >>>> effect on the resultant interference pattern which is observed. But isn't >>>> this what the phase angles are supposed to effect? AG >>>> >>>> >>>> The screen pattern is determined by *relative phase angles for the >>>> different paths that reach the same point on the screen*. The >>>> relative angles only depend on different path lengths, so the overall >>>> phase >>>> angle is irrelevant. >>>> >>>> Brent >>>> >>> >>> >>> *Sure, except there areTWO forms of phase interference in Wave >>> Mechanics; the one you refer to above, and another discussed in the >>> Stackexchange links I previously posted. In the latter case, the wf is >>> expressed as a superposition, say of two states, where we consider two >>> cases; a multiplicative complex phase shift is included prior to the sum, >>> and different complex phase shifts multiplying each component, all of the >>> form e^i (theta). Easy to show that interference exists in the latter case, >>> but not the former. Now suppose we take the inner product of the wf with >>> the ith eigenstate of the superposition, in order to calculate the >>> probability of measuring the eigenvalue of the ith eigenstate, applying one >>> of the postulates of QM, keeping in mind that each eigenstate is multiplied >>> by a DIFFERENT complex phase shift. If we further assume the eigenstates >>> are mutually orthogonal, the probability of measuring each eigenvalue does >>> NOT depend on the different phase shifts. What happened to the interference >>> demonstrated by the Stackexchange links? TIA, AG * >>> >>> Your measurement projected it out. It's like measuring which slit the >>> photon goes through...it eliminates the interference. >>> >>> Brent >>> >> >> *That's what I suspected; that going to an orthogonal basis, I departed >> from the examples in Stackexchange where an arbitrary superposition is used >> in the analysis of interference. Nevertheless, isn't it possible to >> transform from an arbitrary superposition to one using an orthogonal basis? >> And aren't all bases equivalent from a linear algebra pov? If all bases are >> equivalent, why would transforming to an orthogonal basis lose >> interference, whereas a general superposition does not? TIA, AG* >> >> >> I don’t understand this. All the bases we have used all the time are >> supposed to be orthonormal bases. We suppose that the scalar product (e_i >> e_j) = delta_i_j, when presenting the Born rule, and the quantum formalism. >> >> Bruno >> > > *Generally, bases in a vector space are NOT orthonormal. For example, in > the vector space of vectors in the plane, any pair of non-parallel vectors * > *(excluding anti parallel vectors) form a basis. AG*
*Same for any general superposition of states in QM. HOWEVER, > eigenfunctions with distinct eigenvalues ARE orthogonal. I posted a link to > this proof a few months ago. IIRC, it was on its specifically named thread. > AG* > *Posted on July 25, 2018:* Proof; Eigenfunctions having different eigenvalues are orthogonal AG > >> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To post to this group, send email to [email protected]. >> Visit this group at https://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> >> >> -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
