On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal wrote: > > > On 17 Jan 2019, at 09:33, [email protected] <javascript:> 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 form a basis. 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* > > -- > 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] <javascript:>. > To post to this group, send email to [email protected] > <javascript:>. > 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.
