On Saturday, February 2, 2019 at 1:40:29 AM UTC-7, Bruno Marchal wrote: > > > On 1 Feb 2019, at 21:29, [email protected] <javascript:> wrote: > > > > On Friday, February 1, 2019 at 5:55:30 AM UTC-7, Bruno Marchal wrote: >> >> >> On 31 Jan 2019, at 21:10, [email protected] wrote: >> >> >> >> On Thursday, January 31, 2019 at 6:47:12 AM UTC-7, Bruno Marchal wrote: >>> >>> >>> On 31 Jan 2019, at 01:28, [email protected] wrote: >>> >>> >>> >>> On Wednesday, January 30, 2019 at 2:38:58 PM UTC-7, [email protected] >>> wrote: >>>> >>>> >>>> >>>> On Wednesday, January 30, 2019 at 5:16:05 AM UTC-7, Bruno Marchal wrote: >>>>> >>>>> >>>>> On 30 Jan 2019, at 02:59, [email protected] wrote: >>>>> >>>>> >>>>> >>>>> On Tuesday, January 29, 2019 at 4:37:34 AM UTC-7, Bruno Marchal wrote: >>>>>> >>>>>> >>>>>> On 28 Jan 2019, at 22:50, [email protected] wrote: >>>>>> >>>>>> >>>>>> >>>>>> On Friday, January 25, 2019 at 7:33:05 AM UTC-7, Bruno Marchal wrote: >>>>>>> >>>>>>> >>>>>>> On 24 Jan 2019, at 09:29, [email protected] wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Sunday, January 20, 2019 at 11:54:43 AM UTC, [email protected] >>>>>>> wrote: >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> On Sunday, January 20, 2019 at 9:56:17 AM UTC, Bruno Marchal wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> On 18 Jan 2019, at 18:50, [email protected] wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> On Friday, January 18, 2019 at 12:09:58 PM UTC, Bruno Marchal >>>>>>>>> wrote: >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> On 17 Jan 2019, at 14:48, [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. * >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> Right. But we can always build an orthonormal base with a decent >>>>>>>>>> scalar product, like in Hilbert space, >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> *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.* >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> Absolutely. And when choosing a non degenerate >>>>>>>>>> observable/measuring-device, we work in the base of its >>>>>>>>>> eigenvectors. A >>>>>>>>>> superposition is better seen as a sum of some eigenvectors of some >>>>>>>>>> observable. That is the crazy thing in QM. The same particle can be >>>>>>>>>> superposed in the state of being here and there. Two different >>>>>>>>>> positions of >>>>>>>>>> one particle can be superposed. >>>>>>>>>> >>>>>>>>> >>>>>>>>> *This is a common misinterpretation. Just because a wf can be >>>>>>>>> expressed in different ways (as a vector in the plane can be >>>>>>>>> expressed in >>>>>>>>> uncountably many different bases), doesn't mean a particle can exist >>>>>>>>> in >>>>>>>>> different positions in space at the same time. AG* >>>>>>>>> >>>>>>>>> >>>>>>>>> It has a non null amplitude of probability of being here and there >>>>>>>>> at the same time, like having a non null amplitude of probability of >>>>>>>>> going >>>>>>>>> through each slit in the two slits experience. >>>>>>>>> >>>>>>>>> If not, you can’t explain the inference patterns, especially in >>>>>>>>> the photon self-interference. >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> Using a non orthonormal base makes only things more complex. >>>>>>>>>> >>>>>>>>> *I posted a link to this proof a few months ago. IIRC, it was on >>>>>>>>>> its specifically named thread. AG* >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> But all this makes my point. A vector by itself cannot be >>>>>>>>>> superposed, but can be seen as the superposition of two other >>>>>>>>>> vectors, and >>>>>>>>>> if those are orthonormal, that gives by the Born rule the >>>>>>>>>> probability to >>>>>>>>>> obtain the "Eigen result” corresponding to the measuring apparatus >>>>>>>>>> with >>>>>>>>>> Eigen vectors given by that orthonormal base. >>>>>>>>>> >>>>>>>>>> I’m still not sure about what you would be missing. >>>>>>>>>> >>>>>>>>> >>>>>>>>> *You would be missing the interference! Do the math. Calculate the >>>>>>>>> probability density of a wf expressed as a superposition of >>>>>>>>> orthonormal >>>>>>>>> eigenstates, where each component state has a different phase angle. >>>>>>>>> All >>>>>>>>> cross terms cancel out due to orthogonality,* >>>>>>>>> >>>>>>>>> >>>>>>>>> ? Sin(alpha) up + cos(alpha) down has sin^2(alpha) probability to >>>>>>>>> be fin up, and cos^2(alpha) probability to be found down, but has >>>>>>>>> probability one being found in the Sin(alpha) up + cos(alpha) down >>>>>>>>> state, >>>>>>>>> which would not be the case with a mixture of sin^2(alpha) proportion >>>>>>>>> of up >>>>>>>>> with cos^2(alpha) down particles. >>>>>>>>> Si, I don’t see what we would loss the interference terms. >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> *and the probability density does not depend on the phase >>>>>>>>> differences. What you get seems to be the classical probability >>>>>>>>> density. >>>>>>>>> AG * >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> I miss something here. I don’t understand your argument. It seems >>>>>>>>> to contradict basic QM (the Born rule). >>>>>>>>> >>>>>>>> >>>>>>>> *Suppose we want to calculate the probability density of a >>>>>>>> superposition consisting of orthonormal eigenfunctions, * >>>>>>>> >>>>>>> >>>>>>> Distinct eigenvalue correspond to orthonormal vector, so I tend to >>>>>>> always superpose only orthonormal functions, related to those >>>>>>> eigenvalue. >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> *each multiplied by some amplitude and some arbitrary phase shift. * >>>>>>>> >>>>>>> >>>>>>> like (a up + b down), but of course we need a^2 + b^2 = 1. You need >>>>>>> to be sure that you have normalised the superposition to be able to >>>>>>> apply >>>>>>> the Born rule. >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> *If we take the norm squared using Born's Rule, don't all the cross >>>>>>>> terms zero out due to orthonormality? * >>>>>>>> >>>>>>> >>>>>>> ? >>>>>>> >>>>>>> The Born rule tell you that you will find up with probability a^2, >>>>>>> and down with probability b^2 >>>>>>> >>>>>>> >>>>>>> >>>>>>> *Aren't we just left with the SUM OF NORM SQUARES of each component >>>>>>>> of the superposition? YES or NO?* >>>>>>>> >>>>>>> >>>>>>> If you measure in the base (a up + b down, a up -b down). In that >>>>>>> case you get the probability 1 for the state above. >>>>>>> >>>>>>> >>>>>>> >>>>>>> * If YES, the resultant probability density doesn't depend on any of >>>>>>>> the phase angles. AG* >>>>>>>> >>>>>>> >>>>>>> *YES or NO? AG * >>>>>>> >>>>>>> >>>>>>> >>>>>>> Yes, if you measure if the state is a up + b down or a up - b down. >>>>>>> No, if you measure the if the state is just up or down >>>>>>> >>>>>>> Bruno >>>>>>> >>>>>> >>>>>> *I assume orthNORMAL eigenfunctions. I assume the probability >>>>>> densities sum to unity. Then, using Born's rule, I have shown that >>>>>> multiplying each component by e^i(theta) where theta is arbitrarily >>>>>> different for each component, disappears when the probability density is >>>>>> calculated, due to orthonormality. * >>>>>> >>>>>> >>>>>> >>>>>> That seems to violate elementary quantum mechanics. If e^I(theta) is >>>>>> different for each components, Born rule have to give different >>>>>> probabilities for each components---indeed given by the square of >>>>>> e^I(theta). >>>>>> >>>>> >>>>> *The norm squared of e^i(thetai) is unity, except for the cross terms >>>>> which is zero due to orthonormality. AG * >>>>> >>>>>> >>>>>> *What you've done, if I understand correctly, is measure the >>>>>> probability density using different bases, and getting different values. >>>>>> * >>>>>> >>>>>> >>>>>> The value of the relative probabilities do not depend on the choice >>>>>> of the base used to describe the wave. Only of the base corresponding to >>>>>> what you decide to measure. >>>>>> >>>>>> >>>>>> >>>>>> *This cannot be correct since the probability density is an objective >>>>>> value, and doesn't depend on which basis is chosen. AG* >>>>>> >>>>>> >>>>>> Just do the math. Or read textbook. >>>>>> >>>>> >>>>> *Why don't YOU do the math ! It's really simple. Just take the norm >>>>> squared of a superposition of component eigenfunctions, each multiplied >>>>> by >>>>> a probability amplitude, and see what you get ! No need to multiply each >>>>> component by e^i(thetai). Each amplitude has a phase angle implied. This >>>>> is Born's rule and the result doesn't depend on phase angles, contracting >>>>> what Bruce wrote IIUC. If you would just do the simple calculation you >>>>> will >>>>> see what I am referring to! AG* >>>>> >>>>> >>>>> >>>>> Bruce is right. Let us do the computation in the simple case where >>>>> e^i(theta) = -1. (Theta = Pi) >>>>> >>>>> Take the superposition (up - down), conveniently renormalised. If I >>>>> multiply the whole wave (up - down) by (-1), that changes really nothing. >>>>> But if I multiply only the second term, I get the orthogonal state up + >>>>> down, which changes everything. (up +down) is orthogonal to (up - down). >>>>> >>>>> Bruno >>>>> >>>> >>>> *Fuck it. You refuse to do the simple math to show me exactly where I >>>> have made an error, IF I have made an error. You talk a lot about Born's >>>> rule but I seriously doubt you know how to use it for simple >>>> superposition. AG * >>>> >>> >>> *If you take the inner product squared (Born's rule) using an >>> orthonormal set of eigenfunctions, you get a sum of the form (a_j)^2 + >>> (b_j) ^2 where A_j is the complex probability amplitude for the jth >>> component, A_j = a_j + i * b_j. The cross terms drop out due to >>> orthonormality, and the phase angles are implicitly determined by the >>> relative values of a_j and b_j for each j. * >>> >>> >>> If you have prepared the state, so that you know that the state of your >>> object is given by >>> >>> phi = A_1 up + A_2 down, say, then, if you decide to measure the >>> up/down state, and use the device doing that, you do not need to make the >>> inner product between phi and phi, but between the base state up and/or >>> down to get the probability given by the square of phi * up (to get the >>> probability of up) and the square of phi*down, to get the probability of >>> down. They will both depend on the value of A_1 and A_2. They are >>> respectively (A_1)^2 and (A_2)^2. Of course, we suppose that we have >>> renormalised the state so that (A_1)^2 + (A_2)^2 = 1 (which makes them into >>> probability of getting up and down). >>> >>> >>> >>> >>> >>> >>> >>> *The question then becomes how do we calculate the probability density >>> with the phase angles undetermined. Are we assuming they are known given >>> the way the system is prepared? AG* >>> >>> >>> >>> Yes. The Born rule, written simply, is only that if phi = A_1 up + A_2 >>> down, (so the state has been prepared in advance) then if you measure if >>> the object is in up or down, you will find up with a probability given >>> respectively by (A_1)^2 and (A_2)^2. >>> All probabilities are relative to the state of the object and the choice >>> of what you decide to measure. It is always simpler to write the state in >>> the base corresponding to the measurement, so that the “simple” Born rule >>> above can be applied immediately. >>> >>> Bruno >>> >> >> *For reference I repeat my last comment and add a significant point:* >> >> If you take the inner product squared (Born's rule) using an >> orthonormal set of eigenfunctions, you get a sum of the form (a_j)^2 + >> (b_j) ^2 where A_j is the complex probability amplitude for the jth >> component, A_j = a_j + i * b_j. The cross terms drop out due to >> orthonormality, and the phase angles are implicitly determined by the >> relative values of a_j and b_j for each j. The question then becomes how do >> we calculate the probability density with the phase angles undetermined. >> Are we assuming they are known given the way the system is prepared? AG >> >> The question for me is how the phase angles are related to interference. >> >> >> But that is explained by may calculation above. You calculation does not >> make sense to me. You compute an inner product of the wave to itself? I >> don’t see the relation with your problem. >> > > *Obviously, you don't know how to apply the rule you speak so highly of, > Born's rule. To calculate the probability density of wf function psi, you > must calculate <psi, psi>. Do you dispute this? * > > > Yes, you need to put some projection operator (corresponding to some > eigenvalue you intend to measure) in between. > <psi,psi> is the amplitude of probability to go from the psi state to the > psi state, and should be equal to one (psi being normalised). >
*Wrong! Not equal to one. I see you like to talk the talk, but refuse to walk the walk. Just read the 4th paragraph of the Wiki link. I correctly calculated the probability density for orthonormal eigenstates! AG * > > > > *How the phase angles relate to interference is another issue, which I > think Phil explained. AG* > >> >> The calculation above shows that the cross terms drop out due to >> orthonormality. >> >> >> Do it again, explicitly. Take the simple state phi = A_1 up + A_2 down. >> Up and down are orthonormal, >> > > *Up and Dn are NOT orhonormal. AG* > > >> but phi is not orthonormal with either up or down. If “up” means go to >> the left hole, and “down” is go the right hole, the amplitude A_1 and A_2, >> if not null, will interfere, even if only one photon is sent.The wave go >> through both silts, and interfere constructively along some direction and >> destructively along other direction, making it impossible for that photon >> to lend on those last place, like anyway, by the laws of addition of >> sinus/wave. >> >> But IIUC these are the terms which account for interference. >> >> >> I am not sure what you say here. The interferences comes only from the >> fact that we have a superposition of two orthogonal state, and that >> superposition is a new state, which is not orthogonal to either up or down. >> >> Thus, applying Born's rule to a superposition of states where the >> components are orthonormal, leaves open the question of interference. >> >> >> That does no make sense. The Born rule just say that if you measure >> (up/down) on phi = A_1 up + A_2 down, you get up with probability (A_1)^2 >> and down with probability (A_2)^2. But if you do any measurement, the state >> beg-have like a wave, and the amplitudes add up, constructively or >> destructively. >> >> If you don’t understand that, it means you begin to understand quantum >> mechanics, as nobody understand this, except perhaps the Mechanist >> Philosophers …(which predicts something at least as weird and >> counter-intuitive). >> >> Bruce wrote that the phase angles are responsible for interference. I >> doubt that result. Am I mistaken? AG >> >> >> Yes, I’m afford you are. The relative phase (in a superposition) angles >> are responsible for the interference. A global phase angle changes nothing. >> > > *If I am wrong, it's just because I assumed all interference comes from > the interactions due to the cross terms -- which cancel out for orthonormal > component states. Also, I never introduced a global phase angle in my > calculation. If you would do my calculation, or at least understand it, > you'd understand Born's rule. I don't need to read Albert's book to > understand Born's rule. AG* > > > > Once you say that up and down are not orthonormal, I am not sure you have > studied the QM formalism correctly. Any two distinguishable eigenstates of > any observable are orthogonal (and normalised). > *Right. I was mistaken. AG * > > I have no clue what you don’t understand in my use of the Born rule. You > definitely need to study Albert’s book, I think. > *Wiki shows I correctly calculated the probability density. Also I agree with Phil, and noted the error I made (not in any calculation, but in interpretation). Didn't you read it? AG * In your other post you mention wikipedia. No problem there? Actually you > can see that they do put the projection operator at the right place. You > can help yourself with a dictionary, but books and papers are better. > *You could help yourself by reading plain English. SEE PARAGRAPH 4 OF WIKI LINK. THEY CALCULATED THE PROBABILITY DENSITY AND DIDN'T PUT IN THE PROJECTION OPERATOR! AG* > > Bruno > > > > > > >> I really wish you to read the first 60 pages of David Albert’s book. Its >> exposition of the functioning of the interferometer is crystal clear. I am >> still not sure if you have a problem with the formalism or with the >> weirdness related to it. Read that piece of explanation by Albert, and if >> you still have problem, we can discuss it, but it would be too long (here >> and now) to do that here. >> >> Bruno >> >> >> >> >> >> >> >> -- >> 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] <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. 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