On Monday, February 11, 2019 at 2:20:25 AM UTC, [email protected] wrote: > > > > On Tuesday, February 5, 2019 at 8:43:59 PM UTC, [email protected] wrote: >> >> >> >> On Monday, February 4, 2019 at 8:56:57 AM UTC-7, Bruno Marchal wrote: >>> >>> >>> On 3 Feb 2019, at 00:03, [email protected] wrote: >>> >>> >>> >>> On Saturday, February 2, 2019 at 2:59:30 PM UTC-7, [email protected] >>> wrote: >>>> >>>> >>>> >>>> On Saturday, February 2, 2019 at 1:40:29 AM UTC-7, Bruno Marchal wrote: >>>>> >>>>> >>>>> On 1 Feb 2019, at 21:29, [email protected] 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, agrays...@ >>>>>>> gmail.com 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, agrays...@ >>>>>>>>>>> gmail.com 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 * >>>> >>> >>> I understand that if you have a plane wave, say, psi(x,t) = Ae^i(kx-wt), >>> the probability density if given by the square of the square of the modulus >>> of A. >>> I am not sure why you need this, and why this should be problematic with >>> Bruce’s (or mine) explanation. >>> >> >> *I started with a more general case; namely, writing psi as a >> superposition of orthonormal states, each with its own implied phase shift >> incorpororated in different A's,. and found that the resultant probability >> density didn't depend on any cross terms, which zero out due to the >> orthogonality. (Actually, I started with an explicitly different phase >> shift of the form e^i(theta), but later omitted that since the phase shifts >> can be assumed as different and incorporated in the A's.) I thought >> interference depended on the result for cross terms, but as Phil pointed >> out, that's not the case. Why I thought this follows from something Feynman >> wrote in his lectures. I will explain this in another post. Thanks for your >> help. AG* >> > > *To reiterate and clarify; you got your result by taking the inner product > of psi with itself, which is what I did, but starting with a more general > initial state; namely, a superposition of orthonormal states, each > multiplied by a presumably different probability amplitude A_j. Now, if you > go back to my earlier comments, you will see the final result; namely, a > sum of terms of the form || A_j ||^2, where the cross terms drop out due to > orthonormality. So what's my problem? Simply this; somewhere in Feynman's > Lectures, he wants to show how quantum probabilities differ from classical > probabilities. He shows the difference is between taking the classical > probability for say the EM double slit experiment as ||A||^2 + ||B||^2 > (where one adds the intensity squared for each wave), and the quantum > calculation, ( ||A + B|| )^2, where one first sums the amplitudes *before* > taking the resultant norm squared. Why then do I get the first result for > the quantum probability -- which presumably is the classical result -- when > the initial quantum psi is taken as a sum of orthonormal states? TIA, AG * >
*Maybe the mistake I am making is confusing the norm squared of a single wf, with that of two distinguishable wf's, one for each slit. 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 * >>>> >>> >>> >>> OK. Good. >>> >>> >>> >>> >>>>> 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 * >>>> >>> >>> That was not enough clear, sorry. >>> The wiki is also rather unclear on the Born Rule. I mean that there are >>> clearer exposition. >>> >>> >>> >>> >>>> 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. >>>>> >>>> >>> *If you put in the projection operator, you're calculating the >>> probability of getting some eigenvalue, not the probability density of the >>> position. AG * >>> >>> >>> >>> You lost me. I was just explaining why the relative phase does play a >>> role for the probability of finding specific values. Bruce was correct, and >>> I still don’t know if you agree on this or not. >>> I am not sure that I understand what is your problem. >>> >>> Bruno >>> >>> >>> >>> >>> >>> >>> >>> >>>> >>>> *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 everything-li...@googlegroups.< >>>>>> >>>>>> -- 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.
