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Apr
17
comment Looking for a simple example of generating unequal probabilities in QM
@Rococo, pinging you once in case the chat doesn't notify you of a message on my end. I don't think it does...
Apr
17
reviewed Approve Why do we get the same result using different ensembles?
Apr
13
comment Looking for a simple example of generating unequal probabilities in QM
@Rococo, thanks, I'm not sure either but I created one for us here
Apr
12
comment Looking for a simple example of generating unequal probabilities in QM
@Rococo, I am awarding you the bounty because you've been helpful, although I'm still not satisfied. I don't see how Schrodinger evolution can split a wave packet in two equal parts where the amplitudes are not one half the original, because of linearity. I thought superposition was linear. If I imagine it in reverse, you have two wave packets, each of amplitude 1/sqrt(2), and when they combine by linear addition the amplitude is 2/sqrt(2) and unitarity is violated. What am I not understanding correctly?
Apr
12
accepted Looking for a simple example of generating unequal probabilities in QM
Apr
12
comment Looking for a simple example of generating unequal probabilities in QM
@adipy, yes I am equating the two, intentionally, in the sense that it seems like one implies the other. Maybe see my question here. That my thinking here is not obviously stupid is justified by Gleason's theorem. Normalization is related to Born in that it is saying that the squared amplitudes sum to probability = 100%. While I may ultimately be wrong, this is the heart of my question, and it cannot be so trivially dismissed without significantly more engagement and explanation.
Apr
9
comment Looking for a simple example of generating unequal probabilities in QM
@adipy, as for your second point, yes, I understand that. What I don't understand is the connection between the amplitude and the probability (Born rule) via branch counting. It's a bit subtler than I think you give credit for, and I think you don't understand my confusion, again see the edit to my original question.
Apr
9
comment Looking for a simple example of generating unequal probabilities in QM
@adipy, I think it would help you understand where I'm coming from if you read the edit below my original question, where I state a confusion that is even simpler than the one you are pointing to.
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Timaeus, I've edited my question to reflect the fact that I now understand at least the kernel of my confusion. Maybe that will help clarify what is going wrong in my thinking.
Apr
7
revised Looking for a simple example of generating unequal probabilities in QM
Based on comments added clarification
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
it pains me to say that you are still not understanding me. I will edit my original question based on the recent conversation with you and Rococo, which might help clarify.
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Timaeus, (continued). Note there is nothing "wrong" with that if it is what is done, but I just want to very clearly understand which it is. Can the $1/\sqrt{2}$ amplitude when you split a beam 50-50 be derived from Schrodinger evolution, or are you just renormalizing it in to enforce that you get 50-50 probabilities?
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Timaeus, (continued), unless you are making an implicit assumption about how those amplitudes relate to probability? In other words you are assuming the Born rule when positing that when you split the initial beam in two, each amplitude goes as $1/\sqrt{2}$. You are essentially enforcing unitarity by fiat by declaring (or renormalizing) the state of equal amplitudes so that the sum of probabilities remains 1. Unless there is some rule I'm not aware of by which Schrodinger evolution unitarily evolves the state to the $1/\sqrt{2}$ amplitudes without you having to make that declaration.
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Timaeus, I never said my 66% calculation is correct, you are completely and totally misunderstanding me if you think I am insisting that I am correct. But you don't seem to have grokked my misunderstanding, and as a result I am still confused. This is the rub: "When someone says they split a beam into two equal parts they mean they split the probability into two equal probabilities." This is the source of my confusion. How can you split 100% of something (the amplitude, not the probability) into two equal piles, and say that the size of each pile is not 50% of the original, unless (continued)
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Timaeus, I understand, but you are still I think not understanding the source of my confusion, which is the same as my comment to Rococo above. If you use, say, SG to split a first beam 50-50 and then a second beam 50-50, it seems self-evident that by simple "counting" the amplitudes should be 1/2, 1/4, 1/4, but like you point out this violates unitarity. Obviously there is something that gives. I suspect you both are tautologically assuming the Born rule in a way to keep unitarity, otherwise I don't see how you can possibly split a beam 50-50 and result in $\sqrt{2}:1$ ratio of amplitudes.
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Rococo, thanks we are getting close, but now my issue is how you arrived at the amplitudes when you write "The state coming out of this modified Mach-Zender interferometer is $\frac{1}{\sqrt{2}}|1>+\frac{1}{2}|2>+\frac{1}{2}|3>$." Since, if you really are splitting each beam 50-50, shouldn't the amplitudes actually be 1/2, 1/4, and 1/4. I understand that this violates unitarity, but clearly I'm missing something. How can you split a beam 50-50 and result in a ratio of amplitudes that is $\sqrt{2}:1$? I understand everything else in your post. In comes down to this.
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Timaeus, (continued) but that seems totally counter intuitive since you should just be able to split one wave function into 50%-50% amplitude using one SG, and then split one of the branches with another SG to create 50%, 25%, 25% amplitude. Which by the above violates unitarity. I'm super confused. Where is my thinking going wrong there?
Apr
7
comment Looking for a simple example of generating unequal probabilities in QM
@Timaeus, let me make sure I understand you. You are saying that if I say the probability for |detector1 you1> is 0.5$^2$/$\sqrt{0.5^2+0.25^2+0.25^2}$ that you can see I must have gone wrong because I'm normalizing the wave function with the $\sqrt{0.5^2+0.25^2+0.25^2}$, and doing so means I've violated unitarity. If that's what you are saying then at least I understand finally your attempt to help me! But if so, do you understand why I did what I did? I'm looking for an experiment that splits the wave function into three paths with amplitude 25%, 25%, and 50%. You say impossible by unitarity.
Apr
6
comment Looking for a simple example of generating unequal probabilities in QM
@MarcelKöpke, yes, but only using something like SG at 90 degrees that split the wave function into halves for example so that branch counting can be followed directly. Punk_Physicist gave an example with SG rotated at $\theta$<90 degrees that violates the spirit of what I'm asking for.
Apr
6
comment Looking for a simple example of generating unequal probabilities in QM
So what I still don't understand is that in your example by branch counting I would derive that the experimental probabilities should be 50% |detector1 you1> and 25% for each of the others. But the Born rule would tell you than the probability for |detector1 you1> is actually 2$^2$/($\sqrt{2^2+1^2+1^2}$) = 66%, and 17% for each of the others. So somewhere there is a contradiction, no?