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Mar
24
comment Looking for a simple example of generating unequal probabilities in QM
The rest of your paragraphs I think extend from the misinterpretation of my question explained above. Thank you so much for engaging though! I really hope that you will understand my question better given the above comments and help me understand how to figure out the expected probabilities for these kinds of situations. Assuming the wave function stays coherent until the end, I still don't know how to handle this situation, because like I said in the OP, for example when the particle detectors register different observables are being measured simultaneously depending on the outcome.
Mar
24
comment Looking for a simple example of generating unequal probabilities in QM
Fourth paragraph, my understanding is that the Born rule is built into the definition of a Hilbert space through definition of wave function normalization. Basically if you define the norm of your wave function of "1" to correspond to 100% probability, then the Born rule is unavoidable given that the norm is the sqrt of sum of squared amplitudes. This can be seen in Gleason's thoerem or Everett's "derivation" of Born in his thesis.
Mar
24
comment Looking for a simple example of generating unequal probabilities in QM
Regarding your third paragraph, I understand that what you say is correct if a S-G truly makes a measurement, but my operating assumption is that the system can be made coherent until the final particle detectors measure the presence of an out-going particle. In other words I'm assuming that each S-G is isolated from the outside environment. So each S-G does not represent a "measurement", just a pathway for the schrodinger evolution to divede up the wave function and cause interference before measurement. Does that make sense?
Mar
24
comment Looking for a simple example of generating unequal probabilities in QM
Regarding your second paragraph, isn't it a very common and mostly agreed-upon complaint of the MWI that the Born rule implies a non-intuitive non-uniform measure for branch counting? The point is not to show that MWI is internally inconsistent. The point is just that ultimately the Born rule is not "derived from simple branch counting", since you ultimately need to axiomatically assume a weird counting measure in order to agree with the Born rule.
Mar
24
comment Looking for a simple example of generating unequal probabilities in QM
To answer your first question, it's because those "two separate final outcomes" result in the exact same experimental signature. The two beams of atoms are combined using S-G and end up at a single particle detector. The two outcomes I listed were for clarity of what was going on -- the +Z and -Z are combined by S-G into a single beam. Regarding "I want to find a flaw in the MWI", that is a misunderstanding of my intention. I want to find an example that intuitively exemplifies the common complaint that the "branch counting measure" is non-intuitive basically due to the square in the Born rule
Mar
24
comment How to use the Born rule to find the expected outcome of this simple Stern-Gerlach experiment
That makes sense, but can you do something analogous instead with halfsilvered mirrors? The context of my question was the one here. Basically I'm trying to find a realistic example where you can split and recombine/interfere equal-sized parts of the wave function so that the Born rule generates probabilities that suggest a non-uniform branch-counting measure. I just wanted to check this criticism of the MWI for myself, and intriguingly can't find a physical example.
Mar
15
reviewed Approve If an object rests on a table, not accelerating, how much work do both the object and the table do?
Mar
14
comment How to use the Born rule to find the expected outcome of this simple Stern-Gerlach experiment
Thanks, and if I combined +|x+> and +|x+> (by just turning one of the S-G upside-down in my above picture) then the probabilities would be 1/6, 4/6, 1/6, right?
Mar
14
comment How to use the Born rule to find the expected outcome of this simple Stern-Gerlach experiment
Thanks! Your edit helps a lot (although I think the c|+010> should be c|-010>, right?). To make sure I understand, the probabilities would be 1/6, 1/4, 1/6 if I turned one of the S-G upside down so that instead of feeding |X+> and |X-> to the last S-G I fed |X+> and |X+> to it, right? Then the naive intuition would be wrong.
Mar
14
comment How to use the Born rule to find the expected outcome of this simple Stern-Gerlach experiment
So the general procedure is piecemeal, there is no way to just write the final state down and use the amplitudes to calculate the probabilities for each outcome at once? What if I sent |X+> and |X+> into the last S-G instead of |X+> and |X->, then would the final probs be 1/6, 4/6, 1/6, since the the amplitude at C is now double and with Born gets squared ?
Mar
14
comment How to use the Born rule to find the expected outcome of this simple Stern-Gerlach experiment
What I'm interested in is the "proper" method you first mention, but then only later give a vague account of. Do you mind explaining in more detail? What is the observable being measured here really (that gives A B or C), and why can't I just write the wave function in a basis of its eigenvalues and use the amplitudes to find the probabilities? Naively the probabilities should be 1/4, 1/2, 1/4, but on the other hand if the amplitudes add when feeding the 2 results into the same machine, then you'd think by the Born rule the probs would actually be 1/, 4/6, 1/6 because the amplitude is squared.
Mar
13
asked How to use the Born rule to find the expected outcome of this simple Stern-Gerlach experiment
Mar
13
comment Looking for a simple example of generating unequal probabilities in QM
continued... to divide the wave function into four equal parts then recombine half of them so that the amplitudes are 1/$\sqrt(6)$(1|a>+2|b>+1|c>) so that the probabilities are 1/6, 4/6, 1/6. Then you can examine the contradiction, ie that half of the branches combined and yet ended up accounting for more than half the total probability. The strange issue is that I cannot seem to be able to find an actual physically possible realization of this! Does my issue at least make sense to you?
Mar
13
comment Looking for a simple example of generating unequal probabilities in QM
continued, Your rotated Stern-Gerlach answer is good but unfortunately it doesn't allow me to do the branch counting experiment I'm trying to do, since you can't count the fraction of the wave function going one way or the other and compare to the cos$^2$ and sin$^2$ probabilities from the Born rule, not without circularly making some assumption about how the wave function scales upon being projected into some rotated basis in order to preserve the norm. In other words I'm searching for something like: use some number of Stern-Gerlachs...
Mar
13
comment Looking for a simple example of generating unequal probabilities in QM
One issue I have is that sin$^n$($\theta$) and cos$^n$($\theta$) also satisfy your requirements stated above in your edited answer, and I think it is only the inner-product space that uniquely chooses n=1 so that the state is properly normalized. But this is beside the point (and I'm not 100% sure I'm right about it anyways) because the bigger problem for me is that I didn't pose the question well enough. I think regardless I basically need a system of orthogonal Stern-Gerlachs (or something similar) that allows me to work with combining some countable number of branches.
Mar
12
comment Looking for a simple example of generating unequal probabilities in QM
How do you calculate <+|+z>=cos($\theta$) without assuming that <+|+z>$^2$+<-|+z>$^2$=1, ie that the Born probabilities sum to 1? This is a subtle issue related to the fact that as Gleason showed, the Born rule is inevitable given the Hilbert inner-product space. This is basically the point of my question. I want to find an example that allows me to count branches without making circular assumptions. One way would be to combine orthogonal Stern-Gerlach experiments in ways like I described in my post.
Mar
12
revised Looking for a simple example of generating unequal probabilities in QM
addendum in response to an answer
Mar
12
comment Looking for a simple example of generating unequal probabilities in QM
Thanks, this does answer my question as-stated, but it shows that I didn't state my question clearly enough, which is my fault. The problem with your above solution (and why I was trying to focus on combining orthogonally oriented Stern-Gerlach magnets) is that the amplitudes you propose are tautologically the result of assuming the Born rule, since we know that sin$^2$($\theta$)+cos$^2$($\theta$)=1
Mar
12
reviewed Approve What happens if a free radical is placed in absolute vacuum?
Mar
12
reviewed Approve Why do the radii decrease when we move left to right in the periodic table?