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Siupa
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Stuck on (c), part (i). Any ideas about what is the most elegant way to prove it, maybe using Mandelstam variables since this exercise is supposed to be about them?
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Unless you assume the CoM frame, then how can that equation hold?Siupa said:Stuck on (c), part (i). Any ideas about what is the most elegant way to prove it, maybe using Mandelstam variables since this exercise is supposed to be about them?
Are you suggesting this as an hint on how to prove it, or are you suggesting that the question is wrong?PeroK said:Unless you assume the CoM frame, then how can that equation hold?
I guess you have to assume the CoM frame, otherwise ##|\mathbf p_1| \ne |\mathbf p_2|##.Siupa said:Are you suggesting this as an hint on how to prove it, or are you suggesting that the question is wrong?
I believe this is what the absolute value signs in the OP are intended to mean, since the vectors in question are in bold, which normally indicates a 3-vector. But the OP should clarify.Vanadium 50 said:magnitude of the 3 vector
Yes, with the extra assumption that ##|\bf{p}_1| = |\bf{p}_2|## I can solve this. Let’s call ##\bf{p}_i := \bf{p}_1 = \bf{p}_2##. Then, ##\bf{p}_1 + \bf{p}_2 = 0 = \bf{p}_3 + \bf{p}_4## implies ##|\bf{p}_3| = |\bf{p}_4|##, and we can also call either of these ##\bf{p}_f##. To get our result we just need to show ##|\bf{p}_i| = |\bf{p}_f|##. But this follows from conservation of energy:PeroK said:I guess you have to assume the CoM frame, otherwise ##|\mathbf p_1| \ne |\mathbf p_2|##.
Do you see how to show the full equation in that case?
What are you referring to? Can you clarify what would have been more clear about my question, had I followed a homework template?Vanadium 50 said:This ls a question where the homework template sure would have helped.
The absolute value of a 4-vector can mean one of two different things: its norm, or the magnitude of the 3 vector. Which one do you meant? Hint: only one is valid in all frames.
It can't hold in the rest frame of either particle!Siupa said:However the text doesn’t make the assumption that ##|\bf{p}_1| = |\bf{p}_2|##. Do you think it’s not possible to show it in a general case? Is the question wrong?
In the center of mass frame that is true by definition, as part (b) of the problem says. Part (c) apparently wants you to continue to assume you are in the center of mass frame, as in part (b), but doesn't say so, which it should for clarity.Siupa said:the text doesn’t make the assumption that ##|\bf{p}_1| = |\bf{p}_2|##.
You’re right, that was easy to check in hindsight. There’s at least one other frame where the process can happen with all equal masses, yet that relationship doesn’t hold. Thanks for your help!PeroK said:It can't hold in the rest frame of either particle!
that’s probably it yes, the author must have imagined it as a continuation of the premises of the previous question, but it’s not clear given that point (c) isn’t intended/nested hierarchally under (b), but instead is presented in parallel as “the next question”.PeterDonis said:In the center of mass frame that is true by definition, as part (b) of the problem says. Part (c) apparently wants you to continue to assume you are in the center of mass frame, as in part (b), but doesn't say so, which it should for clarity.
Let me give you another lesson. It pays to be nice to people who are trying to help you.Siupa said:There’s no need for that condescending lesson about being afraid to confront authority though
I’ve been nothing but thankful to everyone who answered here, I’ve thanked him multiple times in that same message you quoted. His comment about beginner students being unwilling to accept that sometimes books might be wrong was indeed condescending and out of context, even if in good faith. There’s nothing rude in clarifying and explaining that his impression was wrong. You might even say that this is what “confronting authority” looks like, right? Or is it only valid when you’re not on the receiving end of it?Vanadium 50 said:Let me give you another lesson. It pays to be nice to people who are trying to help you.
First, I have no authority. I'm just an amateur who has taught myself physics during my retirement.Siupa said:I’ve been nothing but thankful to everyone who answered here, I’ve thanked him multiple times in that same message you quoted. His comment about beginner students being unwilling to accept that sometimes book might be wrong was indeed condescending and out of context, even if in good faith. There’s nothing rude in clarifying and explaining that his impression was wrong. You might even say that this is what “confronting authority” looks like, right? Or is it only valid when you’re not on the receiving end of it?
How you interpreted that as condescending is beyond me.PeroK said:You must have confidence in your own knowldege and intelligence.
That looks like you were still very reluctant to accept that the book had a simple omission.Siupa said:However the text doesn’t make the assumption that ##|\bf{p}_1| = |\bf{p}_2|##. Do you think it’s not possible to show it in a general case? Is the question wrong?
Well, whatever you and I are in real life, in this context you’re in a position of authority with respect to me, as you’re a respected member of the community who’s giving me help and I’m a relatively new member with little questions asked and no help given on the forum. But that’s besides the point, you’re right I don’t think it’s relevant.PeroK said:First, I have no authority. I'm just an amateur who has taught myself physics during my retirement.
Comparing me to a stubborn student that refuses to accept that authority could be wrong and doesn’t have confidence in his own positions feels quite condescending to me: the underlying assumption being that I lack critical thinking skills and I mindlessly follow what other people tell me is true, without questioning. Maybe this is a harsh read and you didn’t mean this, but this is what it felt like.PeroK said:How you interpreted that as condescending is beyond me.
Fair, nothing to say here, my bad.PeroK said:Third I never used the word "beginner". I just said students. There were not all beginners, which is very much my point.
It wasn’t clear to me at all that it could not be true. Just because one can prove a statement by making it more specific by adding an extra assumption, it doesn’t mean that it can’t also hold in general without that assumption: it could just mean that there is a more general proof you’re taking a specific case of (A+B -> C doesn’t mean that A -> C is wrong). I kept asking you if *you* thought that the book was wrong because you alluded to it, and I was open to the possibility. But I didn’t have any good reason to think that it could be wrong at that time.PeroK said:Fourth, it was definitely not out of context. The context of this thread is that the book appears to be asking you to prove something that is clearly not generally true. And by post #9, you were still unsure:
That looks like you were still very reluctant to accept that the book had a simple omission.
I think problem c wants you to use the center of mass frame just like it did in (b).Siupa said:Stuck on (c), part (i). Any ideas about what is the most elegant way to prove it, maybe using Mandelstam variables since this exercise is supposed to be about them?View attachment 341640