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  1. J

    I Can this particular method solve these quadratic equations?

    When substituting the solution for ##x## in the other equation, I get: $$y^2 \bigg(a'\cdot \frac{(-2h\pm s)^2}{4a^2}+2h'\cdot \frac{-2h\pm s}{2a}+b'\bigg)+c'=0$$ Where ##s=\sqrt{4h^2-4ab+\frac{4ac}{y^2}}##. Not sure how such an equation can be solved. Also, shouldn't a quartic equation have a...
  2. J

    I Can this particular method solve these quadratic equations?

    Why does this solution for ##x## not have the form of a fraction? It's missing a factor of ##\frac{1}{2a}## as mentioned in my opening post, isn't that how the solution of a quadratic function is defined according to the abc-formula?
  3. J

    I Can this particular method solve these quadratic equations?

    Thanks, but wouldn't this lead to another method to solve these equations? I was wondering whether the quoted method can actually be used to solve this.
  4. J

    I Can this particular method solve these quadratic equations?

    You mean as in dividing ##S_1## and ##S_2## by either ##x^2## or ##y^2##? I have tried that but I'm not sure how to continue to get ##x## or ##y## on one side.
  5. J

    I Can this particular method solve these quadratic equations?

    Given are two equations: $$S_1 = ax^2+2hxy+by^2 + c=0$$ $$S_2 = a'x^2+2h'xy+b'y^2 + c'=0$$ This source states that there are several methods to solve for ##x## and ##y##. One of them is the following quote:"Treat equation ##S_1## as a quadratic equation in ##x## and solve it for ##x## in terms...
  6. J

    I Correlation between Symmetry number & Total wavefunction

    "proton or neutron" as in exchange of a proton pair "or" neutron pair? If yes, could you please elaborate how this quote is compatible with this: I am a bit confused.
  7. J

    I Correlation between Symmetry number & Total wavefunction

    I understood that specific phrase "won't have to be" as in "not obliged to be" which made me think that it is also possible for such a model to be antisymmetric under exchange of a proton and neutron as well, just as in the exchange of a proton pair or neutron pair. But I assume you meant that...
  8. J

    I Correlation between Symmetry number & Total wavefunction

    Does "won't have to" imply that it is still possible to stay antisymmetric? Because in that case, the classical symmetry number would still not have a clear link (to me) since it relies on the physical indenticality of particles.
  9. J

    I Correlation between Symmetry number & Total wavefunction

    But whether the total wavefunction is symmetric/anti-symmetric depends on the spins of the nuclei, not on whether the nuclei are identical or not. It goes even further by the fact that the total wavefunction of e.g. dihydrogen can be symmetrical even if the two atoms have opposed spins. So it...
  10. J

    I Correlation between Symmetry number & Total wavefunction

    Some rotational quantum states are not allowed for a rotating particle. At quantum level, these "forbidden" quantum states is based on the requirement of the total wavefunction being either symmetrical or anti-symmetrical, depending on whether the particle is a fermion or boson. The particle's...
  11. J

    I Accuracy of the Density of States

    I'm trying to understand the detailed concept of why the density of states formula is accurate enough to calculate the number of quantum states of an energy level, including degeneracy, within a small energy interval of ##dE##. The discrete energie levels are calculated by $$E = \frac{h^2 \cdot...
  12. J

    Why is the most probable energy different from the speed?

    Never mind, I figured it out, thanks. My calcuations in post #30 are wrong, I didn't pay attention when doing them. I was overlooking the fact that the relation between ##dE## and ##dv## is a function of ##v## while the probability itself is also a function of ##v##. If the relation between...
  13. J

    Why is the most probable energy different from the speed?

    @Ibix & @PeroK I think using your info's helped me a bit by explaining it as follows: For a certain ##E_0## and a corresponding ##v_0##, the following must be valid:$$f(E_0)\cdot dE = f(v_0) \cdot dv$$ For ##f(E)##, the maximum number of particles lies within the range ##\frac{k_BT}{2} \geq...
  14. J

    Why is the most probable energy different from the speed?

    You asked about what I based it on and I took the time to explain to you how I understood it so you could perhaps pinpoint how I should understand it with ##dE = mvdv##. Merely substituting my whole explanation with "...the wrong thing" and that he meant something that I already mentioned...
  15. J

    Why is the most probable energy different from the speed?

    The fact that he, as well as another user, liked my post #16 containing that very description of ##dE## as a statement I asked about would make me assume that my understanding about ##dE## was correct. Hence me asking about it after. Perhaps I based my description on his mathematical fact...
  16. J

    Why is the most probable energy different from the speed?

    How an answer "looks" is also influenced by how one understands it. I'm not solely basing the difference on the "looks" of my answers, but on my own understandings when writing them. As I see them, there is a difference since one answer says "covers a portion" and the other says "covers all"...
  17. J

    Why is the most probable energy different from the speed?

    I don't think so because I said in post #16 that ##dE## covers only a portion of higher speeds because of the spread, but if ##dE## gets wider at higher speeds, it would compensate for that spread and thus cover all of them anyway. I.e. the "density" within ##dE## stays the same.
  18. J

    Why is the most probable energy different from the speed?

    @PeroK I noticed an inner clash afterwards, because I also know that ##dE = mv \cdot dv##. Doesn't this mean that ##dE## get wider at higher speeds and thus covers the increasing spread of the speed anyway?
  19. J

    Why is the most probable energy different from the speed?

    Ah, so in the context of my reasoning. A ##dE## starting from low energy values (thus low speeds) would cover more particles of the same speed and a ##dE## starting from higher energy (higher speeds) covers only a portion of the particles with the same speed because they are more spread out when...
  20. J

    Why is the most probable energy different from the speed?

    I understand that the formulas are different mathematically, but my question is how this can be imagined physically, for example using my scenario to distribute the particles in such a way to fulfill that criteria. If that's not possible then how about the following reasoning?
  21. J

    Why is the most probable energy different from the speed?

    That's what I'm saying. My question in that regard is how it can be physically explained that the most probable energy does not correspond to the most probable speed that belongs to that energy. @PeroK Yes
  22. J

    Why is the most probable energy different from the speed?

    They can be found here and here. For the 2nd link, please search for the terms "most probable". @vanhees71 @Ibix @Vanadium 50 @phyzguy I'm aware that the average speed is not the same as the root-mean-square speed. I don't see how this has a link with the most probable speed. Please see @PeroK...
  23. J

    Why is the most probable energy different from the speed?

    It appears that the most probable energy level according to the Maxwell-Boltzmann distribution is not equal to the most probable speed squared multiplied by ##\frac{1}{2}m##. The most probable speed has a different value. $$E_{max} = \frac{k_BT}{2}$$ $$v_{max} = \sqrt{\frac{2k_BT}{m}}$$ I am...
  24. J

    Influence of vacuum in the Conservation of energy

    @Cutter Ketch I have been thinking this through and I am curious about the validity of my deduced integral to calculate the work done on the small piston. The pressure force ##F_P## is the difference between the two pressures halves in the container times the surface of the small piston. These...
  25. J

    Influence of vacuum in the Conservation of energy

    Thank you so much for your time and your clear explanations. This helped me
  26. J

    Influence of vacuum in the Conservation of energy

    Ah, my bad. I was taking piston 2 's larger inertia to decelerate into account but I forgot that its larger area of force at pressure equilibrium would also compensate for that. One last question; if friction from the pistons on the walls are taken into account, would the distances swept out by...
  27. J

    Influence of vacuum in the Conservation of energy

    Assuming both pistons would swipe the same volume to reach equlibrium, does this mean that piston 2 would move a distance that is a factor ##\frac{r^2_2}{r^2_1}## smaller than piston 1, even though they have the same initial acceleration? (again, no resisting forces added).
  28. J

    Influence of vacuum in the Conservation of energy

    Ah ok. You said furthermore that it will show an oscillation when there's no friction. In the presence of friction, would this mean that the pistons would stop at positions where the pressure difference is zero?
  29. J

    Influence of vacuum in the Conservation of energy

    Because you said in post #26 that the initial accelerations would be different in this scenario. Perhaps I have missed your point but I'm not sure how.
  30. J

    Influence of vacuum in the Conservation of energy

    Could you please tell me what I'm donig wrong here? In the case of a pressure difference while holding the pistons and then releasing them to relax (without applying any additional forces) the pressure force on piston ##1## would be ##F_1 = (P_2 - P_1) \cdot \pi r^2_1 = m_1 \cdot a_1##. The...
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