studiot Posted December 12, 2023 Share Posted December 12, 2023 4 hours ago, Alysdexic said: fast -> swift Thank you. Yes indeed I misspelled 'to fast' it should have been too fast. 5 hours ago, Luc Turpin said: Linear versus applied maths is not familiar to me As in many subjects, we distinguish between Pure Maths and Applied Maths. The pure subject is all about the theory of that subject, for its own sake. This is regardless of whether this theory is of any use or not. The applied subject takes that theory and uses it to study, analyse explain and predict phenomena in the real world to create things. For example the theory of structures is applied maths. For the next page we will need to know and understand a few technical terms The difference between an equation and an expression. The mathematical form of something. What functions , operators, mappings are. What a polynomial is. Please indicate if you are unsure of any of these so I can expand the explanation for this as I go along. Link to comment Share on other sites More sharing options...

Luc Turpin Posted December 12, 2023 Author Share Posted December 12, 2023 1 hour ago, joigus said: I think we're losing the patient. Good one😑 Link to comment Share on other sites More sharing options...

Luc Turpin Posted December 13, 2023 Author Share Posted December 13, 2023 3 hours ago, studiot said: Thank you. Yes indeed I misspelled 'to fast' it should have been too fast. As in many subjects, we distinguish between Pure Maths and Applied Maths. The pure subject is all about the theory of that subject, for its own sake. This is regardless of whether this theory is of any use or not. The applied subject takes that theory and uses it to study, analyse explain and predict phenomena in the real world to create things. For example the theory of structures is applied maths. For the next page we will need to know and understand a few technical terms The difference between an equation and an expression. The mathematical form of something. What functions , operators, mappings are. What a polynomial is. Please indicate if you are unsure of any of these so I can expand the explanation for this as I go along. All good! Confirms what applied Maths is all about. When you have time, I would like for you to substantiate on the predictive phenomena of applied maths. This is getting good for me; hope you don't get bored with it? And the answer to the kjw's riddle, is it model development, observation, conclusions? Link to comment Share on other sites More sharing options...

studiot Posted December 13, 2023 Share Posted December 13, 2023 10 hours ago, Luc Turpin said: All good! Confirms what applied Maths is all about. When you have time, I would like for you to substantiate on the predictive phenomena of applied maths. This is getting good for me; hope you don't get bored with it? And the answer to the kjw's riddle, is it model development, observation, conclusions? No it's the plus sign. So far we have only scratched the surface of 'linear'. We have only dealt with expressions that are linear in one single variable. The plus sign is how we combine variables or other mathematical objects (which may or may not be linear in themselves) To form expressions that are called linear combinations. Look back at KJW's expression - it contains a plus sign. Linear combinations have the form ax + by + cz + ...... where x, y, z are variables and a, b, c are constants. This is the underlying reason why linear mathematics is so wide, rich and varied. It covers maths from differential and integral calculus, differntial equations, matrices, fourier methods, linear simultaneous equations, and just so much more. We are off to the seaside for the rest of today so I will produce the next sheet tonight or tomorrow morning. Keep smiling. Link to comment Share on other sites More sharing options...

Luc Turpin Posted December 13, 2023 Author Share Posted December 13, 2023 26 minutes ago, studiot said: No it's the plus sign. So far we have only scratched the surface of 'linear'. We have only dealt with expressions that are linear in one single variable. The plus sign is how we combine variables or other mathematical objects (which may or may not be linear in themselves) To form expressions that are called linear combinations. Look back at KJW's expression - it contains a plus sign. Linear combinations have the form ax + by + cz + ...... where x, y, z are variables and a, b, c are constants. This is the underlying reason why linear mathematics is so wide, rich and varied. It covers maths from differential and integral calculus, differntial equations, matrices, fourier methods, linear simultaneous equations, and just so much more. We are off to the seaside for the rest of today so I will produce the next sheet tonight or tomorrow morning. Keep smiling. Failed my first test; probably looking at the wrong post. Got everything; understand My epiphany moment is keeping me focused at least for now. Was trying to choose a 'sick' smiley face for joigus's response, but ended up with a mad one. Waiting in expectation for the next post; thanks for all of this 🤢maybe this one is better Link to comment Share on other sites More sharing options...

studiot Posted December 13, 2023 Share Posted December 13, 2023 In English and in French we have letters, words, phrases, clauses, sentences and paragraphs, in order of complexity. That is we build up more and more complicated structures by combining simpler ones. In mathematics we really only have two basic building blocks; the symbol and the expression. Expressions possess several subtypes such as equations, statements of combination (like linear combination) and so forth. If an equation is roughly equivalent to a full sentence, then linked or simultaneous equations are equivalent to several sentences with a common link ie a paragraph. On 12/8/2023 at 9:44 PM, KJW said: Just in case you don't, an operator L() is linear if and only if it satisfies: L(ψ+ϕ)=L(ψ)+L(ϕ) Linearity is essential to QM because quantum superposition demands it. So ax + by + cz is not an equation but is a statement of combination of 3 different variables, combined to form one unit. each constant may be a different number, so the 'operation' (multiplication by a, b or c) carried out on each variable is different, but still linear in itself by our definition. But considered as a whole we have that if we multiply the whole by another number, say 2, we obtain 2[ ax + by + cz ] = 2ax + 2by + 2cz = (a+a)x + (b+b)y + (c+c) = (ax + by + cz) + (ax + by + cz) or twice our 'considered as a whole' combination. So if we condense this and say C = ax + by + cz then 2C = C + C So C is a linear combination. The point of my square brackets (did you nitice them ?) is that if we can show that [some expression] + [some expression] = 2 [some expression] Then the combination is linear. Next time we will explore the significance of this in relation to all those very important areas of applied maths I mentioned. L(ψ+ϕ)=L(ψ)+L(ϕ) Link to comment Share on other sites More sharing options...

Luc Turpin Posted December 14, 2023 Author Share Posted December 14, 2023 50 minutes ago, studiot said: In English and in French we have letters, words, phrases, clauses, sentences and paragraphs, in order of complexity. That is we build up more and more complicated structures by combining simpler ones. In mathematics we really only have two basic building blocks; the symbol and the expression. Expressions possess several subtypes such as equations, statements of combination (like linear combination) and so forth. If an equation is roughly equivalent to a full sentence, then linked or simultaneous equations are equivalent to several sentences with a common link ie a paragraph. So ax + by + cz is not an equation but is a statement of combination of 3 different variables, combined to form one unit. each constant may be a different number, so the 'operation' (multiplication by a, b or c) carried out on each variable is different, but still linear in itself by our definition. But considered as a whole we have that if we multiply the whole by another number, say 2, we obtain 2[ ax + by + cz ] = 2ax + 2by + 2cz = (a+a)x + (b+b)y + (c+c) = (ax + by + cz) + (ax + by + cz) or twice our 'considered as a whole' combination. So if we condense this and say C = ax + by + cz then 2C = C + C So C is a linear combination. The point of my square brackets (did you nitice them ?) is that if we can show that [some expression] + [some expression] = 2 [some expression] Then the combination is linear. Next time we will explore the significance of this in relation to all those very important areas of applied maths I mentioned. L(ψ+ϕ)=L(ψ)+L(ϕ) For KJW's riddle, I was not even on the right post. The analogy was useful. All is well in math-land for me so far. It is with great humility that I am learning basic-math once again; but this time with purpose Yes, noticed square brackets; understand their meaning; noticed also a small mistake in one equation. I am sticking to the program! At a proper tme, I would like for you to substantiate on the predictive phenomena of applied maths. Also, Linearity is essential to QM because quantum superposition demands it. Why is it so? But can wait if we need to stick to the basics for now! I will be posting something on Mind in the Medical Science-neuroscience section tomorrow. It might be of interest to you. Again, appreciate your time and attention! Link to comment Share on other sites More sharing options...

Genady Posted December 14, 2023 Share Posted December 14, 2023 (edited) 19 hours ago, studiot said: if we can show that [some expression] + [some expression] = 2 [some expression] Then the combination is linear. The above ^^^ does not look right to me. Do I understand correctly that "the combination" refers to the "[some expression]"? Edited December 14, 2023 by Genady 1 Link to comment Share on other sites More sharing options...

studiot Posted December 14, 2023 Share Posted December 14, 2023 52 minutes ago, Genady said: The above ^^^ does not look right to me. Do I understand correctly that "the combination" refers to the "[some expression]"? Well Luc did better than you then because 18 hours ago, Luc Turpin said: noticed also a small mistake in one equation. I assume that the small mistake was the missing z in 19 hours ago, studiot said: 2[ ax + by + cz ] = 2ax + 2by + 2cz = (a+a)x + (b+b)y + (c+c) = (ax + by + cz) + (ax + by + cz) or twice our 'considered as a whole' combination. As for you question, this is very fundamental and basic so I will labour the point. Let someexpression≡5 then we have 5 + 5 = 2 * 5 Yes ? Which is trivially linear. But the kicker is Let someexpression≡x4 Then [x^{4}] + [x^{4}] = 2 * [x^{4}] even though x^{4} itself is decidedly non linear. Link to comment Share on other sites More sharing options...

Luc Turpin Posted December 14, 2023 Author Share Posted December 14, 2023 11 minutes ago, studiot said: Well Luc did better than you then because Impossible 🙂 12 minutes ago, studiot said: I assume that the small mistake was the missing z in Yes! 13 minutes ago, studiot said: then we have 5 + 5 = 2 * 5 Yes ? Yes! 14 minutes ago, studiot said: But the kicker is Let someexpression≡x4 Then [x^{4}] + [x^{4}] = 2 * [x^{4}] even though x^{4} itself is decidedly non linear. Non linear expression becoming linear? So, you can "package" non linear as an expression and place it in a linear equation? On 12/1/2023 at 7:21 PM, studiot said: Posted; would be honoured to obtain comments from you, Joigus, Genady, etc. Link to comment Share on other sites More sharing options...

Phi for All Posted December 14, 2023 Share Posted December 14, 2023 40 minutes ago, Luc Turpin said: Posted; would be honoured to obtain comments from you, Joigus, Genady, etc. ! Moderator Note Please don't use one thread to advertise another. Thanks for understanding. Link to comment Share on other sites More sharing options...

sethoflagos Posted December 14, 2023 Share Posted December 14, 2023 On 11/30/2023 at 3:56 AM, swansont said: Entanglement does not cause a wave function to collapse. The individual states in an entangled state are undetermined, and superposition does not require entanglement. Went back to the start of the thread to see how far I could follow it before I got lost. The following list a a sequence of what appear to be logical inferences that spring to me from the above post. At which step does the logic break down? 1) The evolution of the individual quantum states in a superposition are accurately described by an appropriate wave equation such as (eg for Dirac fermions) the Dirac equation. 2) Although the Dirac equation maps onto our R^{3+1} spacetime, it is expressed in terms of a 4 dimensional complex vector space that is not defined within our R^{3+1} spacetime. 3) It is unclear whether or not quantum states in superposition have an individual material presence, but either way, they appear to be unmeasurable while they remain in superposition. 4) If indeed these individual quantum states have no concrete, physical existence within R^{3+1}, then they appear to be only abstract objects existing outside of R^{3+1}. 5) Side note: Point 4) is consistent with the possible emergence of our R^{3+1} universe from an initial quantum state that was not a part of our R^{3+1} universe. 6) As an abstract form, a quantum state need have no energy content, and therefore may coexist with an infinite multiplicity of alternate quantum states without conflict with the 1st Law of Thermodynamics. 7) As a uniquely defined state, it has a single permutation and therefore zero entropy. It may therefore coexist with an infinite multiplicity of (independent, non-interacting) alternate quantum states without conflict with the 2nd Law of Thermodynamics. 😎 Side note: if a quantum state in superposition did have a concrete physical presence in R^{3+1}, then points 6) & 7) become difficult to explain away. 9) As any specific solution to the Dirac equation is bidirectional in time (explicitly includes an advanced wave acting on the state in the time reversed direction), using point 7), no information is exchanged in this process and properties apparently exhibiting computation of eg Feynmann's 'sum over histories' concept may evolve without conflict with the principle of Causality. 10) It appears that an unambiguous measurement within our R^{3+1} universe requires an action originating from somewhere external to it. 11) It is really difficult to refer to a point 😎 on this site. Link to comment Share on other sites More sharing options...

Luc Turpin Posted December 14, 2023 Author Share Posted December 14, 2023 29 minutes ago, Phi for All said: ! Moderator Note Please don't use one thread to advertise another. Thanks for understanding. Apologies; did not know; will not do it again! Link to comment Share on other sites More sharing options...

studiot Posted December 14, 2023 Share Posted December 14, 2023 My apologies to @Genady for being so short just now. I was yet again loosing the tussle with the world's worst input editor, whils overcooking the dinner. Anyway thanks for quite rightly raising the question as it does point to our destination and prmpted me to elaborate more fully. 1 hour ago, Luc Turpin said: So, you can "package" non linear as an expression and place it in a linear equation? Exactly so. You have got it in one. So back to the program. Please remember that if you can add two things this way you can add three or four or....... 14 minutes ago, sethoflagos said: As an abstract form, a quantum state need have no energy content Can you give an example ? Do you for instance mean an unoccupied state such as an anti bonding orbital ? Link to comment Share on other sites More sharing options...

sethoflagos Posted December 14, 2023 Share Posted December 14, 2023 1 minute ago, studiot said: Do you for instance mean an unoccupied state such as an anti bonding orbital ? Wouldn't that be a physical state? The mathematical description of it would be an abstraction without substance, Link to comment Share on other sites More sharing options...

studiot Posted December 14, 2023 Share Posted December 14, 2023 (edited) 17 minutes ago, sethoflagos said: Wouldn't that be a physical state? The mathematical description of it would be an abstraction without substance, Clearly I am not following you meaning. What about zero point energy ? please note I am working towards developing the LCAO bonding method as an final answer to Luc's question about QM and Superposition. Edited December 14, 2023 by studiot Link to comment Share on other sites More sharing options...

Genady Posted December 14, 2023 Share Posted December 14, 2023 13 minutes ago, sethoflagos said: individual quantum states in a superposition If a state is 'in a superposition' or not depends on basis in which it is expressed. The same state is 'individual' in one basis and 'in a superposition' in another. Link to comment Share on other sites More sharing options...

swansont Posted December 14, 2023 Share Posted December 14, 2023 22 minutes ago, sethoflagos said: As an abstract form, a quantum state need have no energy content, and therefore may coexist with an infinite multiplicity of alternate quantum states without conflict with the 1st Law of Thermodynamics. Do you have an example? Quantum states I can think of will have an energy, and you need to add energy to put a system in the ground state into a superposition of energy eigenstates (unless they’re degenerate) Link to comment Share on other sites More sharing options...

sethoflagos Posted December 14, 2023 Share Posted December 14, 2023 (edited) 44 minutes ago, swansont said: Do you have an example? Quantum states I can think of will have an energy, and you need to add energy to put a system in the ground state into a superposition of energy eigenstates (unless they’re degenerate) I was hoping to keep this general and not get sidetracked into the details of specific cases. Maybe the superposition of free neutron and its decay products. Obviously both states have a particular defined and equal total energy as isolated states in our physical universe. However, in superposition, these states do not exist as the simple sum which amongst other things would double the total energy content. Does the superposition actually have a concrete form within our universe or does it imply some degree of real existence in a space outside of it as does its abstract complex mathematical description? It's that abstract form of existence that I'm asking about. Not massless neutrons. 1 hour ago, Genady said: If a state is 'in a superposition' or not depends on basis in which it is expressed. The same state is 'individual' in one basis and 'in a superposition' in another. As in my above response. It's the unmeasurable nature of the superposition - the coexistence of multiple versions of a single system - that's of interest to me here. @studiot, @Genady & @swansont I can understand the attraction of dogpiling onto point 6) Please confirm your position on points 3) & 4) to help me understand where we have parted in the logical progression. Edited December 14, 2023 by sethoflagos Link to comment Share on other sites More sharing options...

swansont Posted December 14, 2023 Share Posted December 14, 2023 38 minutes ago, sethoflagos said: However, in superposition, these states do not exist as the simple sum which amongst other things would double the total energy content. No, because each state has an amplitude, and the probabilities of being in the states add to 1. Link to comment Share on other sites More sharing options...

Genady Posted December 14, 2023 Share Posted December 14, 2023 3 hours ago, sethoflagos said: 3) It is unclear whether or not quantum states in superposition have an individual material presence, but either way, they appear to be unmeasurable while they remain in superposition. 4) If indeed these individual quantum states have no concrete, physical existence within R^{3+1}, then they appear to be only abstract objects existing outside of R^{3+1}. 2 hours ago, sethoflagos said: Please confirm your position on points 3) & 4) My position is that the component states are an expansion of the entire state in a specific basis. Link to comment Share on other sites More sharing options...

sethoflagos Posted December 15, 2023 Share Posted December 15, 2023 12 minutes ago, Genady said: My position is that the component states are an expansion of the entire state in a specific basis. Whatever that means. Forgive me but it comes over as avoiding the question. Link to comment Share on other sites More sharing options...

Genady Posted December 15, 2023 Share Posted December 15, 2023 (edited) 20 minutes ago, sethoflagos said: Whatever that means. Forgive me but it comes over as avoiding the question. I agree. I don't understand the question. But I am sure that you understand my answer. It is not different from, e.g., Fourier transform. Edited December 15, 2023 by Genady Link to comment Share on other sites More sharing options...

Luc Turpin Posted December 15, 2023 Author Share Posted December 15, 2023 16 hours ago, studiot said: please note I am working towards developing the LCAO bonding method as an final answer to Luc's question about QM and Superposition. What is LCAO bonding method? In the double-slit experiment with photons being fired one at the time, is there superposition? if so? when or where? Link to comment Share on other sites More sharing options...

Genady Posted December 15, 2023 Share Posted December 15, 2023 (edited) 1 hour ago, Luc Turpin said: In the double-slit experiment with photons being fired one at the time, is there superposition? if so? when or where? Yes, superposition of states resulting from a photon passing through different slits. If the state after passing through slit 1 is \(\psi_1\) and the state after passing through slit 2 is \(\psi_2\), then after passing the screen the state of the photon is the superposition, \(\frac 1 {\sqrt 2}\psi_1 +\frac 1 {\sqrt 2}\psi_2\). Edited December 15, 2023 by Genady Link to comment Share on other sites More sharing options...

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