KJW

If a particle is in a state of definite momentum, it is in a superposition of position states. And if a particle is in a state of definite position, it is in a superposition of momentum states. This is a mathematical consequence of conjugate variables.

Sounds like measurements interfere with the state. But how? I should point out that some years ago, I simulated the quantum Zeno effect on an Excel spreadsheet, so I know that this effect is not in any way magical, but is a consequence of the nature of quantum mechanical measurements.

Proven??? How would you explain the quantum Zeno effect?

That was addressed to you. I believe I gave a clear answer to the OP, and until the OP comes back with a request for clarification, I see no reason to say anything different.

It's not an "approach", it's a skill that chemists, especially organic chemists, need to master. I don't see the problem with the ammonium ion - the nitrogen atom has a +1 charge. As for the carbonate ion, each resonance structure has its own Lewis structure with a –1 charge on the two singly bonded oxygen atoms. Or one could use a single abbreviated structure that manifests the symmetry with a –2/3 charge on all three oxygens (averaging the three resonance structures). As for nitric acid, dissociates in what way? Anyway, the products of dissociation have their own Lewis structures requiring formal charge allocation.

Looking at Google, I came across the following formula for the formal charge of an atom in a molecule: Formal charge of atom = (Number of valence electrons in free atom) – (Number of non-bonding electrons) – (Half the number of bonding electrons) Admittedly, I had not encountered this explicit formula before. Instead, I would work out the formal charge by some ad hoc calculation that would ultimately lead to the same answer. The formula works by starting from an atom which has had all its valence electrons removed, adding the number of non-bonding electrons, then adding the number of bonding electrons that contribute to the formal charge of the atom, which is half the number of bonding electrons, the other half contributing to the formal charge of the other atom. Note that the subtractions in the formula are because electrons are negatively charged. Although I didn't use an explicit formula to calculate the formal charges of the atoms of ozone, the above description corresponds to the formula: Formal charge of atom = (Number of valence electrons in free atom) – (Number of electrons) + (Half the number of bonding electrons) Because (Number of electrons) = (Number of non-bonding electrons) + (Number of bonding electrons), these two seemingly different formulae will give the same answer. However, by adding (Half the number of bonding electrons) instead of subtracting, conceptually I'm removing electrons that do not contribute to the formal charge of the atom.

Given that the triple bond of carbon monoxide is the strongest known chemical bond, I would say "no". The Wikipedia articles on quadruple bonds and quintuple bonds seem to focus more on bond length than on bond strength, so I don't know if these bonds are particularly strong.

Even quintuple bonds exist. These involve all five d orbitals on each metal, participating in one sigma, two pi, and two delta bonds.

Quadruple bonds are possible, although they are quite rare, found in some transition metal complexes. According to Wikipedia, the first chemical compound containing a quadruple bond to be synthesised was chromium(II) acetate, Cr2(μ-O2CCH3)4(H2O)2, which contains a Cr–Cr quadruple bond. It's worth noting that delta bonds are also possible. Similar to the way pi bonds can be formed from overlapping p orbitals, delta bonds can be formed from overlapping d orbitals. In the case of chromium(II) acetate, the quadruple bond is considered to arise from the overlap of four d orbitals on each metal with the same orbitals on the other metal: the dz² orbitals overlap to give a sigma bonding component, the dxz and dyz orbitals overlap to give two pi bonding components, and the dxy orbitals give a delta bond.

Yes, but I'm actually talking about charge bookkeeping.

I'll explain O3 then leave CO22– and HNO3 for you to explain. A neutral oxygen atom has 6 electrons in its valence shell. If it gains two electrons, completing the octet, it will have a charge of –2. Each of the three oxygen atoms of ozone has a complete octet. Now consider that each of the oxygen atoms has donated electrons to the oxygen atoms to which it is bonded. Each donated electron changes the charge by +1. The oxygen atom on the left has donated 2 electrons to the central oxygen atom, and therefore its charge is neutral. The oxygen atom on the right has donated 1 electron to the central oxygen atom, and therefore its charge is –1. The central oxygen atom has donated 3 electrons to the two bonded oxygen atoms (2 to the left oxygen atom and 1 to the right oxygen atom), and therefore its charge is +1. This representation is missing a negative charge on the carbon atom, and a positive charge on the oxygen atom.

The Wikipedia article "Nitrogen oxide" lists the following: Nitric oxide (NO), nitrogen(II) oxide, or nitrogen monoxide Nitrogen dioxide (NO2), nitrogen(IV) oxide Nitrogen trioxide (NO3), or nitrate radical Nitrous oxide (N2O), nitrogen(0,II) oxide Dinitrogen dioxide (N2O2), nitrogen(II) oxide dimer Dinitrogen trioxide (N2O3), nitrogen(II,IV) oxide Dinitrogen tetroxide (N2O4), nitrogen(IV) oxide dimer Dinitrogen pentoxide (N2O5), nitrogen(V) oxide, or nitronium nitrate [NO2]+[NO3]− Nitrosyl azide (N4O), nitrogen(−I,0,I,II) oxide Nitryl azide (N4O2) Oxatetrazole (N4O) Trinitramide (N(NO2)3 or N4O6), nitrogen(0,IV) oxide Some of these have somewhat complicated structures. It's worth noting that the electron configuration of nitrogen is [He] 2s2 2p3. This means that nitrogen can donate up to 5 electrons and accept up to 3 electrons. Note that chemical bonding usually involves both donating electrons to the atoms to which it is bonded and accepting electrons from those atoms. Thus, donating up to 5 and accepting up to 3 electrons impacts on the possible structures that nitrogen can form. Contrast this with phosphorus whose electron configuration is [Ne] 3s2 3p3. Again, this can donate up to 5 electrons, but with empty 3d orbitals available to accept electrons, expanding the octet allows phosphorus to have structures that are not available to nitrogen.

For quite a long time, I've thought the Bible would be more convincing as the word of God if it had said something factually correct about the surface of Venus.

It is understood that map vs territory is an analogy of mathematical model vs physical reality, and that saying "the map is not the territory" is analogous to saying that mathematical models of physics are not complete or accurate descriptions of physical reality. So this is not about literal maps and territories. Anyway, the assertion I'm making, based on the assumption that the laws of physics are not ad hoc, is that it is possible in principle to construct a perfect mathematical description of physical reality, that physical reality does not contain features that are indescribable. But note that "perfect" respects fundamental limitations imposed by physical reality itself. However, I am considering a perfect description of physical reality as a hypothetical notion that is amenable to mathematical analysis in a way that is not available to physical reality itself. And because of the correspondence between physical reality and a perfect description of it, any results obtained from the mathematical analysis of the description will apply to physical reality once the correspondence has been realised. On this basis, it can be said that the laws of physics correspond to the mathematical properties of descriptions. And because symmetry plays such a huge role in the mathematical properties of descriptions, symmetry plays a huge role in the laws of physics. Can you please elaborate. It's not clear to me what you are asking. Have you not seen a globe of the world?

But all you have done is restate the conservation of energy and momentum using different words. You have not actually explained why energy and momentum are conserved. You have stated that the medium can't create or destroy its own net motion or stored work but have not explained why it can't. I've said that purely physical explanations ultimately lead to the notion that the laws of physics are ad hoc as if provided from above, and you've said nothing to dispel that. The problem with that is that mathematical principles do have their consequences whether they are understood or not. To ignore the mathematics is a failure to understand the laws of physics. And it appears to me that you do not understand the role symmetry plays in the laws of physics. One example concerns the cosmological constant which is a contribution to the spacetime curvature that is not energy-momentum but behaves like dark energy. At every location in spacetime, the cosmological constant is not only constant over spacetime but also invariant to Lorentz transformations. Note that this is a mathematical symmetry of the cosmological constant. A consequence of this symmetry is that one cannot measure one's velocity relative to this cosmological constant field. Although it is a spacetime curvature with measurable properties associated with that curvature, it otherwise behaves like empty spacetime. It is worth noting that the only reason we can measure our velocity at all is that the things relative to which we measure our velocity break the Lorentz symmetry of the spacetime. So what is it? You don't trust mathematics? Or you don't accept the symmetry of the laws of physics? Well I'm not suggesting that physics can be derived purely from mathematics without any physics. It is through observing physical reality that provides the connection between the mathematical quantities and the corresponding physical quantities. But mathematics has to be taken as the cause of physics because it is mathematics that provides the underlying logic of physics. Physics itself has no logic. It should also be noted that mathematics does not derive from physics. Thus, if one mathematically derives a relationship between physical quantities, then that derivation is self-contained in that no other physical quantities not mentioned in the derivation are involved in the derived relationship. For example, the derivation of the relationship between time dilation and familiar gravity is based on geometry and provides the correct relationship, as determined by the Pound-Rebka experiment. There is no room for any other cause of familiar gravity. Mathematical notation is the language, mathematics itself is an underlying logic that transcends physical reality. You don't know what the universe is made of. Nobody does. That's because we examine the universe from the inside and therefore can only determine the universe in terms of itself. I come across this statement every now and then. And I reject it. Suppose one has a map that perfectly describes the territory. Then, for every thing that is in the territory, there is a corresponding thing in the map; and for every thing that is in the map, there is a corresponding thing in the territory. Thus, everything we want to know about the territory can be found out by examining the map. "Medium" is just a word. It means nothing without a description of its properties and behaviour. And only mathematics is precise enough to provide such a description. Thus, you need to be in the mathematical realm, anyway. Then the issue becomes how the mathematical description of the properties and behaviour is to be obtained.

This also applies to artificial gravity. Again, time dilation and artificial gravity could be considered to be concomitant. But because artificial gravity can be controlled, it becomes more reasonable to say that artificial gravity causes time dilation. Nevertheless, the relationship between time dilation and artificial gravity is the same as the relationship between time dilation and familiar gravity.

What about the mass of a muon, or tauon?

I'm not thinking backwards. Time dilation and familiar gravity could be considered to be concomitant. But because time dilation is about the structure of spacetime, and familiar gravity is about how objects respond to the structure of spacetime, it is indeed natural to say that time dilation causes familiar gravity rather than the other way around. The relationship between time dilation and familiar gravity can be mathematically derived. And the reason I say familiar gravity is because full gravitation is more complicated, as I alluded to when I said that time dilation only provides half the deflection of starlight by the sun. But time dilation does completely provide the gravity described by Newton. The problem with pure physicality of physics is that it becomes impossible to truly justify the behaviour of physical objects. The laws of physics become ad hoc as if provided from above. For example, in purely physical terms, explain why energy and momentum are conserved. Noether's theorem does provide an explanation, but that's mathematical.

There are no paradoxes in relativity, only misunderstanding of it. So, retaining all the good bits of relativity means retaining relativity. The gravity with which we are familiar is not caused by the curvature of three-dimensional space. It is caused by time dilation. However, half of the deflection of starlight by the sun is caused by the curvature of three-dimensional space, the other half predicted by Newtonian gravity is caused by time dilation. Is it the Einstein-Hilbert action? The Schwarzschild metric, which describes the geometry surrounding a spherically symmetric distribution of matter, reproduces the Newtonian equations of motion in the weak-field limit. However, strictly speaking, Newtonian gravitation contradicts itself, so although the Newtonian equations of motion are reproduced in the weak-field limit, Newtonian theory of gravitation itself is not reproduced. It's curvature in four dimensions. However, curvature does involve distorting the metric away from a rectangular coordinate system or any coordinate transformation of a rectangular coordinate system. Although this might be considered to be "stretching or compressing something", the notion of stretching or compressing something places too much physicality on the notion of spacetime and might not be correct in its finer details.

[math]\text{mathbb: } \mathbb{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math] [math]\text{mathrm: } \mathrm{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math] [math]\text{mathcal: } \mathcal{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math] [math]\text{mathfrak: } \mathfrak{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math] [math]\text{mathscr: } \mathscr{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math] [math]\text{mathbb: } \mathbb{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math] [math]\text{mathrm: } \mathrm{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math] [math]\text{mathcal: } \mathcal{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math] [math]\text{mathfrak: } \mathfrak{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math] [math]\text{mathscr: } \mathscr{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math]

@Dhillon1724X , When it comes to solving the geodesic equation, one normally obtains the solution: [math]x^0 = x^0(\lambda)\\x^1 = x^1(\lambda)\\x^2 = x^2(\lambda)\\x^3 = x^3(\lambda)[/math] where there are four functions of the parameter instead of the three functions: [math]x^0 = t\\x^1 = x^1(t)\\x^2 = x^2(t)\\x^3 = x^3(t)[/math] from my previous post. That is, [math]x^0(\lambda)[/math] is now a function to be solved instead of a predetermined function [math]t[/math] that is forced upon the solution. This extra degree of freedom allows one to choose an affine parameter by solving the standard geodesic equation: [math]\dfrac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\sigma} \dfrac{dx^\nu}{d\lambda} \dfrac{dx^\sigma}{d\lambda} = 0[/math] If one was to force the parametrisation [math]x^0 = t[/math], then one would have to solve the equation: [math]\dfrac{d^2 x^\mu}{dt^2} + \Gamma^\mu_{\nu\sigma} \dfrac{dx^\nu}{dt} \dfrac{dx^\sigma}{dt} = f(t) \dfrac{dx^\mu}{dt}\ \ \ \ \ \text{where:}\ \ \ \ \ \dfrac{d^2 \lambda}{dt^2} - f(t) \dfrac{d\lambda}{dt} = 0[/math] including solving [math]f(t)[/math] to make up for the loss of a degree of freedom that resulted from forcing the parametrisation. Note that each function of the solution: [math]x^0 = x^0(\lambda)\\x^1 = x^1(\lambda)\\x^2 = x^2(\lambda)\\x^3 = x^3(\lambda)[/math] has two arbitrary constants because the one is solving a system of second-order ordinary differential equations. One arbitrary constant specifies the initial location of the solution curve. The other arbitrary constant specifies the direction of the solution curve [math]dx^\mu\!/d\lambda[/math] at the initial location. [If the above LaTex doesn't render, please refresh browser.]

Not long ago, I saw on (Australian) 60 Minutes about people who are willingly having romantic relationships with AI.

Elementary operations between transcendental numbers will likely produce at least an irrational. But it's a case study. The transcendental numbers [math]2^{\sqrt{2}}[/math], [math]i^{\>\!i}[/math], and [math]e^\pi[/math] I mentioned are known to be transcendental by the Gelfond-Schneider theorem, which states: If [math]a[/math] and [math]b[/math] are algebraic numbers with [math]a \notin \left\{0, 1\right\}[/math] and [math]b[/math] not rational, then any value of [math]a^b[/math] is a transcendental number. And in the case of [math]i^{\>\!i}[/math] and [math]e^\pi[/math], there is also the identity: [math]e^{i\pi} = -1[/math] However, there is no known corresponding identity for [math]\pi^e[/math], and thus it is not known whether or not this number is transcendental. But Schanuel's conjecture, if proved, would establish many nontrivial combinations of [math]e[/math], [math]\pi[/math], algebraic numbers and elementary functions to be transcendental, including [math]\pi^e[/math]. [If the above LaTex doesn't render, please refresh the browser.]

When you outsource your brain to a computer, isn't that inevitable?

[math]2^{\sqrt{2}}[/math] is known to be transcendental. [math]i[/math] is not rational, but is algebraic. [math]i^{\,i}[/math] is known to be transcendental. [math]e^\pi[/math] is known to be transcendental. It is not known whether or not [math]\pi^e[/math] is transcendental.