The question of multiple dimensions in quantum mechanics:

[quant]

Quantum mechanics is based almost entirely Schrodinger’s wave equation and later the Schrodinger wave function. What is the difference? A wave function is a mathematical function of the state coordinates of some system that represents a “complete description” of that system in quantum mechanics. The Schrödinger equation is a differential equation that describes how the wave function evolves in time. At the time that Schrodinger got interested in quantum mechanics the big question of the day was how atoms could exist. According to calculations made by Larmor an Irish physicist, the electron which is a charged particle should radiate away all of its energy and fall into the nucleus in 10 pico seconds (Ten trillionths of a second) as a corollary to this: matter, you and I, the solar system, the Universe itself should not exist. It was a problem. Around this time a young physics graduate from France by the name of Louis De Broglie introduced the idea of wave-particle duality. Schrodinger was immediately smitten by this idea and began to formulate an equation that would depict the electron as a standing wave (i.e., a wave-packet).

The wave packet model of the electron suggested by Schrodinger enjoyed a runaway success when it was first introduced. However, it was soon realised that Schrodinger’s wave equation could only describe the hydrogen atom with its single electron. When attempts were made to expand the theory to the atoms of other elements, disaster struck. This was because, a multi-dimensional space was required for this 'standing wave' model of the electron. Helium with its two electrons required a 6-dimensional space, lithium with three electrons got 9 dimensions and uranium with 92 electrons needed 276 dimensions. These were real spatial dimensions not degrees of freedom. What is the difference? There can be an infinite number of degrees of freedom, for instance the distance between the base of the nose and the lip (philtrum) is used as a degree of freedom in face recognition software. Returning to the 3n dimensions required by the Schrodinger equation it is mind boggling to think that if Uranium with its 92 electrons needs 276 dimensions to describe, what would happen when the same Schrodinger wave function is used to describe the propagation of light (n.b: it is so used). The inescapable conclusion is that an infinite number of dimensions would be needed, if one is tempted to think in terms of degrees of freedom. Here is a quote from Max Born: The Father of quantum mechanics:

“We have two possibilities. Either we use waves in space of more than three dimensions…..or we remain in three dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude , and replace it with a purely mathematical concept into which we cannot enter.” Yet one has to wonder how something that can be ethically unacceptable in the ‘real’ world can be perfectly justifiable in the abstract ‘mathematical’ world. …. " Max Born

Surely, such insanely difficult obstacles should convince any reasonable sane person to try other solutions? Also think about how acceptable the'abstract 'mathematical world' described by Max Born is to describe reality.

The point is no matter how many and how varied are the solutions suggested to solve this problem such as the Fock-Hartree theory or the DFT (Density Functional Theory) nothing lessens the immensity of infinite multiple dimensions. Surely other better solutions must exist. But physicists smug in their own invincibility do not wish to examine other possibilities.

 

[exchemist]

Quantum mechanics is based almost entirely Schrodinger’s wave equation and later the Schrodinger wave function. What is the difference? A wave function is a mathematical function of the state coordinates of some system that represents a “complete description” of that system in quantum mechanics. The Schrödinger equation is a differential equation that describes how the wave function evolves in time. At the time that Schrodinger got interested in quantum mechanics the big question of the day was how atoms could exist. According to calculations made by Larmor an Irish physicist, the electron which is a charged particle should radiate away all of its energy and fall into the nucleus in 10 pico seconds (Ten trillionths of a second) as a corollary to this: matter, you and I, the solar system, the Universe itself should not exist. It was a problem. Around this time a young physics graduate from France by the name of Louis De Broglie introduced the idea of wave-particle duality. Schrodinger was immediately smitten by this idea and began to formulate an equation that would depict the electron as a standing wave (i.e., a wave-packet).

The wave packet model of the electron suggested by Schrodinger enjoyed a runaway success when it was first introduced. However, it was soon realised that Schrodinger’s wave equation could only describe the hydrogen atom with its single electron. When attempts were made to expand the theory to the atoms of other elements, disaster struck. This was because, a multi-dimensional space was required for this 'standing wave' model of the electron. Helium with its two electrons required a 6-dimensional space, lithium with three electrons got 9 dimensions and uranium with 92 electrons needed 276 dimensions. These were real spatial dimensions not degrees of freedom. What is the difference? There can be an infinite number of degrees of freedom, for instance the distance between the base of the nose and the lip (philtrum) is used as a degree of freedom in face recognition software. Returning to the 3n dimensions required by the Schrodinger equation it is mind boggling to think that if Uranium with its 92 electrons needs 276 dimensions to describe, what would happen when the same Schrodinger wave function is used to describe the propagation of light (n.b: it is so used). The inescapable conclusion is that an infinite number of dimensions would be needed, if one is tempted to think in terms of degrees of freedom. Here is a quote from Max Born: The Father of quantum mechanics:

“We have two possibilities. Either we use waves in space of more than three dimensions…..or we remain in three dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude , and replace it with a purely mathematical concept into which we cannot enter.” Yet one has to wonder how something that can be ethically unacceptable in the ‘real’ world can be perfectly justifiable in the abstract ‘mathematical’ world. …. " Max Born

Surely, such insanely difficult obstacles should convince any reasonable sane person to try other solutions? Also think about how acceptable the'abstract 'mathematical world' described by Max Born is to describe reality.

The point is no matter how many and how varied are the solutions suggested to solve this problem such as the Fock-Hartree theory or the DFT (Density Functional Theory) nothing lessens the immensity of infinite multiple dimensions. Surely other better solutions must exist. But physicists smug in their own invincibility do not wish to examine other possibilities.

Can you provide a reference and context for the quote from Born? It looks to me as if it may be no more than his realisation that the the wave function does not itself have a physical meaning. It is a mathematical object that, when operated on by the operator for a physical property, yields a result for that property.

 

[QuarkHead]

physicists smug in their own invincibility do not wish to examine other possibilities.

A person who voluntarily joins a science disussion forum cannot expect to make many friends there with comments like this.

But for your information, since the 1920s and thanks to P.A.M Dirac, the wave function is modeled as an elemnt in an infinite-dimensional Hilbert space i.e a vector space. And the term "dimension" in this context doesn't mean something wierd like you might have seen on Star Trek. It is really rather straightforward (but that persumably won't interest you)

Learn some respet or go elsewhere for your kicks

 

[exchemist]

A person who voluntarily joins a science disussion forum cannot expect to make many friends there with comments like this.

But for your information, since the 1920s and thanks to P.A.M Dirac, the wave function is modeled as an elemnt in an infinite-dimensional Hilbert space i.e a vector space. And the term "dimension" in this context doesn't mean something wierd like you might have seen on Star Trek. It is really rather straightforward (but that persumably won't interest you)

Learn some respet or go elsewhere for your kicks

Yes I wondered whether Hilbert space might be where Born was, at least by implication, going, in the quoted passage. But It’s impossible to know without context. They were feeling their way at that time.

Obviously in mathematics one can have as many dimensions as one likes. It is only when trying to attach a physical meaning to each that one is limited.

 

[QuarkHead]

Well, I amnot the right person to try and attach physical meaning to anything.

But to continue with Max Born - he saw it made litle sense in the socalled "real world" for a physical function to return values between -1 and 1, which is what a wave function does. Rather, he suggested (as you youself know better than anyone) that a real meaning can be attached to its square which, having values between 0 and1 can be taken as a probility amplitude.

And again, as you well know, a simple argument leads from this to the so-called ubcetainty principle.

P.S. I once had the privilige of attending a lecture by you old mentor (?) Peter Atkins, where he explained this beautifully to a non-chemist audience

 

[exchemist]

Well, I amnot the right person to try and attach physical meaning to anything.

But to continue with Max Born - he saw it made litle sense in the socalled "real world" for a physical function to return values between -1 and 1, which is what a wave function does. Rather, he suggested (as you youself know better than anyone) that a real meaning can be attached to its square which, having values between 0 and1 can be taken as a probility amplitude.

And again, as you well know, a simple argument leads from this to the so-called ubcetainty principle.

P.S. I once had the privilige of attending a lecture by you old mentor (?) Peter Atkins, where he explained this beautifully to a non-chemist audience

Atkins was a genius at lecturing. My maths tutor told us Atkins was a real performer - would get really worked up before he went on to lecture. The series he ran on Quantum Chemistry, which I took as a supplementary subject, was the best thing in the four years I was there. But also by far the hardest.

But Atkins was not my tutor - thank God. He had a reputation for being a real tartar. My physical chemistry tutor was Richard Wayne, now deceased, who was a photochemist and far more genial. But he was pretty strong on molecular QM.

Back to the science, yes Born’s insight was that the square modulus (the wave function, being complex, has to be multiplied by its complex conjugate) corresponded to a probability density function, which you could integrate over a volume of space to determine the probability of detecting the entity within it. Which is sort of like the way the intensity of light radiation is the square of the amplitude.

 

[Pinball1970]

P.S. I once had the privilige of attending a lecture by you old mentor (?) Peter Atkins, where he explained this beautifully to a non-chemist audience

Got his phys Chem book still

 

[Pinball1970]

Quantum mechanics is based almost entirely Schrodinger’s wave equation and later the Schrodinger wave function. What is the difference? A wave function is a mathematical function of the state coordinates of some system that represents a “complete description” of that system in quantum mechanics. The Schrödinger equation is a differential equation that describes how the wave function evolves in time. At the time that Schrodinger got interested in quantum mechanics the big question of the day was how atoms could exist. According to calculations made by Larmor an Irish physicist, the electron which is a charged particle should radiate away all of its energy and fall into the nucleus in 10 pico seconds (Ten trillionths of a second) as a corollary to this: matter, you and I, the solar system, the Universe itself should not exist. It was a problem. Around this time a young physics graduate from France by the name of Louis De Broglie introduced the idea of wave-particle duality. Schrodinger was immediately smitten by this idea and began to formulate an equation that would depict the electron as a standing wave (i.e., a wave-packet).

The wave packet model of the electron suggested by Schrodinger enjoyed a runaway success when it was first introduced. However, it was soon realised that Schrodinger’s wave equation could only describe the hydrogen atom with its single electron. When attempts were made to expand the theory to the atoms of other elements, disaster struck. This was because, a multi-dimensional space was required for this 'standing wave' model of the electron. Helium with its two electrons required a 6-dimensional space, lithium with three electrons got 9 dimensions and uranium with 92 electrons needed 276 dimensions. These were real spatial dimensions not degrees of freedom. What is the difference? There can be an infinite number of degrees of freedom, for instance the distance between the base of the nose and the lip (philtrum) is used as a degree of freedom in face recognition software. Returning to the 3n dimensions required by the Schrodinger equation it is mind boggling to think that if Uranium with its 92 electrons needs 276 dimensions to describe, what would happen when the same Schrodinger wave function is used to describe the propagation of light (n.b: it is so used). The inescapable conclusion is that an infinite number of dimensions would be needed, if one is tempted to think in terms of degrees of freedom. Here is a quote from Max Born: The Father of quantum mechanics:

“We have two possibilities. Either we use waves in space of more than three dimensions…..or we remain in three dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude , and replace it with a purely mathematical concept into which we cannot enter.” Yet one has to wonder how something that can be ethically unacceptable in the ‘real’ world can be perfectly justifiable in the abstract ‘mathematical’ world. …. " Max Born

Surely, such insanely difficult obstacles should convince any reasonable sane person to try other solutions? Also think about how acceptable the'abstract 'mathematical world' described by Max Born is to describe reality.

The point is no matter how many and how varied are the solutions suggested to solve this problem such as the Fock-Hartree theory or the DFT (Density Functional Theory) nothing lessens the immensity of infinite multiple dimensions. Surely other better solutions must exist. But physicists smug in their own invincibility do not wish to examine other possibilities.

So what is your (concise) question?

 

[exchemist]

Got his phys Chem book still

And I still keep his "Molecular Quantum Mechanics" handy, though I've forgotten a lot of the maths. It was 50 years ago now......

 

[quant]

It is difficult to be concise when dealing with multiverses that are considered by current physics to be real and stem from conclusions made in the OP about multiple dimensions. I will pose a question to you. If an alternate to the electron cloud theory is suggested, would you be willing to hear it?

In 1947, physicist Willis Eugene Lamb and his partner Retherford investigated the energy levels of atoms exposed to microwave frequencies. By plotting the critical magnetic field strength against various microwave frequencies, they found an unexpected energy difference in hydrogen atoms, revealing the Lamb shift. This shift arises from interactions between virtual photons from vacuum energy fluctuations and the electron orbiting the hydrogen nucleus. Their work not only confirmed the existence of virtual particles but also solidified quantum electrodynamics (QED) as a key theory in quantum physics. Why didn’t this concept of ‘virtual particles gain wider coverage? The answer is that security was at a maximum, the discovery of the Lamb shift was made just one year after Hiroshima and the exploding of the first atomic Bomb. Therefore the concept of virtual particles did not gain traction and remained within the purview of the few scientists involved in the project.

When an electron emits and absorbs a virtual photon, these interactions are significant and occur across different atomic regions, from the nucleus to the valence shell. One possibility is that this process stabilizes the electron's orbit by requiring it to emit and absorb specific energies, rather than random fluctuations. This interaction explains why an accelerating electron does not spiral into the nucleus; by regulating its energy through the emission and absorption of virtual photons, it minimizes the chance of collapsing into the nucleus.

The staggering implications of the conclusion that electrons are undergoing constant virtual interactions through the emission and absorption of 'virtual photons' is that the whole notion of wave particle duality is false. Wave-particle duality is no longer needed to explain the question of how the electron maintains its stability within the atom: which is the primary reason that wave-particle duality had been introduced in the first place. By constantly emitting and absorbing virtual ‘photons’ as it orbits the nucleus, the electron is in effect self-stabilizing its energy and the possibility of its spiraling into the nucleus no longer arises. Physics is all about not only explaining reality but also accepting it. Like it or not, this new explanation for the stability of the electron around the nucleus not only does away with wave particle duality but also brings the explanation for electron stability around the nucleus in keeping with the explanations for the stability of the nucleus itself, where wave-particle duality was not resorted to but directly replaced with the concept of virtual particle exchange.

 

[exchemist]

It is difficult to be concise when dealing with multiverses that are considered by current physics to be real and stem from conclusions made in the OP about multiple dimensions. I will pose a question to you. If an alternate to the electron cloud theory is suggested, would you be willing to hear it?

In 1947, physicist Willis Eugene Lamb and his partner Retherford investigated the energy levels of atoms exposed to microwave frequencies. By plotting the critical magnetic field strength against various microwave frequencies, they found an unexpected energy difference in hydrogen atoms, revealing the Lamb shift. This shift arises from interactions between virtual photons from vacuum energy fluctuations and the electron orbiting the hydrogen nucleus. Their work not only confirmed the existence of virtual particles but also solidified quantum electrodynamics (QED) as a key theory in quantum physics. Why didn’t this concept of ‘virtual particles gain wider coverage? The answer is that security was at a maximum, the discovery of the Lamb shift was made just one year after Hiroshima and the exploding of the first atomic Bomb. Therefore the concept of virtual particles did not gain traction and remained within the purview of the few scientists involved in the project.

When an electron emits and absorbs a virtual photon, these interactions are significant and occur across different atomic regions, from the nucleus to the valence shell. One possibility is that this process stabilizes the electron's orbit by requiring it to emit and absorb specific energies, rather than random fluctuations. This interaction explains why an accelerating electron does not spiral into the nucleus; by regulating its energy through the emission and absorption of virtual photons, it minimizes the chance of collapsing into the nucleus.

The staggering implications of the conclusion that electrons are undergoing constant virtual interactions through the emission and absorption of 'virtual photons' is that the whole notion of wave particle duality is false. Wave-particle duality is no longer needed to explain the question of how the electron maintains its stability within the atom: which is the primary reason that wave-particle duality had been introduced in the first place. By constantly emitting and absorbing virtual ‘photons’ as it orbits the nucleus, the electron is in effect self-stabilizing its energy and the possibility of its spiraling into the nucleus no longer arises. Physics is all about not only explaining reality but also accepting it. Like it or not, this new explanation for the stability of the electron around the nucleus not only does away with wave particle duality but also brings the explanation for electron stability around the nucleus in keeping with the explanations for the stability of the nucleus itself, where wave-particle duality was not resorted to but directly replaced with the concept of virtual particle exchange.

Why are you talking about multiverses? They form no part of the theory of Quantum Mechanics.

And this stuff about virtual photons from vacuum fluctuations stablising electrons and doing away with wave-particle duality is ridiculous. The Lamb Shift is a tiny effect. The interactions are not "significant". And crucially, no mechanism is offered for how this alleged process would work to achieve the alleged result. It seems to be no more than handwaving, using sciency words. And I notice no explanation is offered for the Uncertainty Principle, which is in many ways one of the most fundamental predictions of QM and which is observed experimentally.

We have a perfectly good theory for electron orbitals and have had for a century now. It is amazingly successful, correctly predicting everything from the structure of the Periodic Table and atomic spectra through to chemical bonding and molecular spectra.

So the first task for anyone wanting to promote rival theory is to answer the question: what problem would a new theory solve that QM cannot address?

P.S. I look forward to you providing the reference and context for that quote from Born which I asked you for earlier. I do not want to take any of this further until that is nailed down, because I suspect you have misunderstood what he was saying.

 

[quant]

P.S. I look forward to you providing the reference and context for that quote from Born which I asked you for earlier. I do not want to take any of this further until that is nailed down, because I suspect you have misunderstood what he was saying.

exchemist: The quote: “We have two possibilities. Either we use waves in space of more than three dimensions…..or we remain in three dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude , and replace it with a purely mathematical concept into which we cannot enter.” Yet one has to wonder how something that can be ethically unacceptable in the ‘real’ world can be perfectly justifiable in the abstract ‘mathematical’ world. …. " Max Born
Is from: Principles of Optics By Max Born and Emil Wolf. First Published in 1959. The 6th Edition of the book has been published by the Pergamon Press.

 

[James R]

Quantum mechanics is based almost entirely Schrodinger’s wave equation and later the Schrodinger wave function.

No. The Schrodinger wave function is an early description of quantum mechanics. Even in the early days, there were alternative formations such as the Heisenberg picture.

Later on, quantum field theories were developed, which do not make any reference to Schrodinger wave functions.

Also, the Schrodinger equation is non-relativistic. There are other equations of quantum physics. For instance, the Dirac equation is a relativistic replacement for the Schrodinger equation.

The wave packet model of the electron suggested by Schrodinger enjoyed a runaway success when it was first introduced. However, it was soon realised that Schrodinger’s wave equation could only describe the hydrogen atom with its single electron. When attempts were made to expand the theory to the atoms of other elements, disaster struck. This was because, a multi-dimensional space was required for this 'standing wave' model of the electron. Helium with its two electrons required a 6-dimensional space, lithium with three electrons got 9 dimensions and uranium with 92 electrons needed 276 dimensions. These were real spatial dimensions not degrees of freedom.

That is incorrect. Larger atoms than hydrogen can be (and are) described in the Schrodinger picture. Only the usual 3 space + 1 time physical dimensions are required. The wave functions are more complicated than for hydrogen, but can - to a good approximation - be written as a set of hydrogen-like wavefunctions. These multi-electron solutions are precisely what allows us to understand the electron configurations of multi-electron atoms, for example.

Here is a quote from Max Born: The Father of quantum mechanics:

“We have two possibilities. Either we use waves in space of more than three dimensions…..or we remain in three dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude , and replace it with a purely mathematical concept into which we cannot enter.” Yet one has to wonder how something that can be ethically unacceptable in the ‘real’ world can be perfectly justifiable in the abstract ‘mathematical’ world. …. " Max Born

And then Born went on to suggest a probabilistic interpretation of the wave function. Did he not?

And that is stock standard quantum physics in the modern era. Is it not?

The point is no matter how many and how varied are the solutions suggested to solve this problem such as the Fock-Hartree theory or the DFT (Density Functional Theory) nothing lessens the immensity of infinite multiple dimensions. Surely other better solutions must exist. But physicists smug in their own invincibility do not wish to examine other possibilities.

It sounds like you're not familiar with the peer reviewed Physics literature. Take a look some time. There, you'' find tons of physicists happily engaged in detailed examinations of "other possibilities".

Where are you getting your information from? youtube videos?

 

[exchemist]

exchemist: The quote: “We have two possibilities. Either we use waves in space of more than three dimensions…..or we remain in three dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude , and replace it with a purely mathematical concept into which we cannot enter.” Yet one has to wonder how something that can be ethically unacceptable in the ‘real’ world can be perfectly justifiable in the abstract ‘mathematical’ world. …. " Max Born
Is from: Principles of Optics By Max Born and Emil Wolf. First Published in 1959. The 6th Edition of the book has been published by the Pergamon Press.

OK and how does the book continue, after quoting this passage? What conclusion does it draw from this quotation? Can you post a copy of the relevant page, so we can see the context?

 

[quant]

OK and how does the book continue, after quoting this passage? What conclusion does it draw from this quotation? Can you post a copy of the relevant page, so we can see the context?

That is beyond my scope at present, I had come across this quote many years ago and had noted it, verified that it was a genuine Max Born quote and that is the limit of what I know. The exact context in which it was used is beyond me, although I am sure a similar quote also occurs in Max Born's published 'Letters to Einstein".

 

[quant]

That is incorrect. Larger atoms than hydrogen can be (and are) described in the Schrodinger picture. Only the usual 3 space + 1 time physical dimensions are required. The wave functions are more complicated than for hydrogen, but can - to a good approximation - be written as a set of hydrogen-like wavefunctions. These multi-electron solutions are precisely what allows us to understand the electron configurations of multi-electron atoms, for example.

Surely a simplification, when it is well known that when used to describe larger atoms Schrodinger's equation becomes so bulky and unwieldy as to be unusable. What is the point in describing one aspect and not the other, it (Schrodinger's equation) is also inaccurate when describing larger atoms.
Finally let me state that maths is a language like any other and like any other language it [maths] can say things that are not true. When a statement like 'Pigs can fly." is made it does not mean that, just because it has been said it is true.

 

[exchemist]

That is beyond my scope at present, I had come across this quote many years ago and had noted it, verified that it was a genuine Max Born quote and that is the limit of what I know. The exact context in which it was used is beyond me, although I am sure a similar quote also occurs in Max Born's published 'Letters to Einstein".

Ah well then, you should be careful how you interpret it. "Quote mining", the taking of a snippet of text out of its context and attaching a meaning to it that may not be what the speaker intended, is a dubious technique, sometimes used by people who are willing to use dishonest means to score a rhetorical point.

As QuarkHead , James and I suspect, it seems likely that Born was describing the thought process by which he came to give the meaning to the wave function that it has had ever since, viz. a sort of square root of probability density. He was evidently struggling to give physical meaning to the amplitude of the wave function and regretfully concluding it does not have one! But, if you multiply it by its complex conjugate and integrate over a region of space, then it has meaning: the probability of detecting the quantum entity in that region.

And as QuarkHead reminds us, solutions to the Schrödinger equation can be represented as mutually orthogonal vectors* in Hilbert space, which implies an unlimited number of mathematical dimensions. However these cannot be associated with the 3 dimensions of physical space, obviously.

*https://en.wikipedia.org/wiki/Orthogonal_functions

4.5: Eigenfunctions of Operators are Orthogonal

The eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position …

chem.libretexts.org

 

[exchemist]

Surely a simplification, when it is well known that when used to describe larger atoms Schrodinger's equation becomes so bulky and unwieldy as to be unusable. What is the point in describing one aspect and not the other, it (Schrodinger's equation) is also inaccurate when describing larger atoms.
Finally let me state that maths is a language like any other and like any other language it [maths] can say things that are not true. When a statement like 'Pigs can fly." is made it does not mean that, just because it has been said it is true.

But this is not just a QM problem. It is impossible to derive an exact solution of a multi-body problem even in classical physics. So even if you were to use a Bohr model of the atom, with electrons as Newtonian "particles", travelling along classical defined paths, you would not be able to get exact solutions for anything more complex than the hydrogen atom.

As it is, in QM, approximate methods are used for multi-electron atoms. I quote the relevant section of the Wiki article on atomic orbitals:

Formal quantum mechanical definition​

Atomic orbitals may be defined more precisely in formal quantum mechanical language. They are approximate solutions to the Schrödinger equation for the electrons bound to the atom by the electric field of the atom's nucleus. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, is approximated by an expansion (see configuration interaction expansion and basis set) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their spin component, one speaks of atomic spin orbitals.) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this independent-particle model of products of single electron wave functions.[8] (The London dispersion force, for example, depends on the correlations of the motion of the electrons.)

In atomic physics, the atomic spectral lines correspond to transitions (quantum leaps) between quantum states of an atom. These states are labeled by a set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s2 2s2 2p6 for the ground state of neon-term symbol: 1S0).

This notation means that the corresponding Slater determinants have a clear higher weight in the configuration interactionexpansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given transition. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are fermions ruled by the Pauli exclusion principle and cannot be distinguished from each other.[9] Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when electron correlation is large.

Fundamentally, an atomic orbital is a one-electron wave function, even though many electrons are not in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital visualization heavily influenced by the Hartree–Fock approximation, which is one way to reduce the complexities of molecular orbital theory.

And, as I said previously, this technique is a triumphant success, underpinning the whole of modern chemistry. Anyone proposing an alternative model has to be able to do as least as well as the current one - and then show some advantage over the current model. That's quite a tall order.

 

[quant]

And as @QuarkHead reminds us, solutions to the Schrödinger equation can be represented as mutually orthogonal vectors* in Hilbert space, which implies an unlimited number of mathematical dimensions. However these cannot be associated with the 3 dimensions of physical space, obviously.

There seems to be a sort of schism here, which is precisely the point of interest. According to your thinking the Multiverse, which is a direct consequence of the wave function, or more accurately a direct consequence of the collapse of the wave function, is a purely abstract formulation, having no existence in reality. Here is a list of leading scientists (together with citations) who believe in the existence of multiple Universes and are not afraid to state their belief. Remember these multiple universes, each supposedly slightly different from the one we live in. come into existence every time light travels from A to B and the wave function collapses. (Hard interpretation of quantum mechanics formulated at the Copenhagen Conference of 1927.) Here is a list of scientist who favour the theory of Multiple Universes.

1. Max Tegmark - A physicist known for his work in cosmology and the multiverse hypothesis. In his paper "The Multiverse Hierarchy" (2003), Tegmark categorizes different types of multiverses.

Citation: Tegmark, M. (2003). "The Multiverse Hierarchy." Physical Review D, 67(4), 043508.

2. Sean Carroll - A theoretical physicist who has discussed the implications of the multiverse in various articles and talks. His book, “From Eternity to Here”, explores concepts related to time and the multiverse.

Citation: Carroll, S. (2010). ‘From Eternity to Here: The Quest for the Ultimate Theory of Time’. Dutton.

3. Brian Greene - An influential physicist and string theorist who has popularized multiverse concepts in his books, including “The Hidden Reality”.

Citation: Greene, B. (2011). ‘The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos’. Knopf.

4. Andrei Linde - One of the key figures in inflationary cosmology, Linde has proposed models that lead to a multiverse. His work on chaotic inflation has been foundational in this area.

Citation: Linde, A. (1983). "Chaotic Inflation." Physical Review D, 49(2), 748-754.

5. Leonard Susskind - A theoretical physicist who contributed to string theory and the concept of the multiverse. He discusses these ideas in his book, “The Cosmic Landscape”.

Citation: Susskind, L. (2005). ‘The Cosmic Landscape: String Theory and the Illusion of Intelligent Design’. Little, Brown and Company.

6. David Deutsch - A physicist and pioneer of quantum computation, Deutsch argues for the multiverse interpretation of quantum mechanics in his book “The Beginning of Infinity”.

Citation: Deutsch, D. (2011). ‘The Beginning of Infinity: Explanations that Transform the World’. Viking.

These scientists have made significant contributions to the multiverse discussion through their research, books, and public lectures. I have gone through the Wikipedia reference you had quoted but this is the reality

 

[exchemist]

There seems to be a sort of schism here, which is precisely the point of interest. According to your thinking the Multiverse, which is a direct consequence of the wave function, or more accurately a direct consequence of the collapse of the wave function, is a purely abstract formulation, having no existence in reality. Here is a list of leading scientists (together with citations) who believe in the existence of multiple Universes and are not afraid to state their belief. Remember these multiple universes, each supposedly slightly different from the one we live in. come into existence every time light travels from A to B and the wave function collapses. (Hard interpretation of quantum mechanics formulated at the Copenhagen Conference of 1927.) Here is a list of scientist who favour the theory of Multiple Universes.

1. Max Tegmark - A physicist known for his work in cosmology and the multiverse hypothesis. In his paper "The Multiverse Hierarchy" (2003), Tegmark categorizes different types of multiverses.

Citation: Tegmark, M. (2003). "The Multiverse Hierarchy." Physical Review D, 67(4), 043508.

2. Sean Carroll - A theoretical physicist who has discussed the implications of the multiverse in various articles and talks. His book, “From Eternity to Here”, explores concepts related to time and the multiverse.

Citation: Carroll, S. (2010). ‘From Eternity to Here: The Quest for the Ultimate Theory of Time’. Dutton.

3. Brian Greene - An influential physicist and string theorist who has popularized multiverse concepts in his books, including “The Hidden Reality”.

Citation: Greene, B. (2011). ‘The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos’. Knopf.

4. Andrei Linde - One of the key figures in inflationary cosmology, Linde has proposed models that lead to a multiverse. His work on chaotic inflation has been foundational in this area.

Citation: Linde, A. (1983). "Chaotic Inflation." Physical Review D, 49(2), 748-754.

5. Leonard Susskind - A theoretical physicist who contributed to string theory and the concept of the multiverse. He discusses these ideas in his book, “The Cosmic Landscape”.

Citation: Susskind, L. (2005). ‘The Cosmic Landscape: String Theory and the Illusion of Intelligent Design’. Little, Brown and Company.

6. David Deutsch - A physicist and pioneer of quantum computation, Deutsch argues for the multiverse interpretation of quantum mechanics in his book “The Beginning of Infinity”.

Citation: Deutsch, D. (2011). ‘The Beginning of Infinity: Explanations that Transform the World’. Viking.

These scientists have made significant contributions to the multiverse discussion through their research, books, and public lectures. I have gone through the Wikipedia reference you had quoted but this is the reality

I repeat: the multiverse idea is not part of QM. It is by no means "a direct consequence of the wave function" as you put it. That is a complete misunderstanding on your part.

Are you confusing it with the "Many Worlds" interpretation of QM, perhaps? That, too, is not part of the theory of QM, but is a metaphysical speculative interpretation, one of many, about what QM may be saying about the nature of the world. It has its supporters and detractors - not to mention those who are utterly indifferent to it - but as it makes no testable predictions it is not part of science.