The Schrödinger time dependent equation (STDE) for the hydrogen atom tells how the wave function of the bound electron moves. Since the equation generates a unitary linear group $U_t=\exp((-i/\hbar)H)$ the energy of the wave-like electron never changes. But the physical fact is that the electron absorbs and radiates energy.
The STDE is so much in contradiction with physical facts that probabilistic quantum jumps had to be invented. Quantum jumps in turn required point-like (infinitely small) electrons, probabilistic interpretation of wave functions, uncertainty and indeterminacy principles, quantum spin (instead of the obvious wave rotations), mysterious collapse of wave packets, mind dependent measurements, entanglements, etc. all in an attempt to rescue the inadequate STDE.
The verdict is clear: The STDE cannot be justified and is therefore a physical mistake. Too much human effort has been wasted in the futile quest of making sense of this colossal theoretical error. Schrödinger himself opposed quantum mechanics, but others succeeded in enhancing and spreading the confusion.
Any appearance of correctness of the STDE is a mirage that arises from the undeniable, miraculous success of the Schrödinger linear self adjoint operator $H$, actually from its eigenvalues $-\lambda_n$, eigenspaces $E_n$ and eigenfunctions $\psi_n$, which are objects much different from an evolution equation. All them, namely $H$, $-\lambda_n$, $E_n$ and $\psi_n$, can be perfectly ingrained to the correct evolution equation. The square of the normalized wave function is the electron charge density, an interpretation originally proposed by Schrödinger.
The correct hydrogen equation is set in a space of states where both electrons and photons coexist. Then the trajectories of the correct evolution equation (not the STDE) can deterministically describe the simultaneous, synchronized movement of both the electron and photon wave functions.
During the synchronized movement an incoming photon, having the correct energy and appropriately located, raises a stationary electron to a new higher stationary configuration while the photon is absorbed. And an excited stationary electron would decay to a lower stationary electron configuration while radiating an outgoing photon.
The movement of the electron configurations and the simultaneous absorption or radiation of photons are continuous, causal, deterministic processes with sensitive dependency on initial conditions, as common sense dictates they should be.
For the correct hydrogen equation see this brief, three page article:
Preprint The Quantum Captivity of Physics
Regards,
Daniel Crespin
Kindly take a look at the following germane links:
Article The time-dependent Schrödinger equation: the need for the Ha...
https://www.researchgate.net/post/What-is-significance-of-Schroedinger-equation-in-quantum-mechanics
and
https://arxiv.org/pdf/1007.4243.pdf
I agree with your argument. What you must keep in mind is that at the time Schrodinger made his equations he was missing information about entanglement with this new information it is clear that his imaginary numbers were needed to make the equations work. Now that we have a new solution that includes entanglement the solution is extremely simple. Shrodinger did amazing work for what he knew. We know can calculate the exact position and velocity of the electron no longer using Schrodinger's fuzzy quantum mechanics. By adding the correct entanglement into the picture we now for the first time can calculate the decoherence cycle or wave collapse. It is found that all the information we were missing is in the wave collapse. I recommend you read Article The discovery of de Broglie and Bohm's pilot wave. The unpre...
Ijaz Durrani
I appreciate your interest.
About the article by Araujo et al that you are referring, it seems reasonable (and familiar) that the energy operator should in general be self adjoint, and that Stone theorem is important in operator theory and quantum mechanics.
Answering the question by Azfal, which is rather a statement of bewilderment, will not be brief. It requires:
1.- A detailed, didactic justification of the Schrödinger self adjoint operator $H$.
In particular, why the operator $H$ which is a linear transformation from a vector space to itself, is called energy, when energies are generally quadratic scalar valued functions. And then
2.- A clear explanation of the relation of $H$ with the Schrödinger time depending equation.
Literate believers of QM should be willing to provide explanations. My own viewpoint is that the time dependent equation is a terrible scientific mistake.
My own papers on quantum mechanics are available here
https://www.researchgate.net/profile/Daniel_Crespin/research
and contain technical descriptions of the STDE, but I am extremely critical of this time dependent equation hence my articles are unsuitable for beginning students that hope to absorb and become followers of Quantism.
Finally, about the (mostly mathematical?) paper by Bondar et al it seems to be written for readers initiated in their specific topic. A critical examination is beyond my present possibilities. Although I am familiar with some distribution theory it is only as explained in the book of Laurent Schwartz and the treatise of Gelfand and Shilov. In my humble opinion the substantial problems of quantum mechanics should be studied giving preference to physical ideas over mathematical abstractions. The physics must be first understood, then the mathematics can be profitably applied. Let me congratulate you for having been able to extract benefit from this article.
Thanks and cordial regards,
Daniel Crespin
The Schrödinger equation can be derived from classical physics, following Hamilton's "wave equation" (which is a rearrangement of energy terms so they are in the form analogous the the form of a wave equation) provided the wavelength of any acceptable wave must correspond to the generation of a quantum of action h. To that extent, the answer to the titled question is yes, of course it can. The problem has always been the equation expresses itself in terms of the major term ψ, which raises the question, what is ψ and what causes it?
The questions raised in the opening of this thread do not comprise any reason to question the equation itself, but rather the use or misuse of it since then. Thus "probabilistic interpretation of wave functions, uncertainty and indeterminacy principles, quantum spin (instead of the obvious wave rotations), mysterious collapse of wave packets, mind dependent measurements, entanglements, etc. all in an attempt to rescue the inadequate STDE." I disagree. These are not required by the basic equation, but rather are clip-ons, many of which are imposed by the assumption that ψ.ψ* is fundamentally a probability. In my guidance wave interpretation, I disagree with that, and consider the probabilities merely emergent. Admittedly, I assume there is a physical wave that causes effects such as diffraction, partial reflection, and other wave-like properties, but assuming ψ is physical hardly contradicts the equation. If you do this, and apply a couple of other constraints (Euler's mathematics and the requirement that the wave pulse actually causes diffraction effects) you can derive the Uncertainty Principle (slightly more uncertain for linear motion), the Exclusion Principle, etc.)
So my answer is we can justify the basic equation, but not necessaril\y the clip--ons, which have to be cpnsidered individually.
Hello Ian Miller and all readers
The STDE or any other imaginable equation that respects certain basic formal rules can be considered mathematically correct. Equations are the substance of mathematics. Illustrious names are associated to a variety of equations and their solutions: Pythagoras, Diophantus, Tartaglia, Cardano, Ricatti, Volterra. The list is very, very long.
I am not questioning the STDE as a mathematical equation. In the case of the quantum hydrogen atom the equation is a linear partial differential equation and mathematically the solutions are contained in the expression $\psi_t=\exp((-i/\hbar)Ht)\psi_0$.
But the behavior of the correct solutions of the mathematically correct STDE contradicts some very basic physics of hydrogen. The correct mathematics turns out to be inadequate when taken as a physical model. Here is why.
1.- The STDE is a mathematically correct equation.
2.- The solutions of the STDE are mathematically correct solutions.
3.- The behavior of an electron bound in hydrogen is physically well understood.
4.- The electron makes transitions between certain states that have energies within a well defined discrete set of energies.
5.- According to the energy conservative STDE such transitions should never happen.
On one hand the energy of the mathematical wave function remains forever constant under the STDE. On the other hand the energy of the physical electron keeps going up and down as it absorbs and radiates photons. Therefore the STDE is dynamically useless. (As a matter of fact the STDE does not even has motionless waves except the zero wave function, $\psi\cong 0$.)
When confronted with this inescapable contradiction Schrödinger said: "If I had known we were going to go on having all this damned quantum-jumping, I would never have got involved in the subject". Is hard to imagine more expressive words.
Quantum mechanics is so far removed from physics that stationary energies and waves are often defined in terms of algebraic notions! The algebraically conceived eigenvalues and eigenfunctions are the stationary physical objects! The usual quantum remedy is to tell that the stationary waves may move but they are standing waves. In such case, for the STDE, all waves are standing waves. It starts to look as if, for the STDE, truly stationary waves do not exists. About this specific inconsistency an effective quantum solution is to introduce Hilbert rays. However, with Hilbert rays the contradiction between energy conservation in the mathematical ray model and change of energy level in the physical electron persists.
Why was it that the STDE was accepted as a rule that mathematically reflects the behavior of hydrogen?
A naive answer can be: Because it was somehow justified.
Then I ask the question of this thread: Can Schrödinger time dependent equation be justified in either quantum or in classical terms?
In view of the inadequacy of STDE, if there was ever a justification (which you and many others may honestly believe there was) then the justification was itself inadequate. By the simple reason that the electron does not do what the STDE predicts it should. The well understood physical phenomenon of electron energy transitions between certain special (stationary) electron states was given a dysfunctional mathematical solution in terms of the STDE.
Also, recall that Hamiltonian optics uses generalized coordinates $q,p$, which are canonically conjugate of each other. With these you express the corresponding Hamilton canonical equations.
If you try to do Hamiltonian wave mechanics for hydrogen you will need, besides the wave variable $\psi$ (which corresponds with $q$) a conjugate variable $\phi$ (which corresponds with $p$, the kinetic energy carrier). Otherwise you cannot write down canonical equations.
In STDE the photon variable $\phi$ is plainly ignored and a Hamiltonian system is surreptitiously introduced using the imaginary unit $i$. But this mathematically acceptable use of $i$ is physically an artificial procedure that leads to all the contradictions, inconsistencies and complications typical of Quantism.
Both $\psi$ and $\phi$ are used in the natural equation.
With great appreciation for your interest and cordial regards
Daniel Crespin
Ss I see it, the Schrödinger equation is also a statement of the law of conservation of energy, therefore in the absence of an external force, the energy can be stationary in quantised states. The reason why a state is stable for an electron in such a state is, in my guidance wave interpretation, the wave only gives physical effects discernible to the outside when it is real, i.e. precisely at the antinode. If the antinode is always in the same place, that has not moved therefor it has not accelerated and Maxwell's relationships do not apply.
The algebraic eigenfunctions are obviously not physically real - but the do account for what you see when they are used in calculations, in other words they represent part of the mathematical description of what is going on, Nevertheless, unless specifically defined otherwise, they are not physical. Problems with the formal mathematics, in my opinion, result simply from using mathematics in ways that they should not be used. If you do not know specifically what every term means outside of the mathematics, there will be rouble.
Ian Miller
Look at it in this way: The guidance wave is in fact the photon. With the natural equation you only need this traditional physical object. The photon is so obvious. Has been around always. Why no one saw it?
If you dismiss the photon the electron cannot radiate energy, cannot absorb energy, is stuck in its initial energy level. As you said, the STDE is "a statement of the law of conservation of energy". But without the photon conservation of energy means that the electron is prisoner in its initial energy. Then to explain energy transitions you have to appeal to probabilities and violate the STDE.
Things get quickly messed up and bright people waste their lives trying to interpret, understand or philosophize, mesmerized by an absurdity. The human cost is enormous. How much creativity goes to waste!
If you like the guidance wave it is OK. Go ahead. But keep in mind that the guidance wave is the photon.
With cordial regards,
Daniel Crespin
@Ian Miller
I just searched the Researchgate page for "Foundations of Quantum Mechanics". The amount of energy and effort dedicated to making sense and understanding Quantism is appalling. All that simplifies once you discard random quantum jumps and bring back determinism.
Daniel Crespin
I am puzzled. Does anyone actually think there are random quantum jumps? There are choices for what level a very excited state may fall to, but the Schrödinger equation, as Schrödinger wrote it, is fully deterministic. You can get choices with various options, such as throwing dice, but that does not mean the physics were not deterministic.
Ian Miller
I agree. STDE equation is fully deterministic, and fails to predict transitions. That is why they invented the probabilistic interpretation, random quantum jumps, uncertainty, indeterminacy, entanglement, and so on. My point is that while the self adjoint operator is correct, they got the quantum evolution wrong. With the natural equation you recover a truly deterministic wave mechanics and can convert back theoretical physics into a rational body of scientific knowledge.
Some chemists are not happy with the complex wave functions required by STDE and have developed an orbital theory that fits better into their more practical work. The energy gradient provides often a practical deterministic evolution equation that fits well enough their needs. Chemists work with matter and radiation, not with the content of fancy ideas. The saying is: "Nobody makes mental experiments in Chemistry".
Daniel Crespin
I am not sure about "Nobody makes mental experiments in Chemistry" since I am a chemist, at least I started my scientific life that way.However, in my opinion, both chemists and physicists get atromic orbitals wrong once we get to lithium or higher. See " I. J. Miller 1987. The quantization of the screening constant. Aust. J. Phys. 40 : 329 -346." or for a more user-frienddly account, and how this can be used to calculate chemical bond properties without even requiring the use of a computer, see my ebook "The Covalent Bond from Guidance Waves".
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