No. The Schrodinger's wave function is just a mathematical representation of the probabilistic potential of appearing an electron at a location or time. But the Schrodinger's wave equation doesn't use time correctly, and it doesn't explain the nature of a quantum wave because it depends on experimental measurements and assumptions. Also, it hypothetically describes only the potential locations and probability of collapsing the electron's structure within the left and right range of its width.

The answer is yes. It is the Euler-Lagrange equation for for the "quantized" Lagrangian so based on stationary action.

See: https://physics.stackexchange.com/questions/385345/the-schrödinger-equation-as-an-euler-lagrange-equation

http://www.scl.rs/papers/10.1007_s10701-014-9810-4.pdf

I second Truman Prevatt's answer. Of course, the Schrödinger equation satisfies the principle of stationary action.

I just would like to point out that there is no principle of least action as a universal law. The law is that the action is stationary. In most cases, it is not a minimum. I give a detailed discussion on this in https://wasd.urz.uni-magdeburg.de/kassner/research_gate_pres/sci_edu_res_gate/stationarity_action.html. In particular, I show that for the simple harmonic oscillator the action is not minimal, if a trajectory involving more than one oscillation is considered.

Ismail Abbas,

Your question reflects your lack of knowledge that the Schrödinger equation is or can be derived from the principle of least action.

This survey is not only about this question but about a whole attitude that is starting to spread in the question/answer sections of research sites.

To be objective and brief, let's refer to my personal experience in the Q/A section of Researchgate:

On October 7, 2023, an aggressive unemployed immigrant in Porto responded as follows:

your answer reflects the fact that your understanding of physics needs to be adjusted. !?

3 weeks later, on October 29, the same evil hunter (I...M...), encouraged by the silence and very probably the green light from the CERTAIN DISHONEST MODERATORS OF RG Q/A team, returned with a wrong answer to my question:

, your question reflects your ignorance.!?

Today, for the third time, the same ugly and ignorant person (I... M...) having received the green light from ONE OR MORE DISHONEST MODERATORS OF the RG Q/A team, came back again and again with irreverent answer:

Your question reflects your lack of knowledge!?

Do you have any suggestions?

Dr. Kassner is absolutely correct - while it is the critical points of a "action" functional might be referred to as least action, the appropriate term is stationary. Even Richard Feynman made the mistake in titling his dissertation.

To understand the quantization of classical concepts such as functionals describing quantum systems, it would be instructive to refer to Ricard Feynman's dissertation.

Feynman’s 1942 Ph.D thesis, entitled “The Principle of Least Action in Quantum Mechanics”

Here he introduced the "path integral" approach to quantization to derive the principles of quantum systems. Of course later John Von Neumann introduced the Hilbert space formulation based on the spectral theorem that tends to be standard today. However, the path integral approach was sufficient to derive QED which led to the Yang-Mills gauge theory.

Of course there have been many different quantizations of classical concepts. A brief history is given here.

https://www.sciencedirect.com/science/article/abs/pii/S0039368122001339#:~:text=In%20physics%20the%20word%20“quantization,from%20classical%20theory%2C%20in%20particular.

The good news is the evolution of more and better quantizations provides PhD students in mathematical physics in mathematics departments around the world a broad assortment of thesis topics. 😉

For a very good reference, see Brian Hall's wonderful text.

https://www.sciencedirect.com/science/article/abs/pii/S0039368122001339#:~:text=In%20physics%20the%20word%20“quantization,from%20classical%20theory%2C%20in%20particular.

Sorry somehow - I did not past the correct reference to Brian Hall's text.

https://www.amazon.com/Quantum-Theory-Mathematicians-Graduate-Mathematics/dp/146147115X/ref=sr_1_11?Adv-Srch-Books-Submit.x=0&Adv-Srch-Books-Submit.y=0&dib=eyJ2IjoiMSJ9.uvQMCsEFooUyOJcC2HembpLpOyKUz9jq5Lua5hVY-Br1ae0ECjkKfC02gDjOpbZ6S8RVRMpxtkdggACFQnhXMWLBCP6f4RqsbKDOt7sz1GXg3dAF7BwpObLjkCr95woAaBdrKfkGlNT2WAVZveaWkB-0711KNAh1U00UqEx0-XFZEbRDpaEAkDDYM8L4GVodLklEXNnyfizDCZTlL5KCy10kozzRWb3zakRhleyrjFE.OXr7ndImcDRBTi4JW4SkPXMJsmlyUEQXQmbka7J8mjY&dib_tag=se&qid=1708553318&refinements=p_27%3Abrian+Hall&s=books&sr=1-11&unfiltered=1

Do Schrödinger's equations include the principle of least action?

Any time-dependent physical process must conform to the principle of least action which is a universal law.

We assume that the Schrödinger PDE (SE) solution in its current form does not automatically follow the principle of least action PLA, but some mathematicians and physicists argue that it should if the terms of the SE and PLA concepts are extended or modified.

On the other hand, the chain mechanics of the matrix B which is the statistical equivalence of SE in its spontaneous form contains intrinsically and explicitly the principle of least action and in addition the normalization condition.

The explanation is:

The quantum mechanical transition matrix Q is an eigenvalue matrix for the space-time solution vector of SE denoted ψ(r,t)^2, i.e.

[Q]. ψ(r,t)^2= λ. ψ(r,t)^2. . . . . (1)

The exceptional importance of equation 1 is that the accepted eigenvalue λ of the matrix chain solution B is equal to 1.

This means that the solution follows a straight line in unit space xt towards its final stationary vector solution.

Here It is clear that this process conforms to the principle of least action!.

We have shown that the chains of matrix B respect the principle of least action.

Since the strings of matrix B are, in some way, the matrix mechanical equivalence of the Schrödinger PDE, then the Schrödinger PDE itself must also conform to the principle of least action.

Now the second question arises:

Does the string mechanics of matrix B automatically satisfy the normalization condition?

The answer is yes..

The area under the first triangular curve for psi^2 is 1/2 and that under the second triangular curve is 1/4. . . etc.

It is clear that 1/2 +1/4+1/8 +1/16 . . etc is equal to 1 which is the normalization condition.

For interested contributors:

Please take a look at,

Cairo Technics Solution of Schrödinger's Partial Differential Equation - Time Dependence, Researchgate, IJISRT Journal, March 24.

We assume there is a detailed answer.

Ismail Abbas, The output energy of an interaction depends on the location of it in the structure/wave of a quantum object. Detector/Observer damages a location in quantum object, causing to start collapsing the processes in them from that location as if appearning a quantum particle. But, the energy produced during the collapse of the structures/waves is just a reconstruction of the quantum structures/waves from the location of their damage. Quantum objects don't contain quantum particles in its structures/waves. Also, the amount of energy produced at the location doesn't represent an existence of a quantum particle at that location. Therefore, the principle of least action is not applicable to the collapse of the wave function, and the probability of appearing a quantum point particle from a quantum structure/wave in Schrodinger's wave equation is just a hypothetical description of appearing energy.

Learn more about matter and energy: Preprint Verifying The Origin Of Everything