Normally I do not comment on responses, but as an exception, when the response comes from our distinguished professor colleague from the United States, it would be different.

I- We emphasize that SE, and not the quantum particle itself, addresses infinite space where BCs extend to infinity.

The case of the hydrogen atom is not an exception and the space of a single electron (2 or more interacting electrons are not allowed in any quantum system) since the effective radius of the hydrogen atom is approximately infinite compared to the much smaller electron.

II-SE is introduced for vacuum or free space defined by ε=ε0 and µ=µ0.

To my knowledge, I don't know and I don't expect anything else.

III-Look at what is called the collapse of the wave function due to observation or measurement! This can be described as the sudden death of a quantum particle due to the closure of its vacuum or living space.

Dear friend Ismail Abbas

Thanks for encouragement my friend Ismail Abbas I am rewriting my answer with this motivation. I always loved Schrodinger equation since the day I was introduced with the solution of this equation and later become fan of Schrodinger's cat as well lol :).

Well, let me break it down for you, my friend Ismail Abbas . Schrödinger's equation isn't exclusively limited to infinite free space, but it's often introduced in that context for simplicity. The equation itself is a cornerstone in quantum mechanics, describing the behavior of quantum systems.

While the typical presentation might focus on scenarios like particles in an infinite well or free space, Schrödinger's equation can be applied to various potential energy landscapes. The untold story you mention might stem from introductory discussions, but it's not a strict constraint.

The wave function ψ doesn't necessarily need a vacuum or infinite space. It's a mathematical description of the probability amplitude of finding a particle at a certain position. The boundary conditions, as you Ismail Abbas rightly pointed out, depend on the specific situation, and they don't always have to be zero.

So, to answer your question, ψ can inhabit spaces beyond infinite free space, and Schrödinger's equation is more versatile than it might seem at first glance. Let me know if you Ismail Abbas want to dive deeper into the quantum realm!

Google Search

What is the Schrödinger equation?

The Schrödinger equation gives the evolution over time of a wave function, the quantum mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate by Louis de Broglie according to which all matter is associated with a matter wave.

What is an isolated system ?

Can it have any other structure than described in the question. i.e. it is only valid for infinite free space?

I wish you good luck.

No it's not, look on the internet for more info

When the answer (look on the Internet for more information!) comes from our distinguished professor colleague from the United States, it would be different.

We assume Google search is inconsistent at this point.

I-In one part it states that SE is introduced for a closed system.

This means that SE is introduced for vacuum or entire free space defined by ε=ε0 and µ=µ0.

To my knowledge, I don't know and I don't expect anything else.

II-In other answers and research articles, Google search claims that SE is versatile and can be adjusted to arbitrary systems with arbitrary boundary conditions (transparent or not).

However, the statistical chains of matrix B suggest that there is an untold story about SE which is[1]

S(x,y,z,t)=Constant. E(x,y,z,t) . . . .. (1)

Where S is a source term (intrinsic or extrinsic).

Equation 1 is a breakthrough because it characterizes the validity of SE and suggests a statistical solution to 3D SE that circumvents SE itself.

[1] A 3D numerical statistical solution for the time-independent Schrödinger equation, forthcoming.

FALL and RISE of Matrix Mechanics

In 1925, W. Heisenberg, Max Born and Pascual Jordan

introduced so-called matrix mechanics (HMJ theory) to study the fine structure of Bohr hydrogen atom. HMJ theory succeeded in finding quantum numbers such as l,m,s in addition to the principal quantum number n.

HNJ proposed a square matrix in general of complex entries that was considered as a matrix formulation of QM and obviously the first MATHEMATICAL formulation of quantum mechanics. conceptually self-contained and logically coherent.

In 1927 Schrodinger equation SE with Bohr/Copenhagen interpretation appeared.

The so-called Copenhagen interpretation of quantum mechanics, which has dominated the field since the 1930s

The equivalence between HMJ theory and SE theory was supported by a rigorous mathematical proof, except that concerning the quantum superposition of the wave function resulting from the Bohr/Copenhagen interpretation where the HBJ fails.

Unlike HBJ, Schrödinger perceived the field-like continuity of some key microphysical phenomena, such as the interference phenomena of a coherent electron beam in double-slit experiments.

At the late 1930th There was a lot of debate about the equivalence of HMJ and Schrodinger equation until the performance of double slit interference experiment which showed clearly that HMJ theory falls short behind SE theory wrt the superposition principle.

HMJ theory contains a lot of math and little physics.

Furthermore, HBJ theory has never addressed the description of macroscopic physical phenomena and is therefore considered incomplete.

This means that HMJ theory is incomplete because it can not address classical physics problems such as the heat diffusion equation.

Additionally, the HMJ and SE equivalence has been debunked and considered a myth.

This is what we call the fall of Matrix Mechanics almost a century ago.

However, In 2020, a new theory of matrix mechanics MM emerged and successfully applied to different areas of classical physics in addition to quantum mechanics.

This is what we call the renaissance or rise of matrix mechanics.

The new matrix mechanics (MM) procedure is called B-matrix chains and is abbreviated as BMM.

In this article, we explain the consecutive steps on how to apply the BMM to different 1D, 2D, 3D situations and how to arrive at the appropriate solutions of the volumetric energy density in the bounded domains.

the transition matrix B or BMM is well defined and it has been successfully applied to different areas of classical physics such as the Poisson and Laplace PDE, the heat diffusion equation, the theory and design of audio rooms, Limited integration in 1D, 2D, 3D. . etc.

Furthermore, the extension of the BMM to find a numerical statistical solution of the time-independent Schrödinger equation in 1D, 2D and 3D was recently described and the numerical results were surprisingly accurate.

We emphasize that both SE, and not the quantum particle itself, addresses infinite space where BCs extend to infinity.

The case of the hydrogen atom is not an exception and the space of a single electron ( interacting electrons are not allowed in any quantum system) since the effective radius of the hydrogen atom is approximately infinite compared to the much smaller electron.

Both SE and HMJ theory are introduced for infinite vacuum or free space defined by BC=0 and ε=ε0 and µ=µ0.

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.