Having pursued all my higher Education in the field of quantum/atomic physics (including my PhD dissertation), I would recommend a good level of understanding of what you refer to as "classic quantum mechanics". For example, in developing my thesis on simulation of fusion reactions (in lab and interstellar media), I did couple classic QT (quantum theory) methods with full quantum theory. Hope this helps

In Niels Bohr original model which was entirely based on Newtons laws of motion supplimented  by the Bohr quantization hypothesis,By classical quantum mechanics we mean N. Bohr's original QM of the hydrogen atom before the Schrödinger equation and the Bohr/Copenhagen superposition interpretation.We assume that classical quantum mechanics is somehow the basis or foundations of modern quantum mechanics and, therefore, a professional mathematician/physicist would have to master classical quantum mechanics before they can start learning modern quantum mechanics, sometimes called complete quantum mechanics.We therefore recommend a good level of understanding of what is called classical quantum mechanics.To be more specific, we provide the following two illustrative examples:Niels Bohr introduced his atomic Hydrogen model in 1913 where he described it as, electrons orbiting  the nucleus in circular orbits similar to solar system.I-Bohr model for Hydrogen atom

mv 2Pi r(n) = n . h ,n=1,2,3,. . .etc.

he did not say a word about electron cloud in a quadratic potential or any of the quantization numbers (n,l,m,s) other than the principal quantum number n.

We present Fig.1 to explain briefly the precedent model.

📷

Fig1. Bohr original description of quantized energy levels in Hydrogen atom.

II- Classical quantum mechanics model for a particle in 1D infinite potential well

It describes a subatomic particle free to move in a small 1D space surrounded by impenetrable barriers.

Here we do not have a precise classical solution but only a Schrodinger equation based solution.

Figure 2 presents a schematic diagram for the time-independent solution of the Schrödinger equation for a quantum particle in a 1D infinite potential well.

📷

Fig.2. A schematic diagram for the time-independent solution of the Schrödinger equation for a quantum particle

in a 1D infinite potential well.

And yet, to our knowledge fixing the minimum energy of a quantum particle in a box (Emin=hf/2) has no solid physical or mathematical basis.

This remains a paradox.

However, the statistical chains of the transition matrices B predict an explanation of the previous problem.

To be continued.

II-Answer II continued.

In fact, the statistical chains of the transition matrices B predict a numerical solution of time-dependent partial differential equations.

It was shown previously that the time-dependent solution of the energy density [U(x,t)] follows the following recurrence relation[1,2,3],

U(x,t+dt)= U(x,t) + B(x,t) . (b+S). . .(1)

where b is the vector of boundary conditions arranged in the correct order and S is the source/sink term at point (x, t).

Equation (1) also applies to the solution of the heat diffusion equation and to the solution of the Schrodiger equation.

However, there are differences in procedural techniques due to the differences in nature between the microscopic quantum world and the world of macroscopic classical physics:

I-The source/sink term in heat diffusion is external and mainly due to the boundary conditions b while the internal source S is neglected or assumed to be zero.

In the Schrödinger equation, the source/sink term is internal and is equal to a constant * V(x,t) at the point in space x while the contribution of the boundary conditions is set to zero since the space extends to infinity.

Note that the latter is an eigenvalue problem and that the constant of proportionality could be imaginary.

II- In heat diffusion, the time-independent steady state is reached when B^N tends to zero while the steady state condition in the Schrödinger equation is reached by convergence of the value of the energy density towards the eigenvalues ​​of equation 1.

From the previous analysis, it is simple to show that the transfer matrix (M) required for the Schrödinger equation is given by:

M=

{1+ ( md^2 V(X1)/h^2 } -1/2 0 0 0 0 0

-1/2 {1+ (md^2 V(X1)/h^2 } -1/2 0 0 0 0........................ .. ...........0 0

0 -1/2 {1+ (md^2 V(X1)/h^2 } -1/2 0 0 0........................ .. ........0 0

.........................

.......................................

0 0 0 0..................................0 0 0 -1 {2+ (2 md^2 V( )/h^2 } -1

0 0 0 0............................................. ..- 1 {2+ (2 md^2 V(X1)/h^2 }

It is worth mentioning that the matrix M is precisely adequate to the matrix obtained for the numerical solution of the time-independent Schrödinger equation and that the matrix M is obtained via statistical methods bypassing the Schrödinger equation itself.

To be continued.

Thanks for sharing these details about the mathematical models. It has been a while since I developed such formulations, however I recall that for the various methods we've used in modeling various atomic and molecular reactions or collisions, we had combined various formulations and often divided the solution space into regions "close to" the nuclei where we applied numerical techniques and others, or far from the nuclei, where we applied various classical (analytical) methods.

Dear Prof. Ismail Abbas

In addition to the interesting previous answers, I guess the first study comes with the Nonrelativistic Quantum Mechanics, finishing this part with the complicated but beautiful subject of Scattering Theory.

The second part arrives with the study of the Relativistic Quantum Mechanics part, and a posteriori the study of the Quantum Electrodynamics.

Best Regards.

I have the same opinion as most contributors here: I basically worked off Feynman's Lectures, and I would not have understood a iota of Volume III (quantum physics) without having studied Volume II (classical electromagnetic theory) in very much detail. Modern quantum physics starts where Maxwell's equations end: they can deal with charge densities only - not with individual charges or discontinuity (charge being either here or there). But so to understand how particles then behave of might behave - through the Planck-Einstein relation and its implications - you need to first understand classical electromagnetic theory. Especially because, in my humble opinion, there may no need to understand or think about other forces than the electromagnetic force (see Preprint Do we need the nuclear force hypothesis?

Here is a more complete and coherent answer published on Researchgate and the IJISRT journal:

A statistical numerical solution for the time-independent Schrödinger equation-II, November 2023.

Your comments on the article are welcome.