We assume that the B-matrix chains, or any other exact statistical string, can introduce a numerical statistical solution for the total quantum energy of a particle in a 3D box.
This may be possible in the same way that the B-matrix chains present a numerical statistical solution for the thermal diffusion energy density without going through the thermal PDE itself.
The quantum particle in the 1D box problem can be expanded to consider a particle within a higher dimensions as demonstrated elsewhere for a quantum particle in a 2D box. Here we continue the expansion into a particle trapped in a 3D box with three lengths Lx, Ly, and Lz. As with the other systems, there is NO FORCE (i.e., no potential) acting on the particles inside the box (Figure 3.9.1
).
📷 Figure 3.9.1
: A particle in a 3-D box scheme with equal lengths on all three dimensions. (CC BY-SA-ND 3.0; www.a-levelphysicstutor.com).
The potential for the particle inside the box
V(r⃗ )=0(3.9.1)
- 0≤x≤Lx
- 0≤y≤Ly
- 0≤z≤Lz
- Lx
The concept you're describing seems to involve using numerical statistical methods to approximate solutions for certain physical problems without explicitly solving the underlying differential equations. While this approach might work in some cases, it's essential to note that the Schrödinger equation in quantum mechanics is a fundamental equation that governs the behavior of quantum systems, and its solutions provide detailed information about the quantum state of a particle.
However, there are numerical techniques that are commonly used to solve the Schrödinger equation for quantum systems, especially when analytical solutions are not feasible. Some common methods include:
While these methods involve numerical computations, they still directly address the Schrödinger equation or its equivalent formulations. It's challenging to bypass the Schrödinger equation entirely while accurately describing the quantum behavior of particles in a 3D box.
If you're exploring alternative numerical/statistical methods, it's crucial to ensure that the proposed approach maintains the essential physical principles and accurately captures the quantum behavior. It's always a good idea to validate any new method against known solutions or benchmark problems to ensure its reliability and accuracy.
Please consider taking a look at the answer in the following articles,
A statistical numerical solution for the time- independent Schrödinger equation, Reseaechgate, IJISRT review, Nov 23.
A numerical statistical solution for the time- independent Schrödinger equation -Part II, Reseaechgate, IJISRT review, Dec 23.
Please take a look at,
A 3D numerical statistical solution for the time-independent Schrödinger equation
Researchgate, IJISRT journal, December 2023.
Please take a look at,
A spontaneous statistical solution of Schrödinger's partial differential equation
tion
February 2024
Researchgate, IJISRT International Journal, February 2024.
We hope you find a detailed and useful answer.
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