YouToks-Instruct-Quantum-Physics-III-Llama-3.2-3B-Instruct

Features

• Ricky is based on Llama-3.2-3B-Instruct from MLX Community
• Fine Tuned on jilp00/YouToks-Instruct-Quantum-Physics-III Dataset
• 3 Billion Parameters
• Finetuned with MLX
• model.safetensors 5.98 GB

Benchmark

MAX TOKENS = 500 Apple Silicon Macbook Pro 18,3 M1 PRO 16 GB RAM

What is the WKB approximation in quantum mechanics, how does it relate to a linear potential, and why is it not valid in certain regions?

========== The WKB (Wentzel-Kramers-Brillouin) approximation is a method used in quantum mechanics to approximate the solutions of the Schrödinger equation in regions where the potential is slowly varying. It is particularly useful for solving problems involving a linear potential, where the potential energy is a linear function of the position. The WKB approximation is not valid in regions where the potential changes rapidly, as it assumes that the wave function can be expanded in a series of plane waves.

In the WKB approximation, the wave function is assumed to be a product of a plane wave and a slowly varying function, often referred to as the "envelope." The plane wave is used to describe the wave-like behavior of the particle, while the envelope function is used to describe the spatial variation of the wave function. The WKB approximation is based on the idea that the wave function can be expanded in a series of plane waves, which are solutions to the free Schrödinger equation.

For a linear potential, the WKB approximation can be used to find the energy eigenvalues and eigenfunctions. The energy eigenvalues are given by the formula:

E_n = (n^2 π^2 ħ^2) / (2mL^2)

where n is an integer, ħ is the reduced Planck constant, m is the mass of the particle, and L is the length scale over which the potential varies. The eigenfunctions are given by the formula:

ψ_n(x) = (2/L)^(1/2) sin(nπx/L)

The WKB approximation is not valid in regions where the potential changes rapidly, such as at the boundaries of the potential well or at the center of a potential barrier. In these regions, the wave function cannot be expanded in a series of plane waves, and the WKB approximation breaks down.

========== Prompt: 64 tokens, 303.930 tokens-per-sec Generation: 380 tokens, 24.398 tokens-per-sec Peak memory: 6.506 GB

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Credits

MCES10 Software

Thanks to:

jilp00 for the dataset MLX-Community for the MLX Converted model Meta Llama AI for the model