Ground State:
1. Prime Mechanic has a built-in invariant system called a Prime Field in where we can determine outcomes. We can pinpoint the location and section off potential energy for use with determination and predictability. Let us find a particle in our Prime Field.
The ground state refers to the lowest energy state of a system and can be described by a wave function Ψ(x), which is a mathematical representation of the probability density of finding the system in a particular state. The wave function of the ground state is a stationary wave meaning that it does not change with time. This is because the ground state has the lowest possible energy, and therefore there is no energy available to cause the system to change or evolve over time. It is a fixed, stable wave that represents the lowest energy state of the system.
Let us create a one-dimensional ground state stationary wave function with zero spins that are stationary between 0 and 1.
Ψ(x) = √(2/L) * sin(πx/L)
This wave function describes a particle that is confined to the region between x=0 and x=L, and has zero spins. The probability density |Ψ(x)|^2 is uniform over the entire length of the box, and the energy of the ground state is given by:
E = (π^2 * h^2) / (2mL^2)
where h is Planck’s constant and m is the mass of the particle.
To apply the wave function Ψ(x) = √(2/L) * sin(πx/L) between 0 and 1, we need to normalize the wave function such that the integral of the probability density over all space is equal to 1. We can do this by using the following normalization condition:
∫0^1 |Ψ(x)|^2 dx = 1
Plugging in the expression for Ψ(x), we get:
∫0^1 |Ψ(x)|^2 dx = ∫0^1 [√(2/L) * sin(πx/L)]^2 dx