The discussion revolves around a problem in quantum mechanics concerning the wave function of a free particle in three-dimensional space, specifically represented by the expression $$ \Psi = Ne^{-ar} $$, where $$ r=| \vec r | $$ at time t=0. Participants are exploring how to express this wave function in terms of the wave vector $$ \hbar \vec k $$ using Fourier transforms.
Some participants are providing insights into the nature of the wave function and its potential origins, while others are exploring the mathematical approach of using Fourier transforms to analyze the situation. The conversation reflects a mix of interpretations regarding the physical context of the wave function.
There is a mention of a specific condition in the problem statement that suggests the wave function describes a free particle, despite its unconventional form. Participants are also considering the implications of a potential transition from a $$1/r$$ potential to a free particle state.
From where did you get this formula? It is not the wave function of a free particle, but for a $$1/r$$ potential.MementoMori96 said:In a 3D space, a free particle is described by :
Ψ=Ne−arΨ=Ne−ar\Psi = Ne^{-ar} with
It is not inconceivable that it describes a free particle at a single time. For example, you could imagine the particle being in the ground state of a ##1/r## potential for times ##t \leq 0## and the potential is turned off at ##t = 0##. This would correspond to a non-adiabatic transition and the state would have to be projected onto the free particle eigenstates in order to determine its time evolution. As alluded to by TSny, this can be done through Fourier transform.eys_physics said:From where did you get this formula? It is not the wave function of a free particle, but for a $$1/r$$ potential.