Intuition behind Green's function (one dimension)

[Ravi Mohan]

I am studying scattering from these notes.
There I came across Green's function in one dimension which is computed as
$$\langle x|G_o|x'\rangle = -\frac{iM}{\hbar ^2k}\exp(ik|x-x'|)$$
I understand Green's function as a sort of propagator from $x'$ to $x$. There are two observations that can be made for this Green's function (leaving aside the oscillatory dependence)
1) More the momentum, less the amplitude to traverse from one point in space to other.
2) Amplitude is independent of the displacement.

Whereas in 3 dimension it is quiet opposite. The amplitude is independent of the momentum but inversely dependent on displacement.

Why is this so?

 

[eljose79]

As a scientist studying scattering, I can offer some insights into the observations you have made about Green's function in one dimension versus three dimensions.

Firstly, it is important to understand that Green's function is a mathematical tool used to solve certain types of differential equations, including those involved in scattering problems. It represents the response of a system to a localized source, in this case a particle at a specific position. The Green's function in one dimension that you have mentioned is specifically for a free particle, meaning there is no potential or force acting on the particle.

Now, let's look at your first observation - that in one dimension, the amplitude of the Green's function decreases as momentum increases. This can be explained by considering the wave nature of particles. In one dimension, the particle has a single degree of freedom - it can only move back and forth along the x-axis. As the momentum increases, the wavelength of the particle also decreases, meaning it is more localized in space. This leads to a decrease in the amplitude of the Green's function, as the particle is less likely to be found at a specific position.

In contrast, in three dimensions, the particle has three degrees of freedom - it can move in any direction. As the momentum increases, the particle has a longer wavelength and is more spread out in space. This results in a larger amplitude for the Green's function, as the particle is more likely to be found at a specific position.

Your second observation, that in one dimension the amplitude of the Green's function is independent of displacement, can also be explained by the wave nature of particles. In one dimension, the particle can only move along the x-axis, so its position is always known. This means that the amplitude of the Green's function is independent of the displacement between the two positions, as the particle is always located at a specific position.

In three dimensions, however, the particle can move in any direction and its position is not always known. This means that the amplitude of the Green's function is inversely dependent on displacement, as the particle has a greater chance of being found at a position closer to the source.

In conclusion, the differences in the amplitude of Green's function in one dimension versus three dimensions can be explained by the wave nature of particles and the number of degrees of freedom they have. It is important to note that these are general observations and may vary depending on the specific problem being studied. it is always important to carefully