Physics Asked by alienare 4422 on January 19, 2021
The Schrodinger equation:
$$-frac{hbar^2}{2m}nabla^2Psi(r)+V(r)Psi(r)=EPsi(r)$$
$$textit{kinetic energy} + textit{potential energy}=textit{total energy}$$
Is one of my favourite equations, but there’s one term I don’t understand: the $V(r)$ term which is supposed to mean potential energy… but what type of potential energy? In the classic 0 potential well example they say that the potential out side the well is infinite but what type of potential energy are they talking about? I googled it and its also called Bohm quantum potential but I really don’t get what it means. Anything would be a great help.
There is not one single special potential function, rather the opposite. The potential function is a placeholder that takes a different functional form depending on what kind of physical situation you want to model. The physics and the system that we want to describe goes into the Schrödinger equation via this potential function.
The only information that the equation gives you written in this way is the fact that it has to be a function depending only on the position variable. The function $V(x)$ may not depend on derivatives of $x$ for example.
Some basic examples for potentials are the particle in a box potential, $$ V(x) = cases{ 0, -L/2 < x < L/2 \infty, textrm {otherwise} } $$ With this we can model situations where a particle can move freely in a certain area, but unable to escape.
Another potential would be a harmonic potential, $$ V(x) = frac{1}{2}momega^2x^2 $$
With this we can model situations where a particle is for example resting in a local minimum that looks like a parabola. This can describe for example molecules in their stable groundstate geometry. Another example that is described by a harmonic potential would be the time dependent amplitudes of the electromagnetic vector potential.
Potential functions are also often so complicated that we are only able to obtain approximate solutions.
Correct answer by Hans Wurst on January 19, 2021
The potential energy in Schrödinger equation is the electrostatic one. Here are a few points to note:
Answered by Vadim on January 19, 2021
It's just a normal potential energy function. It could be gravitational potential energy, electric potential energy, or any other kind of potential energy from classical mechanics. The way that the particle reacts to the potential energy will be different, but the form of $V(x)$ is exactly the same.
For the infinite square well, we are not really concerned about what is causing the potential. It's mostly used as a teaching example, but it could provide a simplistic model for a particle strongly confined to a region by any type of potential energy.
Answered by JoshuaTS on January 19, 2021
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