Does this type of potential have a name?

[Darrenm95]

A particle with mass m and electric charge e is confined to move in one dimension along the x -axis. It experiences the following potential: x<0 V(x) = infinity, x>=0 V(x) = e^2/4*pi*ε0*x

 

[Simon Bridge]

Welcome to PF;
What type of potential has what?
Are you asking what sort of potential the one in the description is?
Well - what kinds of potentials are there, and what sort does the described one fit?

 

[Darrenm95]

Yes, I've not come across a potential before the goes from infinity to an actual potential, how would a wavefunction exist in this potential, the wavefunction obviously wouldn't where the potential is infinity and I imagine it would be exponential decay for the second region but I've never come across this and wondered what type of potential is it (i.e. you have potential steps, potential wells) and where would you come across this in nature?

 

[Simon Bridge]

An infinite potential does not exist in nature - it is usually used to model a situation where the potential becomes strongly repulsive over a distance too small to measure... i.e. if you wanted to model a "rigid" or "hard" wall in QM that's how you do it. It is also used as a first approximation upon which other models can be built... such as particle-in-a-box as a model for the atomic nucleus.

Now for your question:
If you think of the x as a radius you may get a clue what is being attempted here: you have seen that potential before.

 

[vanhees71]

Really? I've never seen that potential before; at least I don't know, what it should describe ;-)).

 

[Vanadium 50]

This is the potential for a particle attached to a spring attached to a wall. V(x) = kx on one side, and x is an excluded region on the other.

 

[TonyS]

It looks to me like a repulsive Coulomb potential outside the excluded region.

 

[Vanadium 50]

I guess it's unclear as to whether what follows the / should be in parentheses or not.

 

[Darrenm95]

The next part of the question reads:

For the region x is greater than or equal 0, by substituting in the Schrödinger equation, show that the wave function u(x) = Cxexp(-αx) can be a satisfactory solution of the Schrödinger equation so long as the constant α is suitably chosen. Determine the unique expression for α in terms of m, e and other fundamental constants. Note that C is a normalisation constant.

I'm struggling to visualise the problem, can someone point me in the right direction, I really have no idea where to start? I substituted into the Schrödinger equation both u(x) and its second derivative but I'm not sure where to go from there, any pointers (hints) would be appreciated

 

[Nugatory]

Before you do anything else with this problem, you'll have to clarify the ambiguity that Vanadium50 pointed out. Is the potential where $x\gt{0}$ $V(x) = e^2/4\pi\epsilon_0x$, or is it $V(x) = e^2/(4\pi\epsilon_0x)$, or what? It makes a difference as you can see by comparing Vanadium50's and TonyS's answers.

And with that said, to get help with the second part of the problem we're going to have to ask you to repost in our homework section, and properly fill out the template you'll see when you start the new thread. We aren't doing this to make you jump through hoops, we're doing this because we have better than a decade of experience to show that questions of this sort get better answers when you go through the homework forum process.

 

[vanhees71]

I'd still like to know the physics context of this problem. What should it describe? Is it a 1D problem? I still don't get the physics, if this is really a Coulomb potential for $x>0$ (it seems to be because of the $\epsilon_0$, which appears in the SI units) for a particle restricted to the right half-line (assuming it's meant as a 1D problem). Of course, you can try to solve the corresponding eigenvalue problem for the Schrödinger equation with the appropriate boundary conditions.

 

[Darrenm95]

there's no parentheses after the / and I will repost it in the homework section

 

[vanhees71]

Well, still one must be clear about which potential is meant (in my opinion one should always use parentheses in such cases). As I said, the form the potential is written it is suggestive to be a repulsive Coulomb potential, $V=e^2/(4 \pi \epsilon_0 r)$, between two point particles with equal charges $e$.

 

[Simon Bridge]

Seems to be continued here:
https://www.physicsforums.com/threads/i-am-unsure-as-to-the-nature-of-the-potential.849138/
... the replies are echoing what was covered above.