Does this type of potential have a name?

In summary, the potential is a repulsive Coulomb potential between two point particles with equal charges.
  • #1
Darrenm95
7
0
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
 
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  • #2
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?
 
  • #3
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?
 
  • #4
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.
 
  • #5
Really? I've never seen that potential before; at least I don't know, what it should describe ;-)).
 
  • #6
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.
 
  • #7
It looks to me like a repulsive Coulomb potential outside the excluded region.
 
  • #8
I guess it's unclear as to whether what follows the / should be in parentheses or not.
 
  • #9
The next part of the question reads:

For the region x is greater than or equal 0, by substituting in the Schrodinger equation, show that the wave function u(x) = Cxexp(-αx) can be a satisfactory solution of the Schrodinger 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 Schrodinger equation both u(x) and its second derivative but I'm not sure where to go from there, any pointers (hints) would be appreciated
 
  • #10
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.
 
  • #11
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.
 
  • #12
Theres no parentheses after the / and I will repost it in the homework section
 
  • #13
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##.
 

1. What is a potential in science?

A potential in science refers to the energy that is stored in a system or object due to its position, shape, or state. It is a fundamental concept in physics and is used to describe the behavior of systems, such as the movement of particles or the flow of electricity.

2. What are the different types of potentials?

There are various types of potentials in science, such as gravitational potential, electrical potential, and chemical potential. These potentials differ based on the type of force or energy they describe and the underlying principles that govern them.

3. What does it mean for a potential to have a name?

In science, a potential may have a specific name if it is derived from a particular force or concept. For example, the potential energy associated with gravity is called gravitational potential energy, and the potential energy associated with electric charges is called electrical potential energy.

4. How do scientists name potentials?

Scientists typically name potentials based on the physical quantity they represent or the underlying principle they describe. For instance, the potential due to gravity is named after the force of gravity, and the potential due to electrical charges is named after the concept of electric potential.

5. Can a potential have multiple names?

Yes, a potential can have multiple names depending on the context in which it is used. For example, the potential energy due to an object's position in a gravitational field can be referred to as gravitational potential energy, potential energy, or simply potential. Additionally, some potentials may have different names in different scientific disciplines, leading to multiple names for the same concept.

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