The Potential Well In Quantum Mechanics

[Yonderboy98]

My calculations always come out all right, but I still feel that I need help conceptualizing the potential well.
1.What does the width(or length) of the well represent?
2.What does the depth of the well represent?
Sincerely appreciative,
Yonderboy

 

[FunkyDwarf]

as far as i know, and I am probably wrong, but the width is proportional to the wavelength of the wave function and the depth is the energy

 

[quasar987]

I guess there are two ways to answer your question.

1) What they represent mathematically in the context of the solutions of the Shrödinger equation.

2) What they represent in term of a physical reality.

FunkyDwarf has given an answer to 1) and you probably already knew it if you said you solved wells problems before. My answer to 2) is the following thing.

Specifying a potential V(x,y,z) is the same as specifying a force field, since $\vec{F}=-\nabla V(x,y,z)$, right? Or in just one dimension, $F=-dV(x)/dx$. So at points where V is constant, F=0. And where V varies very fast, F is equally large. Knowing this, a potential well represents an idealized force field in which the particle feels no force when it is in the regions inside and outside the well but as soon as it gets to the borders, it feels an infinitely large* force.

In reality, potentials are continuous, so there is no infinite forces. A particle that charges a well's wall will gradually feel a repulsion force. If its energy it not large enough (i.e if it is lesser that the well's height), it will not escape the well and it will be confined in it forever. If the particle has an energy larger than the well's height, it will escape the well at the cost of some kinetic energy.

Of course this whole interpretation of particles "charging" walls and "being places" is valid only in the classical picture. In quantum, it is dull, you just solve the thing and look at your probability function: oh it has such and such probability of being there, and none there and it's decreasing there.

So to sumarize, the width of the well is the region where a particle of energy lesser than the height is contained.*It is infinite because V varies of a finite amount (or worse, of an infinite amount in the case of the infinite well) in an infinitely small interval, so the slope dV/dx at this point is infinite.

 

[XVX]

Well, in QM, the particle can have less energy then the potential barrier and still pass through it. Only the infinite well DOES NOT have barrier penetration. Thats one thing that makes QM non-classical.

Albeit, the lower the energy of the particle, the less chance for penetration.

 

[quetzalcoatl9]

My calculations always come out all right, but I still feel that I need help conceptualizing the potential well.
1.What does the width(or length) of the well represent?
2.What does the depth of the well represent?
Sincerely appreciative,
Yonderboy

it may be helpful to think about (and graph) a harmonic potential

 

[Manchot]

I think the OP was asking a more general question. In theory, a quantum well is just a construct in which the potential energy of a particle is lower in the well than in the surrounding areas. In practice, a quantum well is formed by putting two dissimilar crystalline materials together with different band gaps. For example, you might sandwich a thin layer of InGaAs into GaAs. For this situation, the width of the well would be the width of the InGaAs layer, and the depth would be the difference between the band gaps of the two materials.