Solving 1D Quantum Potential Well Problem

[December]

Hi

I'm trying to solve a one-dimensional quantum well problem. The problem itself is probably (or: hopefully) not too hard to solve, but I'm having a difficult time to understand how the given potential actually works.

The incident particles is coming from the left, and the potential well is given by:

0 when abs(x) > a
-V0 when abs(x) < a

...Where V0 > 0...

I'm not sure how the particles will interact with this potential. My first thought was that it would act as a "upside down" potential barrier... What I mean is that if the particle's energy E is greater than 0, then it would act as a potential well, and if E < 0, then it would act as a barrier.. But I'm not sure at all if I'm right, or if I have misunderstood the entire thing!

All help is appreciated!
Thanks in advance.

 

[meopemuk]

Hi

I'm trying to solve a one-dimensional quantum well problem. The problem itself is probably (or: hopefully) not too hard to solve, but I'm having a difficult time to understand how the given potential actually works.

The incident particles is coming from the left, and the potential well is given by:

0 when abs(x) > a
-V0 when abs(x) < a

...Where V0 > 0...

I'm not sure how the particles will interact with this potential. My first thought was that it would act as a "upside down" potential barrier... What I mean is that if the particle's energy E is greater than 0, then it would act as a potential well, and if E < 0, then it would act as a barrier.. But I'm not sure at all if I'm right, or if I have misunderstood the entire thing!

All help is appreciated!
Thanks in advance.

Think about what a classical particle would do in this potential. This might help you to better understand the quantum problem.

Eugene.

 

[December]

Thanks for quick your answer.

Classically, I guess that since I can define the zero point of the potential energy myself, it doesn't matter if the potential is defined between 0 and -V0 since it might as well be between V0 and 0. But if that's the case, then this potential wouldn't differ from a regular potential well (for example, if V(x)=V0 when abs(x) > a and V(x)=0 when abs(x) < a).

Furthermore, I guess that it would be pointless to talk about particles "entering from the left" when the energy E is less than 0, since they can only exist in bound states in the well.

Could this be correct, or am I missing something?

 

[meopemuk]

Thanks for quick your answer.

Classically, I guess that since I can define the zero point of the potential energy myself, it doesn't matter if the potential is defined between 0 and -V0 since it might as well be between V0 and 0. But if that's the case, then this potential wouldn't differ from a regular potential well (for example, if V(x)=V0 when abs(x) > a and V(x)=0 when abs(x) < a).

Yes, nothing would change if you simply add a constant to your potential. However, it is conventional to define potentials in such a way that their values at infinity are zero.

Furthermore, I guess that it would be pointless to talk about particles "entering from the left" when the energy E is less than 0, since they can only exist in bound states in the well.

You are right that for negative energies (assuming that the potential at infinity is zero) the particle is confined in a bound state inside the well.

Eugene.

 

[December]

Thank you!

It's always easier to understand something when you actually get to discuss the topic. I've been trying to work this out on my own for some hours now, and I think I even managed to make the problem harder than it really was. It feels as if quantum mechanics slowly starts to make sense to me. ;)

I really appreciated your help.

// Eric

 

[thesaruman]

I'm trying to solve exercise number 2 of Cohen's Quantum Mechanics, vol1.
The item (b) demands the normalization of the functions but I can't do it because I need one more relation between the constants A1 and A1'. Does anyone have any idea?