Bound States in Quantum Mechanics: Confused?

[ehrenfest]

Homework Statement

I am confused about bound states in QM. My book defines bound states as those in which the particle cannot escape to infinite.

It then gives an example of a potential which is infinite when x is less than 0, -V_0 when x is between 0 and a, and 0 when x >= a.

But then it says that "a particle with mass m is in a bound state in this potential with energy <= 0"

How can this particle possibly be in a bound state when it will have a nonzero probability of being at positive infinity?

Homework Equations

The Attempt at a Solution

 

[cristo]

Well, from just reading your definition, I don't see how you draw the conclusion that the particle has a nonzero probability of being at positive infinity when x<0. For the particle to escape to infinity, it must move towards the right, breaking through x=0. However, the potential at x=0 (and anywhere to the left) is infinite, so the particle will not be able to break through this. Therefore, the particle is bound.

 

[ehrenfest]

What do you mean it must break through at x=0?

What do you mean when "being at positive infinity when x< 0"?

The particle is never anywhere where x is less than 0.

 

[ehrenfest]

I see the problem. I said that the particle has non-zero probability of being at infinity. That is never true.

I meant that it has nonzero probability of being at at an arbitrarily large finite positive x-position.

Just consider the finite square well with energy less than the top of the well. The solution in the classically disallowed region is an decreasing exponential. My book says that is bound. That makes no sense to me.