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This is not a homework problem, just a question I encountered I thought I should figure out.

.................__..... _______

....._____...|..|_ ..|-------------Energy

.............|_|.....|_|

........A.....B.C.D..E.....F

Edited due to formatting of my picture. Please ignore the periods I had to use them to preserve my drawing.

A particle with energy E interacts with a 1-D potential energy function V(x) as shown above.

A) For a classical particle, if at time t=0 the particle is found in region B, in what regions is it possible to find the particle at t>0?

B) For a quantum mechanical particle, write down all the regions where you may find the particle with energy E shown. Is the particle in a bound or scattering state?

My question is, what exactly is the condition described by the words "scattering state" and "bound state?" These terms aren't mentioned anywhere in my textbook, and I've had trouble finding a strict definition of them.

Clearly for question a) the particle can reach region A, B, and nowhere else.

Now for question B. I know that there is a nonzero probability to find the particle anywhere except infinitely far away. Now I get confused. If the particle comes from the left, it would be bound in the sense that it decays exponentially outside region I and II. Is that what they mean by bound? There is still a small probability that the particle could be anywhere else, so it is not really "bound". Finally, what conditions would have to be met to make it a scattering state? Would the potential have to be perfectly uniform, with E>V? Why is it called a scattering state?

## Homework Statement

.................__..... _______

....._____...|..|_ ..|-------------Energy

.............|_|.....|_|

........A.....B.C.D..E.....F

Edited due to formatting of my picture. Please ignore the periods I had to use them to preserve my drawing.

A particle with energy E interacts with a 1-D potential energy function V(x) as shown above.

A) For a classical particle, if at time t=0 the particle is found in region B, in what regions is it possible to find the particle at t>0?

B) For a quantum mechanical particle, write down all the regions where you may find the particle with energy E shown. Is the particle in a bound or scattering state?

My question is, what exactly is the condition described by the words "scattering state" and "bound state?" These terms aren't mentioned anywhere in my textbook, and I've had trouble finding a strict definition of them.

## The Attempt at a Solution

Clearly for question a) the particle can reach region A, B, and nowhere else.

Now for question B. I know that there is a nonzero probability to find the particle anywhere except infinitely far away. Now I get confused. If the particle comes from the left, it would be bound in the sense that it decays exponentially outside region I and II. Is that what they mean by bound? There is still a small probability that the particle could be anywhere else, so it is not really "bound". Finally, what conditions would have to be met to make it a scattering state? Would the potential have to be perfectly uniform, with E>V? Why is it called a scattering state?

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