What Are Scattering States and Bound States in Quantum Mechanics?

[Phyisab****]

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

Homework Statement

....__... _______
..._____...|..|_ ..|-------------Energy
....|_|...|_|
...A...B.C.D..E...FEdited 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?

 

[Phyisab****]

Or furthermore, is it possible for the particle to be in a bound state for regions A and B and in a scattering state for region D? I just don't know what these words mean, or I would just do the math and find out.

 

[LiorE]

I think that by scattering state they simply mean that it's a state that isn't bound - for instance, if E>V_F (the potential at F). Because then the idea is that the particle is scattered by the potential.

 

[Phyisab****]

So to answer question "is the particle in a bound or scattering state?" Would you just say that the particle is in a bound state, period?

 

[elduderino]

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.

First, examine the time independent Schrödinger equation. It tells you that E should always be greater than minimum of the potential well (otherwise the function and the second derivative would have the same sign and function would no longer be normalizable).

Now, think in analogy to the classical bound and scattered states. If a particle is trapped in a well with energy lower than the height of the well then it will stay forever oscillating within it. This is called a bound state. If the particle with an energy greater than the height of the well encounters the barrier wall, it will just encounter a "bump" and continue whatever it was doing earlier.

Thus, classically, a particle is bound if its energy E<V_h where V_h is the height of the well. If E>V_h then it is "scattered".

Take this terminology to quantum mechanics. The only difference is that we know from the solution of the finite well that tunneling is a reality. So for a particle to be bound we need only be concerned with the values that a potential assumes at infinity. For a bound state it should have an energy less than the value of the potential at infinity. For a scattered, it should have a value greater than the value of the potential at infinity.

Note that this implies that scattered states are non normalizable. Also, since the harmonic potential and the infinite well have their potentials going over to infinity, they are systems with an infinite number of bound states.

In your question you have attractive wells and repulsive wells. (Bells above the x-axis or repulsive wells cannot have a bound state.)

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?

Depends on the Energy of the particle. If E>V_h (Scattering State) where V_h is the height of the repulsive well above the axis it will not stay in the potential well. So if it was moving the +x direction with energy E then it will continue doing so. If Vmin<E<0 (bound state) then it will forever oscillate within the walls of the well.

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?

You can find the particle everywhere for a scattered state. (Reflection coefficient is non zero in Quantum Mechanics, so the classically forbidden region is now accesible). For a bound state as well, you may find the particle anywhere (This time, the transmission coefficient is finite for a finite barrier potential, so it can leak through the walls which is classicaly forbidden)