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Bound states

Page history last edited by Joe Redish 8 years, 11 months ago

6.4.P4

 

One of the challenging ideas of using energy at the atomic and the molecular level is the idea of bound states.  What this means is that you might start with two objects that have essentially zero kinetic energy, they get close and interact strongly in an attractive way. They find some way to emit energy into another form and wind up being stuck together -- bound. You have to put energy in -- do work on them -- in order to get them apart. Let's work through the language of potential energy to see how to talk about this. (Since we are making a model that looks like atoms and molecules, we will ignore friction.)

 

A. Let's first talk about a simple problem that you have now had some practice with -- the motion of a skateboarder on a track.  Suppose the track looks like a dip in the ground as shown in the figure. A potential shape like this is often referred to as a potential well, since it looks like a dug-out area for a well.


Now suppose that the skateboarder approaches the dip from the left traveling with a positive kinetic energy. The figure at the right shows the skateboarder's total mechanical energy as a solid black line at a PE of 10 units (units unspecified).

 

Describe the motion of the skateboarder and how her potential and kinetic energies change as she moves through the well.

 

B. Now suppose that she starts inside the well at a zero velocity -- say at point x = -2.5 units with a total energy as shown by the heavy solid line.

 

Describe the motion of the skateboarder and how her potential and kinetic energies change as she moves through the well.

 

C. Her total energy is shown is the figure as -10 units. How can this be?  Is it reasonable for the total mechanical energy to be negative?

 

D. If she wants to climb out of the well and be at 0 kinetic energy at the point x = 3 units, how much energy would she need to gain?

 

 

E. The skateboarder is actually just an analogy for the cases we are interested in, which are interacting atoms. This is really too simple a model: the atoms are impenetrable and will repel if pushed too close together. Instead of the simple well shown above, the atom-atom potential looks more like the one shown in the figure at the right. When the atoms are far apart there is little to no interaction.  When they are closer, they are attracted and pulled together.  If they get too close they are pushed apart. The potential energy of the interaction looks like the figure at the right.

 

If the atoms have the energy of -7.5 units as shown by the solid line in the figure, describe their motion and how their potential and kinetic energies change as they move in the well.

 

 

F. If the atoms have an energy of -7.5 units as shown by the solid line in the figure, would you have to put energy in to separate the atoms or by separating them would you gain energy? How much? Explain why you think so.

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