In this unit you will investigate the family of energy functions:
We are particularly interested in the effects that the two parameters a and b have on this function. In the live picture below a = 0 and b = 0 but you can change the values of these two parameters and see the effects of these changes by clicking at the point on the blue graph that represents the new values of a and b. You can fine tune the values of a and b by using the arrows above and to the right of the blue graph. The red curve on the green graph represents the function V(x) and the blue curve represents its derivative.
Experiment with different values of the parameters a and b by clicking at various places in the blue graph below.
Next you will open a new window with an example. After you open the new window arrange these two windows so you can move back-and-forth between them as we discuss this example. Click here to open the new window.
The graphs in the new window show one of the most interesting values of these two parameters. Notice the energy function has one minimum and the derivative of the energy function has two zeros. One of these zeros corresponds to the minimum of the energy function. The other one corresponds to a point of inflection of the energy function that also is a critical point. In the live picture above, replicate the situation you see in the example -- that is, position the cursor on the blue graph at the same point it is positioned in the example. Then click the big left arrow twice and then the big right arrow four times. Notice that the point shown in the example is a dividing point (we will call it a cusp point) where the character of the energy function changes. At some points near this cusp point the energy function has two minimums and at others it has only a single minimum.
Now click the Mark point button. Notice that a blue mark appears at the point indicated by the cursor. Just for fun mark a few other points by moving the cursor and pressing the Mark point button. If you click the Clear marked points button the marks will be erased.
Determine the location of the cusp points that separate two regions in the blue graph experimentally. In one region the energy function has two minimums and in the other it has only one minimum. Determine the location of the cusp points by moving systematically along each of the lines a = -1, a = -0.9, a = -0.8, ... and marking each cusp point that you find.
Determine the location of the cusp points that separate two regions in the blue graph analytically by analyzing the function
to see where it has exactly two zeros. This function will have exactly two zeros when one of its zeros is also a local maximum or minimum, or equivalently a point of inflection for the original function V(x).