Two Interacting Wells

The two wells are identical to the ones in Section 1 except that they are closer.

Section 2 Exercises:

  1. What is the width of the barrier separating the two wells?
  2. What are the energies of the first six eigenstates?  Are any of them degenerate to the accuracy of the measurement?
  3. What is the parity of this lowest lying state?

The Schroedinger equation relates the second derivative of the wave function to the energy times the wave function.  Mathematically speaking, the second derivative is a measure of the curvature of a function.  A line has zero curvature and a parabola has a constant curvature.  An oscillatory function has a curvature that changes from + to negative.  Many texts identify the "curvature" as the second derivative divided by the function.  Some prefer to call this the "waviness" or "curviness" of the function.  With this definition, an oscillatory function has a negative waviness since the second derivative is always opposite in sign to the function at an arbitrary x.  Such functions are called concave functions.  A monotonic function, also called a convex function, has a positive waviness and is either entirely nonincreasing or nondecreasing, i.e., its first derivative (which need not be continuous) does not change sign.   An exponential function is an example of a convex function.

  1. Which of the lowest two states has the least overall waviness?  Does it correspond to the state with the largest binding energy?
  2. Is the probability of finding the electron halfway between the two wells zero for either of the two lowest states?  How might your answer to this question be used to interpret your result in #3 above?
  3. Use the increase in energy splitting between pairs as the quantum number n increases to make a general statement about the waviness of the wave function.

 

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