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Multiple Wells

Warning! The graphs in this problem use color. You cannot do this problem from a black and white hard copy.

The figure below shows the energy levels of a single, double, triple, and quadruple well. Note that the single well has two states, the double well has two doublets, the triple well has two triplets, and the quadruple well has two quadruplets. With more wells the energy levels form bands, and the bands are widely separated compared to the spacing of levels within the bands. The width of the wells is L = 12 nm, the separation between adjacent wells is D = 5 nm. Let the potential, V, be zero inside the well. Outside the the well, V = 0.106 eV. These parameters are taken from an article by Kolbas and Holonyak, Amer. J. Phys. 52 (431)1984, Man-made quantum wells. The semiconductor industry manufactures layered structures which are accurately modeled by multiple wells, e.g. laser emitting diodes. In all the plots below, the origin, x = 0, is at the left edge of the first well.

a) For the single well to the accuracy that you can read the energy level diagram, what is the energy
of the ground state? (you need to be within 0.01 eV of the correct answer)
E1 = eV      

and first excited state?
E2 = eV        

b) For the double well to the accuracy that you can read the energy level diagram, what is the separation of the ground state doublet? (you need to be within 0.001 eV of the correct answer)

DE = eV    

c) If the separation between the wells, D, were increased from 5 nm to 6 nm, would the separation of the ground state doublet increase or decrease?
    


The graph below shows the wave functions of the single well.

d) Identify the ground state wave function. Is it red or blue?
    


The two graphs below show the wave functions of the double well. One graph shows the wave functions of the ground state doublet. The other graph shows the wave functions of the first excited state doublet.

e) Is the ground state doublet shown by the solid or dotted graph?
    

f) With respect to the point x = 14.5 nm, the midpoint of the region between the two wells, is the lowest level of the ground state doublet an even function or an odd function? (That is, is it the red or blue curve in the graph above?)
    


The two graphs below show the wave functions of the triple well. The graph with solid curves shows the wave functions of the ground state triplet. The graph with dotted curves shows the wave functions of the first excited state.

g) Using the key,

1 = solid red ___
2 = solid green ___
3 = solid blue ___
4 = dotted red .....
5 = dotted green .....
6 = dotted blue .....
identify the wave function that corresponds to
the least energetic level of the ground state triplet
the middle level of the ground state triplet
the most energetic level of the ground state triplet
the least energetic level of the excited state triplet
the middle level of the excited state triplet
the most energetic level of the excited state triplet

This page contributed by "SnotNoodle"