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Single, double and triple square well. The potential is always zero except inside the wells where it assumes the constant negative value
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We find the solutions with negative energy therefore inside the well.
All wells have a width of 2a. In the double and triple well the distance between the wells is d.
With negative energy values it is not possible to analytically solve the Schrödinger equation,
analogously to the previous case of the single well.
Since the potential is an even function,
the solutions of Schrödinger equation can only be even and odd functions.
Unless otherwise specified we have set the following values:
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;
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The Mathematica notebook, given a potential V(x),
allows to numerically solve the equation, finding the energies (eigenvalues) and the relative functions (eigenfunctions).
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Single well (with half-width a=3) compared to another single well (with half-width a=6).
Doubling the width of the well approximately doubles the number of solutions
and the distance between the respective eigenvalues approximately halves.
Note that the eigenfunctions are different from zero even within the potential zone,
where classically the particle cannot be found since in that zone its kinetic energy
is lower than that of the potential.
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Single well versus double well. In the double well the solutions double.
Note the slight splitting of the eigenvalues.
If we analyze the double well and we consider a determined eigenvalue, we notice that
the solution in the right branch is practically equal to the solution in the left branch
(up to an insignificant phase factor), and each of them is practically equal to the single well solution.
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Two double well placed at different distances (d=4 versus d=1).
Decreasing the distance between the wells increases the distance of the doubling of the eigenvalues.
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Two double well placed at different distances (d=1 versus d=0.1).
Note that a double well is also shown in the right-hand graph even though the distance between the two wells is very small.
Decreasing the distance between the wells increases the distance of the doubling of the eigenvalues.
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Single well of half-width a=6 versus double well each with half-width a=3 but placed by the very small distance d=0.1.
The eigenvalues and eigenfunctions are almost the same.
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Double well versus triple well. Every single hole has the same semi-width (a=3)
and they are all placed at the same distance (d=1).
Where the single well has only one eigenvalue, in the double hole they become two
and in the triple hole they become three.
The distance of these eigenvalues depends on the distance of the individual holes.
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