page_title: Delta potential
section_title: Double delta potential
hierarchical_section_title: Delta potential / Double delta potential
caption_attribution_description: English: Symmetric (gerade) and Anti-symmetric (ungerade) wavefunctions for the Double-well Dirac delta function model with "internuclear" radius R = 2 units.
context_page_description: In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.
  The delta potential well is a limiting case of the finite potential well, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential.
  This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions.
context_section_description: The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation:
  where the potential is now:
  where  is the "internuclear" distance with Dirac delta function (negative) peaks located at x=±R/2 (shown in brown in the diagram).  Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use atomic units and set . Here  is a formally adjustable parameter. From the single well case, we can infer the "ansatz" for the solution to be:
  Matching of the wavefunction at the Dirac delta function peaks yields the determinant: 
  Thus,  is found to be governed by the pseudo-quadratic equation:
  which has two solutions . For the case of equal charges (symmetric homonuclear case), λ=1  and the pseudo-quadratic reduces to:
  The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram) where A =  B  and is called gerade.  Correspondingly, the "-" case is the wave function that is anti-symmetric about the midpoint where  A = –B is called ungerade (shown in green in the diagram).  They represent an approximation of the two lowest discrete energy states of the three-dimensional  and are useful in its analysis.  Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by:
  where W is the standard Lambert W function.  Note that the lowest energy corresponds to the symmetric solution . In the case of unequal charges, and for that matter the three-dimensional molecular problem, the solutions are given by a generalization of the Lambert W function (see section on generalization of Lambert W function and references herein).
  One of the most interesting cases is when qR ≤ 1, which results in .  Thus,  one  has a non-trivial bound state solution with E=0.  For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the transmission coefficient is unity at zero energy.
caption: The symmetric and anti-symmetric wavefunctions for the double-well Dirac delta function model with "internuclear" distance R=2.
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