In quantum mechanics phenomenon of tunneling is well understood ; we know that there is some finite probability to find the particle in classically forbidden region but potential of this forbidden region should not be infinite (for e.g. In case of infinite potential well, probability to find particle outside well, where potential is infinite, is zero.). Now coming to my question : we get some non zero probability even in the case of Dirac Delta potential well i.e, particle can tunnel through this infinite potential line (at some point). How is it possible?


Dirac potential has infinite energy, but zero width. It can be seen as a limiting case of a potential with finite height and width. For example, one could consider potential $$U(x) = \begin{cases}U_0, |x|<\frac{a}{2},\\ 0, |x|> \frac{a}{2}\end{cases},$$ solve the Schrödinger equation for this potential, and then take the limit $$U_0\rightarrow \pm\infty, a \rightarrow 0, U_0 a \rightarrow const.$$ Other shapes of potential giving Dirac delta-function as a limit (Gaussian, Lorentzian) can be also considered with an obvious observation that solving Dirac potential is easier.

  • $\begingroup$ So, doesn't it implies that particle will have negative kinetic energy at that particular point? $\endgroup$
    – Barry
    Apr 30 '20 at 9:43
  • $\begingroup$ You mean that inside the barrier the wave function is exponentially decaying, rather than a plane wave? $\endgroup$ Apr 30 '20 at 9:48
  • $\begingroup$ Yes, and also for crossing that infinite barrier kinetic energy of particle must be minus infinity( since K+V=E) $\endgroup$
    – Barry
    Apr 30 '20 at 10:08
  • $\begingroup$ The operator of the kinetic energy is $\hat{K}=-\frac{\hbar^2\nabla^2}{2m}$. It is not negative but the energy conservation breaks whenever a particle tunnels under a barrier (not necessarily Dirac barrier). This is why the phenomenon of tunneling is not possible in classical mechanics, where the energy is strictly conserved. $\endgroup$ Apr 30 '20 at 10:12

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