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I'm not sure if this is the right place to ask a question about the Schrödinger equation, but I'll take my chances anyway. Basically, I would like to know how one can set up a potential function that represents a double-slit barrier and then solve the Schrödinger equation for this potential. Of course, according to classical optics, we will obtain an interference pattern, but it would be nice to see a solution entirely within the quantum-mechanical framework. I see this as a problem in mathematical physics, so hopefully someone could kindly provide me with some references.

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migrated from Oct 15 '12 at 18:33

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I suggest that you consult the next reference about quantum potential, in which De Broglie–Bohm theory considers that famous experiment according to its pseudo classic point of view, in which the path of the particle is followed meaningly, explaining its intrinsical behaviour due to the fact that the derived potential $Q(\vec{x},t)$, has to do directly with the quantic conduct (since the center of mass of the particle), independently of the usual $U(\vec{x},t)$. In other words, Bohm considers that wave function can be written like $\Psi(\vec{x},t)=R(\vec{x},t)\cdot\exp{[\frac{i}{\hbar}\cdot S(\vec{x},t)]}$, where $R\doteq{\rho}^{0.5}$ represents the probability density and $S$ is the already known action. After introducing $\Psi$ into the Schrödinger equation, it is obtained that $Q(\vec{x},t)\propto\dfrac{\nabla^2 R}{R}$. At references of the first link, you can find a lot of articles, explaining us how considering Quantum Mechanics in this way, we can appreciate/follow almost deterministically the trajectory of the particle.

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I will only sketch out how one would arrive at the actual equation, as I am rather lazy at the moment.

First, you will have to decide how many spatial dimensions you want to include: Obviously, a double-slit experiment won’t work in one dimension, so we need at least two (which can then be generalised to three rather easily).

You then basically have a two-dimensional scattering problem, for which one would need a certain potential:

Assume waves to arrive from $x = -\infty$ parallel to the $x$ axis, to be scattered at a potential $U(x,y)$. $U$ should be $0$ nearly everywhere except for a certain barrier, possibly located around the $y$ axis. Something like this should do:

$$U_1(x,y) = \theta(y-y_1) \cdot \theta(x) \cdot \theta(1-x) $$ $$U_2(x,y) = \theta(y_0-y) \cdot \theta(-y_0 - y) \cdot \theta(x) \cdot \theta(1-x) $$ $$U_3(x,y) = \theta(-y-y_1) \cdot \theta(x) \cdot \theta(1-x) $$ $$U(x,y) = u_0 \cdot( U_1(x,y) + U_2(x,y) + U_3(x,y) \quad u_0 \textrm{ large or }\infty, \quad y_{0,1} > 0 $$

$U_1$ is meant to describe a potential of height $u_0$ in the area where $0 < x < 1$ and $y > y_1$ (that is, the upper part of the barrier). $U_2$ describes a similar potential for $0 < x < 1$ and $- y_0 < y < y_0$, $U_3$ is similar to $U_1$ in the lower half-plane for $y < -y_0$.

As you can see, this potential lacks basically any symmetry one could remotely hope for. I would try a planar wave ansatz, but it is really not nice.

Maybe someone else has a better idea? :)

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