D.31 Integral Schrö­din­ger equation

In this note, the integral Schrö­din­ger equation is derived from the partial differential equation version.

First the time-independent Schrö­din­ger equation is rewritten in the form

\begin{displaymath}
\left(\nabla^2 + k^2\right)\psi = f
\qquad k = \frac{\sqrt{2mE}}{\hbar} \quad f = \frac{2mV}{\hbar^2}\psi
\end{displaymath} (D.16)

The left equation is known as the “Helmholtz equation.”

The Helmholtz equation is not at all specific to quantum mechanics. In general it describes basic wave propagation at a frequency related to the value of the constant $k$. The right hand side $f$ describes the amount of wave motion that is created at a given location. Quantum mechanics is somewhat weird in that $f$ involves the unknown wave function $\psi$ that you want to find. In simpler applications, $f$ is a given function.

The general solution to the Helmholtz equation can be written as

\begin{displaymath}
\fbox{$\displaystyle
\left(\nabla^2+k^2\right) \psi = f
...
...ec r}^{\,\prime}) {\,\rm d}^3{\skew0\vec r}^{\,\prime}
$} %
\end{displaymath} (D.17)

Here $\psi_0$ is any solution of the homogeneous Helmholtz equation, the equation without $f$.

To see why this is the solution of the Helmholtz equation requires a bit of work. First consider the solution of the Helmholtz equation for the special case that $f$ is a delta function at the origin:

\begin{displaymath}
\left(\nabla^2+k^2\right)G = \delta^3({\skew0\vec r})
\end{displaymath}

The solution $G$ to this problem is called the “Green’s function of the Helmholtz equation.

The Green’s function can be found relatively easily. Away from the origin $G$ is a solution of the homogeneous Helmholtz equation, because the delta function is everywhere zero except at the origin. In terms of quantum mechanics, the homogeneous Helmholtz equation means a particle in free space, $V$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Possible solutions for $G$ are then spherical harmonics times spherical Hankel functions of the first and second kinds, {A.6}. However, Hankel functions of the first kind are preferred for physical reasons; they describe waves that propagate away from the region of wave generation to infinity. Hankel functions of the second kind describe waves that come in from infinity. Incoming waves, if any, are usually much more conveniently described using the homogeneous solution $\psi_0$.

Further, since the problem for $G$ is spherically symmetric, the solution should not depend on the angular location. The spherical harmonic must be the constant $Y_0^0$. That makes the correct solution a multiple of the spherical Hankel function $h_0^{(1)}$, which means proportional to $e^{{{\rm i}}kr}$$\raisebox{.5pt}{$/$}$$r$. You can easily check by direct substitution that this does indeed satisfy the homogeneous Helmholtz equation away from the origin in spherical coordinates.

To get the correct constant of proportionality, integrate the Helmholtz equation for $G$ above over a small sphere around the origin. In the right hand side use the fact that the integral of a delta function is by definition equal to 1. In the left hand side, use the divergence theorem to avoid having to try to integrate the singular second order derivatives of $G$ at the origin. That shows that the complete Green’s function is

\begin{displaymath}
G({\skew0\vec r}) = -\frac{e^{{\rm i}k r}}{4\pi r} \qquad r=\vert{\skew0\vec r}\vert
\end{displaymath}

(You might worry about the mathematical justification for these manipulations. Singular functions like $G$ are not proper solutions of partial differential equations. However, the real objective is to find the limiting solution $G$ when a slightly smoothed delta function becomes truly singular. The described manipulations are justified in this limiting process.)

The next step is to solve the Helmholtz equation for an arbitrary right hand side $f$, rather than a delta function. To do so, imagine the region subdivided into infinitely many infinitesimal volume elements ${\rm d}{\skew0\vec r}^{\,\prime}$. In each volume element, approximate the function $f$ by a delta function spike $\delta({\skew0\vec r}-{\skew0\vec r}^{\,\prime})f({\skew0\vec r}^{\,\prime}){\,\rm d}{\skew0\vec r}^{\,\prime}$. Such a spike integrates to the same value as $f$ does over the volume element. Each spike produces a solution given by

\begin{displaymath}
G({\skew0\vec r}-{\skew0\vec r}^{\,\prime}) f({\skew0\vec r}^{\,\prime}){\,\rm d}{\skew0\vec r}^{\,\prime}
\end{displaymath}

Integrate over all volume elements to get the solution of the Helmholtz equation (D.17). Substitute in what $f$ is for the Schrö­din­ger equation to get the integral Schrö­din­ger equation.