A.13 Integral Schrö­din­ger equation

The Hamiltonian eigenvalue problem, or time-independent Schrö­din­ger equation, is the central equation of quantum mechanics. It reads

\begin{displaymath}
\frac{\hbar^2}{2m} \nabla^2 \psi({\skew0\vec r}) + V({\skew0\vec r}) \psi({\skew0\vec r}) = E \psi({\skew0\vec r})
\end{displaymath}

Here $\psi$ is the wave function, $E$ is the energy of the state described by the wave function, $V$ is the potential energy, $m$ is the mass of the particle, and $\hbar$ is the scaled Planck constant.

The equation also involves the Laplacian operator, defined as

\begin{displaymath}
\nabla^2 \equiv \frac{\partial^2}{\partial x^2}+
\frac{\partial^2}{\partial y^2}+ \frac{\partial^2}{\partial z^2}
\end{displaymath}

Therefore the Hamiltonian eigenvalue problem involves partial derivatives, and it is called a partial differential equation.

However, it is possible to manipulate the equation so that the wave function $\psi$ appears inside an integral rather than inside partial derivatives. The equation that you get this way is called the integral Schrö­din­ger equation. It takes the form, {D.31}:

\begin{displaymath}
\fbox{$\displaystyle
\psi({\skew0\vec r}) = \psi_0({\ske...
...ec r}^{\,\prime}
\qquad k = \frac{\sqrt{2mE}}{\hbar}
$} %
\end{displaymath} (A.42)

Here $\psi_0$ is any wave function of energy $E$ in free space. In other words $\psi_0$ is any wave function for the particle in the absence of the potential $V$. The constant $k$ is a measure of the energy of the particle. It also corresponds to a wave number far from the potential. While not strictly required, the integral Schrö­din­ger equation above tends to be most suited for particles in infinite space.