# PlanetPhysics/Wave Equation of a Particle in a Scalar Potential

In order to form the wave equation of a particle in a potential $V(\mathbf {r} )$, we operate at first under the conditions of the `geometrical optics approximation' and seek to form an equation of propagation for a wave packet $\Psi (\mathbf {r} ,t)$ moving in accordance with the de Broglie theory.
The center of the packet travels like a classical particle whose position, momentum, and energy we shall designate by $\mathbf {r} _{cl.}$, $\mathbf {p} _{cl.}$, and $E_{cl.}$, respectively. These quantities are connected by the relation
   $E_{cl.}=H(\mathbf {r} _{cl.},\mathbf {p} _{cl.})={\frac {p_{cl.}^{2}}{2m}}+V(\mathbf {r} _{cl.})$
$H(\mathbf {r} _{cl.},\mathbf {p} _{cl.})$ is the classical Hamiltonian. We suppose that $V(\mathbf {r} )$ does not depend upon the time explicitly (conservative system), although this condition is not absolutely necessary for the present argument to hold. Consequently $E_{cl.}$ remains constant in time, while $\mathbf {r} _{cl.}$ and $\mathbf {p} _{cl.}$ are well-defined functions of $t$. Under the approximate conditions considered here, $V(\mathbf {r} )$ remains practically constant over a region of the order of the size of the wave packet; therefore
   $V(\mathbf {r} )\Psi (\mathbf {r} ,t)\approx V(\mathbf {r} _{cl.})\Psi (\mathbf {r} ,t)$
On the other hand, if we restrict ourselves to time intervals sufficiently short so that the relative variation of $\mathbf {p} _{cl.}$ remains negligible, $\Psi (\mathbf {r} ,t)$ may be considered as a superposition of plane waves of the type
   $\Psi (\mathbf {r} ,t)=\int F(\mathbf {p} )\exp ^{i(\mathbf {p} \cdot \mathbf {r} -Et)/\hbar }d\mathbf {p}$
whose frequencies are in the neighborhood of $E_{cl.}/\hbar$ and whose wave vectors lie close to $\mathbf {p} _{cl.}/\hbar$. Therefore
$i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)\approx E_{cl.}\Psi (\mathbf {r} ,t)$
   ${\frac {\hbar }{i}}\nabla \Psi (\mathbf {r} ,t)\approx \mathbf {p} _{cl.}(t)\Psi (\mathbf {r} ,t)$
and taking the divergence of this last express ion, one obtains
   $-\hbar ^{2}\nabla ^{2}\Psi (\mathbf {r} ,t)\approx p_{cl.}^{2}\Psi (\mathbf {r} ,t)$
combining the relations (2),(3), and (4) and making use of equation (1), we obtain
$\mathrm {i} \hbar {\frac {\partial }{\partial t}}\Psi +{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi -V\Psi \approx \left(E_{cl.}-{\frac {p_{cl.}^{2}}{2m}}-V(\mathbf {r} _{cl.})\right)\Psi \approx 0$
The wave packet $\Psi (\mathbf {r} ,t)$ satisfies - at least approximately - a wave equation of the type we are looking for. We are very naturally led to adopt this equation as the wave equation of a particle in a potential, and we postulate that in all generality, even when the conditions for the `geometrical optics' approximation are not fulfilled, the wave $\Psi$ satisfies the equation
   $i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right)\Psi (\mathbf {r} ,t)$
It is the Schr\"odinger equation for a particle in a potential $V(\mathbf {r} )$.
[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].