PlanetPhysics/Wave Equation of a Particle in a Scalar Potential

In order to form the [[../TransversalWave/|wave equation]] of a [[../Particle/|particle]] in a potential $V(𝐫)$, we operate at first under the conditions of the `geometrical optics approximation' and seek to form an equation of propagation for a [[../CosmologicalConstant2/|wave]] packet $\mathrm{\Psi }(𝐫,t)$ moving in accordance with the de Broglie theory.

The center of the packet travels like a classical particle whose [[../Position/|position]], [[../Momentum/|momentum]], and [[../CosmologicalConstant/|energy]] we shall designate by ${𝐫}_{cl.}$, ${𝐩}_{cl.}$, and $E_{cl.}$, respectively. These quantities are connected by the [[../Bijective/|relation]]

$E_{cl.}=H({𝐫}_{cl.},{𝐩}_{cl.})=\frac{p_{cl.}^2}{2m}+V({𝐫}_{cl.})$

$H({𝐫}_{cl.},{𝐩}_{cl.})$ is the classical [[../Hamiltonian2/|Hamiltonian]]. We suppose that $V(𝐫)$ does not depend upon the time explicitly (conservative [[../GenericityInOpenSystems/|system]]), although this condition is not absolutely necessary for the present argument to hold. Consequently $E_{cl.}$ remains constant in time, while ${𝐫}_{cl.}$ and ${𝐩}_{cl.}$ are well-defined [[../Bijective/|functions]] of $t$. Under the approximate conditions considered here, $V(𝐫)$ remains practically constant over a region of the order of the size of the wave packet; therefore

$V(𝐫)\mathrm{\Psi }(𝐫,t)\approx V({𝐫}_{cl.})\mathrm{\Psi }(𝐫,t)$

On the other hand, if we restrict ourselves to time intervals sufficiently short so that the relative variation of ${𝐩}_{cl.}$ remains negligible, $\mathrm{\Psi }(𝐫,t)$ may be considered as a superposition of plane waves of the [[../Bijective/|type]]

$\mathrm{\Psi }(𝐫,t)=\int F(𝐩){\exp }^{i(𝐩\cdot 𝐫-Et)/\hslash }d𝐩$

whose frequencies are in the neighborhood of $E_{cl.}/\hslash$ and whose wave [[../Vectors/|vectors]] lie close to ${𝐩}_{cl.}/\hslash$. Therefore

$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(𝐫,t)\approx E_{cl.}\mathrm{\Psi }(𝐫,t)$

$\frac{\hslash }{i}\mathrm{\nabla }\mathrm{\Psi }(𝐫,t)\approx {𝐩}_{cl.}(t)\mathrm{\Psi }(𝐫,t)$

and taking the [[../DivergenceOfAVectorField/|divergence]] of this last express ion, one obtains

$-{\hslash }^2{\mathrm{\nabla }}^2\mathrm{\Psi }(𝐫,t)\approx p_{cl.}^2\mathrm{\Psi }(𝐫,t)$

combining the relations (2),(3), and (4) and making use of equation (1), we obtain

$\mathrm{i}\hslash \frac{\partial }{\partial t}\mathrm{\Psi }+\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }-V\mathrm{\Psi }\approx (E_{cl.}-\frac{p_{cl.}^2}{2m}-V({𝐫}_{cl.}))\mathrm{\Psi }\approx 0$

The wave packet $\mathrm{\Psi }(𝐫,t)$ 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 $\mathrm{\Psi }$ satisfies the equation

$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(𝐫,t)=(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(𝐫))\mathrm{\Psi }(𝐫,t)$

It is the Schr\"odinger equation for a particle in a potential $V(𝐫)$.

[1] Messiah, Albert. "[[../QuantumParadox/|Quantum mechanics]]: [[../Volume/|volume]] I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1].