- #1

Wavefunction

- 99

- 4

## Homework Statement

Find an expression involving the function [itex] ϕ(x_1, x_2, x_3) [/itex] that has a minimum average value of the

square of its gradient within a certain volume V of space.

If ϕ is the electric potential, [itex] \vec{E} = -\nabla ϕ [/itex] is the electric field, and [itex] ρ = \frac{1}{2} ϵ_0|\vec{E}\cdot\vec{E}| [/itex] is the energy density of

the electric field, this result tells us what equation the electric potential must satisfy to minimize

the total energy.

## Homework Equations

(1)[itex] \frac{∂f}{∂ϕ}+\sum_{i=1}^3\frac{∂}{∂x_i}\frac{∂f}{∂ϕ'_i} = 0 [/itex] where [itex] ϕ'_i [/itex] are the partial derivatives of [itex] ϕ [/itex]

(2)min[itex](\frac{1}{V}\iiint ϕ dV) = \nabla ϕ \cdot \nabla ϕ [/itex]

## The Attempt at a Solution

First, since minimizing the average value of [itex] ϕ [/itex] yields a function [itex] \nabla ϕ \cdot \nabla ϕ [/itex] I will use this as my functional since it must satisfy (1). Doing so yields:

[itex] \frac{∂f}{∂ϕ}= 0 [/itex] and [itex] \nabla \cdot [\frac{∂(\nabla ϕ \cdot \nabla ϕ)}{∂(\nabla ϕ)}] = 0 [/itex]

The second part of (1) yields:

[itex] 2\nabla \cdot \nabla ϕ =0 [/itex] or alternatively, [itex] \nabla^2 ϕ = 0 [/itex] (Laplace's Equation)

Now this makes sense to me ( or at least tells me that I'm on the right track) since I know that the electrostatic potential does satisfy this equation; however, I know that Laplace's equation is a simpler case of Poisson's equation and when I plug my result in using [itex] \vec{E} = -\nabla ϕ [/itex] I get [itex] \nabla \cdot \vec{E} = 0 [/itex] which is only true if the charge density inside of the object is zero. So here's my question should I have gotten the more general Poisson equation, or am I overthinking this whole thing?