Show that the Laplace operator is Hermitian

[Lambda96]

Homework Statement

Show that the Laplace operator is hermetian

Relevant Equations

none

Hi,

the task is as follows:

I now have to show the following

$$\begin{align*}

\langle f , \Delta g \rangle &= \langle \Delta f , g \rangle\\

\int_{V} dx^3 \overline{f(x)} \cdot \Delta g&= \int_{V} dx^3 \overline{\Delta} \overline{f(x)} \cdot g\\

\int_{V} dx^3 \overline{f(x)} \cdot \Delta g&= \int_{V} dx^3 \Delta \overline{f(x)} \cdot g\\

\end{align*}$$

Unfortunately, I can't get any further because I don't know how to show that the equations are equal. I assume that Dirichlet and the Neumann boundary conditions must be used, unfortunately I don't know how to include them in my expression above.

 

[TSny]

I now have to show the following
$$\begin{align*}

\langle f , \Delta g \rangle &= \langle \Delta f , g \rangle\\

\int_{V} dx^3 \overline{f(x)} \cdot \Delta g&= \int_{V} dx^3 \overline{\Delta} \overline{f(x)} \cdot g\\

\int_{V} dx^3 \overline{f(x)} \cdot \Delta g&= \int_{V} dx^3 \Delta \overline{f(x)} \cdot g\\

\end{align*}$$

Unfortunately, I can't get any further because I don't know how to show that the equations are equal. I assume that Dirichlet and the Neumann boundary conditions must be used, unfortunately I don't know how to include them in my expression above.

Integration by parts can be used to transfer a derivative acting on $g$ in the integrand to a derivative acting on $\overline f$.

 

[Lambda96]

Thanks TSny for your help and the tip

I started with the integral on the left-hand side:

$$ \int_{V} dx^3 \overline{f(x)} \cdot \Delta g=\int_{\partial V} dS \overline{f(x)} \cdot \partial_n g-\int_{V} dx^3 \Delta \overline{f(x)} \cdot \Delta g =-\int_{V} dx^3 \Delta \overline{f(x)} \cdot \Delta g$$

And the following applies to the right-hand side

$$\int_{V} dx^3 \Delta \overline{f(x)} \cdot g=\int_{\partial V} dS \partial_n \overline{f(x)} \cdot \Delta g-\int_{V} dx^3 \Delta \overline{f(x)} \cdot \Delta g =-\int_{V} dx^3 \Delta \overline{f(x)} \cdot \Delta g$$

Then the following applies to both sides, which I should show:

$$-\int_{V} dx^3 \Delta \overline{f(x)} \cdot \Delta g=-\int_{V} dx^3 \Delta \overline{f(x)} \cdot \Delta g$$

Is that correct?

 

[TSny]

Thanks TSny for your help and the tip

I started with the integral on the left-hand side:

$$ \int_{V} dx^3 \overline{f(x)} \cdot \Delta g=\int_{\partial V} dS \overline{f(x)} \cdot \partial_n g-\int_{V} dx^3 \Delta \overline{f(x)} \cdot \Delta g $$

The integrand of the last integral is not correct. In this integral did you mean to use the notation $\vec \nabla$ instead of $\Delta$?

Note that $$\int_{V} d^3x \overline{f(x)} \cdot \Delta g = \int_0^{L_z} \int_0^{L_y} \int_0^{L_x} \bar f \left( \partial_x^2 + \partial_y^2 + \partial_z^2 \right) g \, dx dy dz $$
Consider the x-integral of the first term in the integrand: $\int_0^{L_x} \bar f \, \partial_x^2 \, g \, dx$. What do you get if you integrate this by parts twice?

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Another approach is to make use of the divergence theorem (Gauss' theorem) and the identity $\vec{\nabla} \cdot (\phi \vec A) = (\vec \nabla \phi)\cdot \vec A + \phi \vec \nabla \cdot \vec A$ where $\phi$ is a scalar function and $\vec A$ is a vector function. I'll let you think about this approach. Maybe this is what you were trying to do in your approach.