PX275 - A4c - the laplacian

cartesian coordinates

the laplacian or the grad-squared operator is given by:

\nabla^{2} f = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}}

$\nabla^{2}$ acts as a scalar, giving a scalar result
eg: in the schrödinger equation:

\frac{- ℏ^{2}}{2 m} \nabla^{2} ψ + V ψ = E ψ

where, $ψ =$ the wavefunction, $E =$ eigen value, $V =$ the potential function
- the hamiltonian operators:
- $\frac{- ℏ^{2}}{2 m} \nabla^{2} ψ$ gives the kinetic energy
- $V ψ$ gives the potential energy

$\nabla^{2}$ can also act on vector quantities:

\nabla^{2} \vec{E} = \frac{1}{c^{2}} \frac{\partial^{2} \vec{E}}{\partial t^{2}}

- this is vector calculus

plane polar coordinates

\frac{\partial^{2}}{\partial x^{2}} = {(\cos θ \frac{\partial}{\partial ρ} - \frac{1}{ρ} \sin θ \frac{\partial}{\partial θ})}^{2}

warning: do not multiply out the bracket, it is the operator syntax

\begin{aligned} \nabla^{2} & = \frac{\partial^{2}}{\partial x} + \frac{\partial^{2}}{\partial y^{2}} \\ = (\frac{\partial^{2}}{\partial ρ^{2}} + \frac{1}{ρ} \frac{\partial}{\partial ρ} + \frac{1}{ρ^{2}} \frac{\partial^{2}}{\partial θ^{2}}) \end{aligned}

spherical polar coordinates

\nabla^{2} = \frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos θ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ϕ^{2}}