# What contributes to the wall of an orbit?

May 29, 2017

I'm not entirely sure what you (the invisible questioner) mean here, but here is an educated guess...

Here's what I think you're referring to.

BOUNDARIES OF AN ORBITAL

Not to be confused with an orbit, an orbital is a region of electron density in the modern model of the atom where the position of the electron is plotted as a probability density, a distribution over many measurements.

This is the $2 s$ orbital:

The darkest regions are where electrons are most likely to be found. They are known as the most probable radial distance [from the nucleus].

Anywhere outside an orbital is where electrons cannot be observed.

This is the result of how the wave function is defined in the first postulate of quantum mechanics (slightly modified from Physical Chemistry: A Molecular Approach, McQuarrie):

The state of a quantum-mechanical system is completely specified by a function $\psi \left(\vec{r}\right)$ that depends on the coordinate of the particle. All possible information about the system can be derived from $\psi \left(\vec{r}\right)$. This function, called the wave function or the state function, has the important property that ${\psi}^{\text{*}} \left(\vec{r}\right) \psi \left(\vec{r}\right) d \tau$ is the probability that the particle lies in the [volume] interval $d \tau$, located at the [radial] position $\vec{r}$.

An important note about this postulate is that $\psi$ is only defined where the system exists... but the orbital is the system here (specified by $\psi$), in which we have the particle, the electron (its position on which $\psi$, and hence the state of the system, depends).

The wave function $\psi$ mathematically vanishes outside the orbital, as formally defined below:

${\overbrace{\psi = \psi \left(\vec{r}\right)}}^{\text{wave function", " "" "overbrace(0 < vecr < r_(max)," "0 < t < oo)^"domain}}$

${\overbrace{\psi \left(0\right) = \psi \left({r}_{\max}\right) = 0}}^{\text{boundary conditions}}$, $\text{ "" } t = 0$

${\overbrace{\textsf{\Psi} \left(\vec{r} , 0\right) = A \psi \left(\vec{r}\right)}}^{\text{initial condition", " "" "overbrace(0 < vecr < r_(max))^"domain}}$

where ${r}_{\max}$ is the radial distance of the orbital past which the orbital ceases to exist, $\textsf{\Psi} = \textsf{\Psi} \left(\vec{r} , t\right)$ is the time-dependent wave function, and $\psi = \psi \left(\vec{r}\right)$ is the time-independent wave function.

$A$ is a normalization constant such that ${\int}_{0}^{{r}_{\max}} {\psi}^{\text{*}} \left(\vec{r}\right) \psi \left(\vec{r}\right) d \tau = 1$.

And thus, the electron only exists until... there is no electron density to speak of, i.e. outside the orbital, the electron does not exist.

Hence, in quantum mechanics, the boundary of the orbital IS the "wall" of the orbital.