# Atoms

## What does an atom look like?

### Or like this?

None of these images depicts an atom as it is. This is because it is impossible to even visualize an atom as it is. Whereas the best you can do with the images in the first row is to erase them from your memory—they represent a way of viewing the atom that is too simplified for the way we want to start thinking about it—the eight fuzzy images in the next row deserve scrutiny. Each represents an aspect of a stationary state of atomic hydrogen. You see neither the nucleus (a proton) nor the electron. What you see is a fuzzy position. To be precise, what you see are cloud-like blurs, which are symmetrical about the vertical and horizontal axes, and which represent the atom's internal relative position—the position of the electron relative to the proton or the position of the proton relative to the electron.

• What is the state of an atom?
• What is a stationary state?
• What exactly is a fuzzy position?
• How does such a blur represent the atom's internal relative position?
• Why can we not describe the atom's internal relative position as it is?

## Quantum states

In quantum mechanics, states are probability algorithms. We use them to calculate the probabilities of the possible outcomes of measurements on the basis of actual measurement outcomes. A quantum state takes as its input

• one or several measurement outcomes,
• a measurement M,
• the time of M,

and it yields as its output the probabilities of the possible outcomes of M.

A quantum state is called stationary if the probabilities it assigns are independent of the time of the measurement.

From the mathematical point of view, each blur represents a density function ${\displaystyle \rho ({\boldsymbol {r}})}$. Imagine a small region ${\displaystyle R}$ like the little box inside the first blur. And suppose that this is a region of the (mathematical) space of positions relative to the proton. If you integrate ${\displaystyle \rho ({\boldsymbol {r}})}$ over ${\displaystyle R,}$ you obtain the probability ${\displaystyle p\,(R)}$ of finding the electron in ${\displaystyle R,}$ provided that the appropriate measurement is made:

${\displaystyle p\,(R)=\int _{R}\rho ({\boldsymbol {r}})\,d^{3}{\boldsymbol {r}}.}$

"Appropriate" here means capable of ascertaining the truth value of the proposition "the electron is in ${\displaystyle R}$", the possible truth values being "true" or "false". What we see in each of the following images is a surface of constant probability density.

Now imagine that the appropriate measurement is made. Before the measurement, the electron is neither inside ${\displaystyle R}$ nor outside ${\displaystyle R}$. If it were inside, the probability of finding it outside would be zero, and if it were outside, the probability of finding it inside would be zero. After the measurement, on the other hand, the electron is either inside or outside ${\displaystyle R.}$

Conclusions:

• Before the measurement, the proposition "the electron is in ${\displaystyle R}$" is neither true nor false; it lacks a (definite) truth value.
• A measurement generally changes the state of the system on which it is performed.

As mentioned before, probabilities are assigned not only to measurement outcomes but also on the basis of measurement outcomes. Each density function ${\displaystyle \rho _{nlm}}$ serves to assign probabilities to the possible outcomes of a measurement of the electron's position relative to the proton. And in each case the assignment is based on the outcomes of a simultaneous measurement of three observables: the atom's energy (specified by the value of the principal quantum number ${\displaystyle n}$), its total angular momentum ${\displaystyle l}$ (specified by a letter, here p, d, or f), and the vertical component of its angular momentum ${\displaystyle m}$.

## Fuzzy observables

We say that an observable ${\displaystyle Q}$ with a finite or countable number of possible values ${\displaystyle q_{k}}$ is fuzzy (or that it has a fuzzy value) if and only if at least one of the propositions "The value of ${\displaystyle Q}$ is ${\displaystyle q_{k}}$" lacks a truth value. This is equivalent to the following necessary and sufficient condition: the probability assigned to at least one of the values ${\displaystyle q_{k}}$ is neither 0 nor 1.

What about observables that are generally described as continuous, like a position?

The description of an observable as "continuous" is potentially misleading. For one thing, we cannot separate an observable and its possible values from a measurement and its possible outcomes, and a measurement with an uncountable set of possible outcomes is not even in principle possible. For another, there is not a single observable called "position". Different partitions of space define different position measurements with different sets of possible outcomes.

• Corollary: The possible outcomes of a position measurement (or the possible values of a position observable) are defined by a partition of space. They make up a finite or countable set of regions of space. An exact position is therefore neither a possible measurement outcome nor a possible value of a position observable.

So how do those cloud-like blurs represent the electron's fuzzy position relative to the proton? Strictly speaking, they graphically represent probability densities in the mathematical space of exact relative positions, rather than fuzzy positions. It is these probability densities that represent fuzzy positions by allowing us to calculate the probability of every possible value of every position observable.

It should now be clear why we cannot describe the atom's internal relative position as it is. To describe a fuzzy observable is to assign probabilities to the possible outcomes of a measurement. But a description that rests on the assumption that a measurement is made, does not describe an observable as it is (by itself, regardless of measurements).