# This Quantum World/Atoms#Quantum states

# Atoms

## What does an atom look like?

### Like this?

Image:Barium\_(Elektronenbesetzung).png Image:Atom.png Image:Stylised
atom with three Bohr model orbits and stylised nucleus.png
Image:Rutherford_atom.svg

### Or like this?

Image:Orbitals1.png\|$\rho_{2p0}$ Image:Orbitals2.png\|$\rho_{3p0}$
Image:Orbitals3.png\|$\rho_{3d0}$ Image:Orbitals4.png\|$\rho_{4p0}$
Image:Orbitals5.png\|$\rho_{4d0}$ Image:Orbitals6.png\|$\rho_{4f0}$
Image:Orbitals7.png\|$\rho_{5d0}$ Image:Orbitals8.png\|$\rho_{5f0}$

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
$\rho(\boldsymbol{r})$. Imagine a small region $R$ 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 $\rho(\boldsymbol{r})$ over $R,$ you obtain the probability
$p\,(R)$ of finding the electron in $R,$ *provided* that the appropriate
measurement is made:

$$p\,(R)=\int_R\rho(\boldsymbol{r})\,d^3\boldsymbol{r}.$$
\"Appropriate\" here means capable of ascertaining the truth value of
the proposition \"the electron is in $R$\", the possible truth values
being \"true\" or \"false\". What we see in each of the following images
is a surface of constant probability density.

\

Image:Orbitals1a.png\|$\rho_{2p0}$ Image:Orbitals2a.png\|$\rho_{3p0}$
Image:Orbitals3a.png\|$\rho_{3d0}$ Image:Orbitals4a.png\|$\rho_{4p0}$
Image:Orbitals5a.png\|$\rho_{4d0}$ Image:Orbitals6a.png\|$\rho_{4f0}$
Image:Orbitals7a.png\|$\rho_{5d0}$ Image:Orbitals8a.png\|$\rho_{5f0}$

\
Now imagine that the appropriate measurement is made. *Before* the
measurement, the electron is neither inside $R$ nor outside $R$. 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 $R.$

Conclusions:

-   Before the measurement, the proposition \"the electron is in $R$\"
    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 $\rho_{nlm}$ 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 $n$), its total
angular momentum
$l$ (specified by a letter, here *p*, *d*, or *f*), and the vertical
component of its angular momentum $m$.

## Fuzzy observables

We say that an observable $Q$ with a finite or countable number of
possible values $q_k$ is **fuzzy** (or that it has a fuzzy value) if and
only if at least one of the propositions \"The value of $Q$ is $q_k$\"
lacks a truth value. This is equivalent to the following necessary and
sufficient condition: the probability assigned to at least one of the
values $q_k$ 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*).

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