### Measurements as observables

We are rarely able to measure directly the quantities that are the final subjects of experiments. Normally, we need to deduce these quantities from other measurable quantities.

These may simply be positions on a dial, scale or ruler. They could also be values that have been assigned arbitrarily to the various outcomes for an experiment.

Consider again the coin, with a state-space (just a set in classical mechanics) comprising of two outcomes, $\left\{\text{Heads}, \text{Tails}\right\}$. In order to make quantifiable statements about this system it is necessary to assign *real* values to each state. For example, $$\begin{align*}F\left(\text{Heads}\right) = 1\\F\left(\text{Tails}\right) = 0\end{align*}$$

Then, to do an experiment we toss the coin and then making a measurement would be writing down the value corresponding to the face shown.

In general, a classical **observable**, $F$, is a function from the state space, $\mathbb{S}$ to the real numbers, $\mathbb{R}$. That is, $$F:\mathbb{S} \to \mathbb{R}$$

A slightly more illuminating example would be a die, with the six outcomes, $\mathbb{S} = \{1, 2, \cdots, 6\}$. Even though the outcomes are already numbers as such, it might still be necessary to assign other values to each state. The experiment might be to determine the probability that the die lands on a multiple of three. In which we case we might define the observable to be $$\forall \:n \in \mathbb{S}, \quad F_n = \left\{\begin{align*}1 &\text{ if } n \in \{ 3, 6 \}\\0 &\text{ otherwise }\end{align*}\right.$$

Both the coin and the die are discrete systems, $\mathbb{S} \subset \mathbb{N}$, but, in general, systems are continuous, $\mathbb{S} \subset \mathbb{R}$.

### Probability

For a given system, each possible of outcome of an experiment, by which we mean a possible state of the system, will have a certain likelihood of occurring. We define the **probability** that a particular event occurs to be the function $$P: \mathbb{S} \to \left [ 0, 1 \right ] $$ such that $$\begin{align*}\mathbb{S} \text{ is discrete} \quad &\Rightarrow \quad \sum_{n \in \mathbb{S}} P_n = 1\\\mathbb{S} \text{ is continuous} \quad &\Rightarrow \quad \int_{x \in \mathbb{S}} P(x) \mathrm{d}x = 1\end{align*}$$

For the coin, $$P\left(\text{Heads}\right) = P\left(\text{Tails}\right) = \tfrac{1}{2}$$ and for the die, $$\forall \: n \in \mathbb{S}, \quad P_n = \tfrac{1}{6}$$

A probability function is usually called a **probability distribution**. We can extend the definition to probabilities associated with observables, for example we could consider the probability that a die lands on a multiple of three.

### Average value

We define the average value, or expectation, of an observable, $F$, written $\bar{F}$ or $\left \langle F \right \rangle$, to be the sum (or integral) of the observable's values weighted by their probabilities of occurring, $$\begin{align*}\bar{F} &= \sum_{n \in \mathbb{S}} P_n F_n & \left( \text{discrete} \right)\\\bar{F} &= \int_{x \in \mathbb{S}} P(x) F(x) \mathrm{d}x & \left( \text{continuous} \right)\end{align*}$$

For the die example above, where $F = 1$ for multiples of three and zero otherwise, then the average for this to occur is $$\bar{F} = \sum_{n \in \{ 3, 6 \}} \tfrac{1}{6} \left( 1 \right) + \sum_{n \not \in \{ 3, 6 \}} \tfrac{1}{6} \left( 0 \right) = \tfrac{1}{3}$$

### Classical vs. quantum observables

In quantum mechanics, observables are not real-valued functions, but a specific set of linear operators called **Hermitian** matrices that act on quantum states and allow us to calculate the probabilities of particular outcomes.

The values that can be *measured* are exactly the eigenvalues of the observable. Futhermore, the eigenvectors of the observable are the states that have 100% certainty of occurring if we are measuring the associated eigenvalue.