## 4.6 Compatible observables and the uncertainty relation

Now that we have explained how observables correspond to normal operators, we can try to understand what implications follow from the fact that matrix multiplication does *not* generally commute: **eigenbasis of A**.

First of all, assume that ^{78} that **simultaneously diagonalisable**: there exists a basis in which both

Conversely, say that

Two operators **compatible**; if **incompatible**.

We have said that eigenvectors *both* observables at once, given by their common eigenbasis, say

If, however, *position* and *momentum* of a particle: taking the momentum measurement “spreads out” the position of the particle throughout space, meaning that a position measurement taken immediately prior will have no reason to be the same as a position measurement taken immediately afterwards.

Incompatible operators turn up all over the place, and actually turn out to be very interesting — sometimes it’s *good* when things don’t work too simply!
One particularly interesting question we can ask is the following: *can we quantify how far away from being compatible two incompatible operators are?*
We can make this question more mathematically concrete by rephrasing it slightly, asking if we can find at least *some* states that are *close* to being common eigenstates.

Imagine preparing a huge number of systems into the same initial state **standard deviation** of these variables, ^{79}
The smaller the standard deviation, the more “well defined” the measurement is.
In particular, given any single operator

The really interesting, purely quantum, phenomena, however, comes when

The **uncertainty principle** for operators **commutator**.

This says that there does not exist *any* state for which

You might recognise the name, having maybe heard elsewhere of **Heisenberg’s uncertainty principle**, which is indeed a special case of this: one can show that the commutator of the (one-dimensional) position and momentum operators is exactly **Planck constant**), whence

We said that

In a way which we shall not make precise, the fact that *discrete*, in contrast to classical physics which treats things like energy *continuously*.
Quite wondrously, it is very often the case that taking a limit **classical limit** or **correspondence principle**.
This isn’t unique to quantum physics: special relativity reduces to classical mechanics if we take all velocities to be much smaller than the speed of light; general relativity reduces to the classical theory of gravity if we take all gravitational fields to be weak enough; statistical mechanics reduces to thermodynamics when we take the number of particles to be large enough; and so on.

This idea, that classical systems can be recovered from quantum ones by taking *can we go in the other direction?*
That is, given some classical theory that we know agrees with physical experiments, can we formulate some corresponding quantum version which we might hope to be correct on much smaller scales?
Trying to answer this question has led to some incredibly deep (and very technical) mathematics known as **quantization theory**, with **geometric quantization** and **deformation quantization** being two key areas.

Before moving on, let us consider one more quantum phenomena that arises when we look at incompatible operators.
Suppose that we have three operators, say

First of all, we know the probability of measuring outcome *given that |a\rangle first evolves into the intermediate state |b_k\rangle*: this is the probability of

But now, if we forget entirely about *not generally equal* to the previous expression for *if and only if*

We briefly discuss an explicit scenario of where three evolutions behave in such a paradoxical way later on in Section 9, when we introduce Bell’s theorem, in what is sometimes known as the **quantum Venn diagram paradox**.

To make this argument fully formal, and to deal with the case where

\lambda is degenerate, isn’t too hard, but we don’t want to get too involved with the necessary linear algebra here.↩︎For example, if the random variable is normally distributed, then around 68% of the results will lie within one standard deviation from the expected value.↩︎