## Quantum physics against intuition – Part II

In the first post we discussed the fact that classical first-order logic is distributive, that is, pizza and (lemonade or water) is the same as (pizza and lemonade) or (pizza and water); or symbolically,

$x \land (y \lor z) = (x \land y) \lor (x \land z)$ .

This time the aim will be to come up with an example demonstrating that this very intuitive identity does not always hold in quantum mechanics. To do that, we will need the uncertainty principle discussed in the previous post.

Quantum cyclist

We are going to use the uncertainty principle for position and momentum to construct a system which does not obey the distributive law. To make the numbers a bit simpler, we take $\hbar=1$, so the uncertainty relation looks like:

$\Delta X \Delta P\geq \frac{1}{2}$ .

Recall the example with a cyclist from the first post, we observed that the cyclist being in some interval and having some velocity is the same as the cyclist being in the first half of the interval with the same velocity or the cyclist being in the second half of the interval with the same velocity. Now consider a (tiny!) quantum cyclist; for concreteness, suppose the cyclist is in the interval $[0,1]$ and has the momentum in the interval $[0,\frac{1}{2}]$. For simplicity, we take the uncertainty to be the length of the interval1, so we are saying that the cyclist is equally likely to be anywhere between $0$ and $1$ and is equally likely to have any momentum between $0$ and $\frac{1}{2}$. Hence we have $\Delta X = 1$ and $\Delta P = \frac{1}{2}$. Now let $x$, $y$ and $z$ be the following statements about our system (i.e. about the cyclist):
$x$ = ‘cyclist has the momentum in $[0,\frac{1}{2}]$
$y$ = ‘cyclist is in $[0,\frac{1}{2}]$
$z$ = ‘cyclist is in $[\frac{1}{2},1]$‘.
The distributive law is:

$x \land (y \lor z) = (x \land y) \lor (x \land z)$ .

Note that the left-hand side of this identity is precisely what we have described above; the cyclist is in $[0,1]$ with momentum in $[0,\frac{1}{2}]$. We calculate $\Delta X\Delta P = 1\cdot \frac{1}{2} = \frac{1}{2}$, which satisfies the uncertainty condition, and so the system is physically possible. On the right-hand side, however, we have $(x \land y)$, that is, the cyclist is in $[0,\frac{1}{2}]$ with momentum in $[0,\frac{1}{2}]$, giving both $\Delta X$ and $\Delta P$ as $\frac{1}{2}$. But this violates the uncertainty bound, since $\Delta X\Delta P = \frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}$, which is certainly smaller than $\frac{1}{2}$! Since $(x \land z)$ gives the same uncertainties, we must conclude that both terms on the right-hand side are physically impossible, and thus false. This makes all of the right-hand side false; we must, therefore, conclude that this identity cannot hold in this case, as it equates a true statement about the physical system with a false one.

The example above raises many questions for classical logic. Must we conclude that its axioms and rules of inference don’t always hold? If yes, what would be the axioms, and how would they account for the fact that classical logic is distributive? If no, how do we account for the anomaly described above? It is not even clear if there should be one formal system of reasoning flawlessly applicable in all situations to all possible systems. No matter the answers to these questions, the example certainly opens up the space for development of a formal system correctly describing the logic of quantum mechanics.2

1This is actually not quite correct, e.g.  $\Delta X$ should really be $\frac{1}{\sqrt{12}}$. We can, however, get the uncertainties we want by scaling the intervals accordingly, but this doesn’t really contribute to the understanding, and so we drop the scaling for clarity.

## Quantum physics against intuition – Part I

In this post I will introduce the basic concepts of quantum mechanics, which we will need later to show that the quantum systems don’t always work according to the classical logic. The amount of concepts introduced may be a bit overwhelming, so it’s in fact enough to concentrate on the significance of the uncertainty principle discussed at the end of the post.1

Quantum states

In quantum mechanics, a physical system (e.g. a particle) is represented by a state, which is usually denoted by $\phi$ or $\psi$. The state contains all the physical information about the system; what this means is best understood by analogy. We can think of a state as being equivalent to specifying the location of a car for any given time; from that we can obtain all the other physical properties, like velocity, which we get by calculating the difference in location between two instants in time.

Measurements and observables

The only way we can get information about a state is to measure it, which should be a fairly uncontroversial statement. Mathematically this is captured by acting on the state we want to measure by an operator, which is perhaps less intuitive. Continuing with the analogy of measuring the velocity of a car, the “velocity operator” for our car would calculate the difference in position of the car in some small time interval and then divide that difference by the time interval, while the “position operator” would simply read off the location of the car. Hence, with each measurable property (called observables), like position, velocity, momentum2 etc. we associate an operator, which is denoted by a capital letter.

An important property of quantum mechanics is that a measurement alters the state. That is, if we start with a state $\phi$, and first measure its position $X$, we end up with the state $X\phi$. Suppose we now want to know the momentum $P$ of the state, measuring $P$ will give the momentum of $X\phi$ rather than that of $\phi$. This is very different from classical physics, where the order of measurements (ideally) doesn’t affect the measured values.

To make things even more complicated, the measurement is always of statistical nature. That is, $X\phi$ doesn’t have a certain value which will be measured every time, instead, we can think that there is a whole range or a set of values associated with $X\phi$. The average of these values is called the expectation value of $X$, it is the statistical average obtained by measuring many identical states $\phi$ and then averaging over the measured values.

Uncertainty

Because of the statistical nature of the measurements, there is a natural uncertainty associated with each observable, denoted by $\Delta A$ for an observable $A$. The uncertainty tells us how the values of the observable are spread around the average; if $\Delta A$ is small, there is almost no variation in the value of $A$, and we are very likely to measure the average value of the observable; on the other hand, if $\Delta A$ is very large, the value of $A$ could be almost anything, which amounts to the system having no information about that observable, as we could equally guess the value instead of measuring it. It is important to note that the uncertainty arises from the fact that an observable has a range of possible values, and is thus an inherent property of the theory, and consequently an inherent property of nature, provided that quantum mechanics is an accurate description of reality3. The quantum mechanical uncertainty therefore has nothing to do with experimental uncertainty or precision of our measurement devices.

It is possible that the measurement of $A$ doesn’t affect the measurement of $B$, in which case $A$ and $B$ are said to commute, and they behave more or less like in classical physics. However, if two observables do not commute, there will be an uncertainty relation between them limiting the precision with which the system can have the properties represented by these observables. It so happens that position $X$ and momentum $P$ do not commute, which gives rise to the most famous uncertainty relation4:

$\Delta X \Delta P\geq \frac{\hbar}{2}$.

This is to be read: the uncertainty in position multiplied with the uncertainty in momentum is always larger than (or equal to) some constant; the actual value of the constant is more or less irrelevant, what is important that it is larger than zero. One way to understand what the uncertainty relation conveys is to consider the extreme cases; suppose the position of a particle is known with a great precision, this means $\Delta X$ becomes very small, but the product $\Delta X\Delta P$ must be greater than a constant no matter what, the only way this can be true is the uncertainty of momentum becoming very large, that is, the system loses all information about its momentum. Similarly, if momentum is known with a very high precision, the system loses all information about the location of the particle. What this means realistically is that there will always be some uncertainty in both position and momentum, and if more information is obtained about one of them, some information must be lost about the other.5

As a conclusion, I recommend this highly entertaining yet informative animation illustrating what kind of weird consequences all of this has.

1Although I promised not to introduce all the technical details, I couldn’t resist adding the mathematical derivation of the uncertainty principle as a separate document for those interested, though it may require some mathematical background.

2momentum is defined as mass times velocity

3All the experiments to date agree with the quantum mechanical predictions, indicating that the theory captures at least some features of nature correctly.

4An elegant way to derive the uncertainty principle using Fourier transforms can be found here. For an elementary derivation see the extra document.

5This simple inequality has far reaching implication is physics, for further reading see e.g. http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/UncertaintyPrinciple.htm.