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 or . 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, momentum^{2} 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 , and first measure its position , we end up with the state . Suppose we now want to know the momentum of the state, measuring will give the momentum of rather than that of . 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, 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 . The average of these values is called the expectation value of , it is the statistical average obtained by measuring many identical states 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 for an observable . The uncertainty tells us how the values of the observable are spread around the average; if is small, there is almost no variation in the value of , and we are very likely to measure the average value of the observable; on the other hand, if is very large, the value of 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 reality^{3}. 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 doesn’t affect the measurement of , in which case and 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 and momentum do not commute, which gives rise to the most famous uncertainty relation^{4}:

.

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 becomes very small, but the product 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.

^{1}Although 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.

^{2}momentum is defined as mass times velocity

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

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

^{5}This 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.