TiP 3: A First Look at Quantum Mechanics

In the last post, we talked about classical mechanics and how time appears there. However, classical mechanics works only for large objects. For very small objects like elementary particles, atoms, or molecules, we need to use a better theory. This theory, called quantum mechanics,¹ illustrates that the microscopic world behaves counter-intuitively, if not to say weird. The reason for the weirdness is the following: Let’s say we have two classical objects that move in one dimension, for example the two runners on a track shown in the left panel of figure 1. The dimension along which they run is, say, the \(x\)-dimension, and the position of the two runners along this dimension is given by two numbers \(x_1\) and \(x_2\). In a more abstract way, we can look at them as two points on a line, as shown in the middle panel of the figure. However, instead of describing them as two points in the one-dimensional \(x\)-space, we can also describe their positions as one point in the \((x_1,x_2)\)-space, as indicated in the right panel. In classical mechanics both representations are equivalent, although the representation of two points in one dimension is much closer to the picture of two runners on a track — it is much closer to our experience of the world.

The correspondence of two points in one dimension vs. one point in two dimensions can still be visualized, but this is about as far as we can go with visualizing this correspondence between the two pictures. Think for example of 100 fish moving in an aquarium. If we describe them as classical particles, each of the fish has three coordinates because it can move left or right, forwards or backwards, and up or down. Hence, we have a correspondence between 100 particles in three dimensions (the fish in the aquarium) and one “fictitious” particle in 300 dimensions. 300 dimensions! Have you experienced 300 dimensions lately?

However, our experience of the world is not necessarily a good guide when it comes to the fundamental laws of nature. It turns out that if the objects that we look at are not “big” things like fish but microscopic things like atoms, we need the high-dimensional space.² This is because real particles obey the rules of quantum mechanics and in quantum mechanics, we do not have just positions (coordinates) like \(x_1\), \(x_2\), etc., but we have to work with functions of these coordinates. Specifically, quantum mechanics works with the probability \(P(x_1,x_2)\) to find one particle at \(x_1\) and the other particle at \(x_2\), when their positions are measured.³ That we work with this probabilities is not a choice but a necessity: When we make a measurement of the position of the particles, we use some apparatus. This apparatus has to interact with the particles, which results in a fundamental uncertainty that is manifest in the probability function \(P(x_1,x_2)\). If we make many measurements of identical systems, we find that the location of the particles is on average distributed according to \(P(x_1,x_2)\). However, for a single measurement we cannot predict with certainty where we would find the particles. All we can know is the probability \(P(x_1,x_2)\) to find one particle at \(x_1\) and the other one at \(x_2\).

Consequently, all we have is \(P(x_1,x_2)\), schematically shown in the left panel of figure 2. There are ways to get functions \(P_1(x)\) and \(P_2(x)\) which correspond to the probability of finding one particle somewhere along one dimension \(x\) and the other particle somewhere along this dimension, but they contain less information than the full function \(P(x_1,x_2)\). If you have \(P(x_1,x_2)\), you can get \(P_1(x)\) and \(P_2(x)\), but if you only have \(P_1(x)\) and \(P_2(x)\) you cannot get \(P(x_1,x_2)\). For example, in the right panel of figure 2 another function \(P'(x_1,x_2)\) is shown. Both \(P(x_1,x_2)\) in the left panel and \(P'(x_1,x_2)\) in the right panel would give the same functions \(P_1(x)\) and \(P_2(x)\) shown in the middle panel, although they look very different!⁴

Hence, while in classical mechanics we freely choose if we want to describe 100 particles in three dimensions or one particle in 300 dimensions, in quantum mechanics we have to work with functions in a 300-dimensional space. Only in certain situations can these functions be replaced by 100 functions in a three-dimensional space without loss of relevant information, and these functions might, if classical mechanics is a good approximation, alsobe replaced with 100 points in a three-dimensional space.

Despite this profound difference, if you look in standard text books you get the impression that time in quantum mechanics works exactly like in classical mechanics. Can this be right? Only if the clock works according to the rules of classical mechanics, as we will discuss next.

To summarize, we found that

• fundamentally, we only know probabilities of finding particles at a certain location if their position is somehow measured,
• these probabilities are functions in the space of the coordinates of all particles, and
• reduction to a space with less dimensions is only possible if we are willing to ignore some information.

Ah, and we also learned that the microscopic world is weird.

1. Quantum Mechanics is more fundamental than classical mechanics in the sense that classical mechanics can be obtained from quantum mechanics, if the considered system is (in a certain mathematical sense) large or heavy enough.
2. BTW, this space is usually called “configuration space” because it is the space of all configurations of the system.
3. What we specifically mean with a measurement will be discussed another time. It is not so easy.
4. The situation is a bit similar to the reconstruction of the shape of an object from its shadow. If you know the picture of the shadow of an object from two angles, you still don’t know its shape.