In the previous post, we looked at a central feature of the theory of the microscopic world of elementary particles, atoms, and molecules, known as quantum mechanics. We learned that in quantum mechanics, we have to work probabilities which are functions of the coordinates of all particles. Consequently, these functions live in a high-dimensional space — for 100 particles, the space is 300-dimensional, with three spatial dimensions per particle. Unlike in classical mechanics, there is no rigorous way to reduce these functions to a three-dimensional picture without some loss of information. This is certainly not very intuitive for humans that are used to experiencing the world in three spatial dimensions.

So, now that we know that space in quantum mechanics is far from being intuitive, what about time? Actually, in standard textbooks on quantum mechanics you find that the probabilities are functions of space and of a parameter \(t\), called the time. And, if you look close enough, you will notice that this parameter \(t\) works exactly as in classical mechanics: It labels changes of the system (here: of the probabilities) but it is not clear where it comes from. The situation is shown schematically in figure 1, where a probability function \(P(x_1,x_2|t)\) is sketched for three different values \(t_1\), \(t_2\), and \(t_3\) of the time parameter \(t\). The solution to the central equation of quantum mechanics, the time-dependent Schrödinger equation, provides us such a probability function \(P(x_1,x_2|t)\) which continuously changes when the parameter \(t\) changes. Again, like in our first discussion of classical mechanics, the time parameter is a Newtonian time that is external to the system and that is assumed to somehow flow without underlying reason.

Aha. Wouldn’t it be better to replace the dubious concept of Newtonian time with an internal correlation of motions, like we did already for classical mechanics? Yes, it would! To do that, we can repeat the program that we did for classical mechanics and explicitly define a clock as part of the system. To make life simple, we do not try to separate the quantum system into two parts (the clock and the actual system of interest), but we add a clock explicitly. Not only that, but we use the same clock that we did when discussing the time parameter in classical mechanics (except for the color): We add a white billiard ball rolling with constant velocity on a billiard table in, say, the \(y\)-direction of the table, as shown in Figure 2. The position of the ball in this direction, \(y_{\rm w}\), can be used as a time parameter for the probability function, \(P(x_1,x_2|y_{\rm w})\).
In real life we don’t use billiard balls, but for example laser pulses to provide a time parameter for the evolution of a quantum system. The main application for what I am usually doing is attoscience, where experimentalists use laser pulses of attosecond duration (\(10^{-18}\) seconds — yes, that is pretty short) to monitor the dynamics of atoms and molecules. Its like a super fast camera, and the laser pulses1 provide a \(t\) as label for the snapshots of the camera.
Why I choose billiard balls as clock here is not because this would be a good idea, but to make one important aspect clear: Time in quantum mechanics is defined via a clock that effectively behaves according to classical mechanics. This is why time in quantum mechanics seems not to be as weird as we would expect it, if we compare it to the idea of working in a high-dimensional space. This also means, however, that the time parameter is something like an intruder in the theory, because the clock from which it originates is not described on the same footing as the particles that constitute the quantum system which we are looking at.
This intruder is welcome for practical application but not so welcome if we want to understand the meaning of time on a fundamental level. To get this understanding, we would have to use a quantum-mechanical description of a clock. Unsurprisingly, this leads to some major problems: For example, a single time parameter cannot be defined anymore because we cannot know where the clock is and how fast it moves. Like for the quantum system, we only have a probability function that tells us where the clock can be found if it is measured. But to measure the position of the clock, we need to interact with it, which leaves some uncertainty in its speed.2 And, more importantly, we need an external apparatus that does this measurement, which somehow collides with the idea that we don’t want an external time.3
It is complicated, but the way how time can be derived mathematically in a quantum system gives us some insights. This will, however, be a topic for another post. What we can already understand from the discussion so far is:
- Time in quantum mechanics is usually treated as external parameter
- but it can be obtained as correlation of changes of the quantum system with changes of a classical system used as clock;
- hence, such a time parameter is not part of quantum mechanics but of classical mechanics — time-dependent quantum mechanics is a semi-classical theory.
Ah, and, by the way, time is not a fourth dimension if we take quantum mechanics into account. Even if we only have two particles an we use one as the actual quantum system and the other one as our clock, we have — guess what! — three spatial dimensions for the quantum system (particle 1) and three spatial dimensions for the clock (particle 2). There is no way to define one time parameter with the position and velocity of the clock, like we did in classical mechanics with the position and velocity of the billiard ball. This is because we only have probability functions for the clock’s position and velocity. These functions are defined on a three-dimensional space which cannot be reduced to only one dimension without loss of information.
- The laser pulses can be described with the equivalent of classical mechanics for light.
- This is Heisenberg’s uncertainty principle, which, for many, is a central feature of quantum mechanics. We will talk about it in the future.
- An external apparatus defines the external time, like ultrashort laser pulses used as a camera to monitor the dynamics of a molecule define a time parameter.