When anybody learns quantum mechanical concepts for the first time it is often hard to visualise what is really going on for a quantum particle: an object neither describable simply in term of a physical wave nor in term of classical mechanics. Now, to be clear, nobody knows what happens “really” to a quantum particle when we don’t observe it but the purpose of quantum mechanics is precisely to figure out, among other things, the likelyhood with which we can find a given quantum particle at a particular location at a particular moment in time. To model this, we use the (in)famous concept of the wave function Ψ: the mathematical object that contains all the possible information there is to know about a quantum particle and whose module squared represents the sought probability of finding a particle at a given place.
The difficulty is at least two-fold for the students: 1) the wave function is an abstract genuine complex mathematical object (having a real and imaginary part) quite far from classical concepts like position and velocity and 2) it evolves according to a somewhat complicated equation called the Schrodinger equation that can only be solved by hand in rare instances.
Particle in a box…
…in one dimension
The first problem often treated in class is that of a quantum particle in a one dimensional unescapable box. The logic of the resolution can seem disconcerting at first sight: one looks at so-called stationary (quantum) wave states and, depending on the state, determine for instance the energy of the particle and its probability to be a certain point in space (the stationary character ensures that this probability does not vary in time). Now, in principle many practical questions can emerge with such an answer provided as is. Here are some of them: since there are infinitely many (countable) possible stationary states, how do we know which ones to look at for a real problem? and how do these stationary states actually come about in practice? In sum, why is it a satisfactory answer to the “particle in a 1D box” problem?
Giving a full answer to these questions might take us too far but we can start understanding part of it by running a little thought experiment. Let us assume that there is a quantum particle (e.g. an atom or an electron for instance) in a box. Let us further assume that at one moment in time we were able to measure quite precisely (with an acceptable uncertainty) the x coordinate of the particle. Now here is the question? What can we say about the future of that very particle?
The answer to that question is given in the video below that shows a numerical solution of the Schrodinger equation corresponding to our problem. This video contains three stages that we will discuss each one at a time.
- We see that the probability density to be at a position x is initially narrow and centered in the middle of the box, in agreement with the beginning of our little thought experiment.
- We then observe something a bit surprising: the probability density spreads like a droplet on a table. Why is that? Here we need to recall that one cannot know accurately both the position and the momentum of a quantum particle. This is the celebrated Heisenberg indeterminacy principle Δx . Δp > h/4Π . What it means is that the more accurately we know the position of an object the less accurately we know its linear momentum/velocity. Since we start in a situation where the position is quite well known, it follows that the momentum is very uncertain. The wave function from the outset thus reads as a superposition of wave functions (a wave packet), each associated to a different definite momentum. In the video this is represented with the particular coloring of the probability curve itself. A color corresponding to a positive (resp. negative) value signifies that this part of the curve wants to go to the right (resp. left). In the jargon of quantum mechanics, this is quantatively described by an object called the probability current j. Basically, if the probability is p = Ψ*.Ψ , then the probability current is given by j ∝ h Im(Ψ*. ∇ Ψ). The net effect is that different parts of the probability function go at different “velocities” and this basically stretches out its shape exactly like when we stretch a pizza dough by pulling on two ends with opposite velocities.
- In the remaining part of the video, we see that the wave packet reaches the walls of the box and starts interfering with itself thus creating regular spatial patterns with crests and troughs. This is where it becomes interesting to us: what we see is that the probability density cycles through various standing wave patterns (those known by every student starting a course in quantum mechanics). So, what it shows is that, in any real setup, a particle in a box is bound to explore in various ways a set of fixed patterns which are the standing waves we learn in school and that is the reason why they are important.
…in two dimensions
We can go a little bit further in our quest for understanding the quantum behaviour of a particle in a box. To this end, let us imagine that we were in fact able to measure quite accurately both the x and y coordinates of the particle but also that the particle has a non-zero momentum (this is fine so long as Heinseberg principle is satisfied). The corresponding probability density is expected to evolve in a fashion demonstrated by the following video.
This video shows all the features we have seen in the one dimensional case: 1) an initially localised probability density, 2) spreading in the whole box after some time and 3) self-interference upon interacting with the walls that leads to a fully delocalised probability distribution that cycles through fixed standing wave patterns.
There is one additional aspect to this video owing to the fact that the particle is now moving in the box: during the first “bounces of the particle on the walls”, the (probability density of the) particle seems to behave almost like a classical particle; in fact the simple fact that we can even talk about “bounces” betrays the fact that the particle does behave a bit like ping-pong ball for some amount of time.
It is not too difficult to understand what is going on. If we start with an initial uncertainty Δp0, then in the long run the rim of the wave packet will expand roughly at a rate Δp0/m. After a time t has elapsed the extansion of the wave packet will have increased by an amount (Δp0/m)t. To put the final touch, we use again Heisenberg relation to infer that Δp0 ~ h/Δx0. So that we get that the size of the wave packet increases as ht/(m Δx0).
This final formula has profound consequences:
- Imagine a hydrogen atom of mass 10-27 kg of which we know the position with the precision of 1 pm (one hundredth of its size). Even, if we don’t do anything to it, within 10 microseconds the uncertainty on its position will be of the order of a metre; at that stage we cannot talk anymore in classical terms about it at all since the particle can literally be anywhere in a radius of one metre.
- Imagine now a pebble on the floor. Let’s assume its mass is 100 g and we know its position to the angstrom. The time we would have to wait for the quantum uncertainty on its position to be 1 metre would amount to 1023 seconds; it is a bigger number than the number of seconds elapsed since the estimated time of the Big Bang!
So, effectively we don’t need quantum mechanics to help us understand the whereabouts of a pebble but it becomes very fast the only tool to describe much lighter objects like elementary particles and atoms.