Properties and observables

Part 3 in the quest for the hydrogen molecule

Last time we talked about how the position and velocity of a particle are not independent properties. In fact, they are totally dependent! Unfortunately this is sometimes explained as that they can not be independently ‘measured’ or that there is ‘uncertainty’. That is absolute rubbish! Quantum mechanics is the most exact and predictive theory there is. However, one cannot extract information from reality that does not exist in that reality. We have to let go of our classical tendencies, and venture into the quantum world and become enlightened.

There are different ways we can postulate this and they are all unfortunately a little magical. Usually textbooks will talk about ‘conjugate variables’ or the ‘uncertaintly principle’, but I would like to try a different approach. It is equally unprovable, but I think it is a rather practical approach, and it will be fun to try to deduce the entire theory of quantum mechanics from it. However we will also have to start doing mathematics now, so let’s get to work!

Let us first focus on superpositions again. Take the case of a particle with a position in one dimension, and take a superposition like this:

$$| \psi \rangle = 0.2 | x=3 \rangle + 0.9| x=5 \rangle.$$

We have said before that a state is the full description of the situation and it can be a superposition of pure states. Then any state describing the position property of a particle is a superposition of the pure states of that particle being at specific locations \(x\).

$$| \psi \rangle = 0.2 | x=0 \rangle + 0.01| x=1 \rangle+0.5| x=2 \rangle+0.4| x=3 \rangle +\ldots.$$

That looks a little awkward. And we have also have to take into account that \(x\) is not discrete but a continuum. So we would also have to include states like \(| x=0.5 \rangle\) and \(| x=0.999 \rangle\). I suppose it is time to do some real mathematics. It makes more sense to replace the sum by an integral:

$$| \psi \rangle = \int_{-\infty}^{+\infty} \psi(x) |x \rangle dx,$$

where \(\psi(x)\) is just a function of \(x\). Note also that \(| x=x \rangle \) was abbreviated to \(| x \rangle \), since it is already clear that \(x\) stands for position.

We can also ask the reverse question: how much does \(| \psi \rangle \) ‘overlap’ with \(| x=3 \rangle \) and how much with \(| x=5 \rangle\)? The answers are 0.2 and 0.9. It works much better with a formula, and we write the question “how much overlap with \(| x=3 \rangle\)?” as “\(\langle x=3|\)“, so that:

$$\langle x=3| \psi \rangle = 0.2,$$

$$\langle x=5| \psi \rangle = 0.9.$$

It is very important to remember that expressions of the form \(\langle a| b \rangle\) are just a number. In fact, we could even decompose our original superposition as:

$$| \psi \rangle = \langle x=3| \psi \rangle | x=3 \rangle + \langle x=5| \psi \rangle| x=5 \rangle.$$

Doing the same with our continuous superposition:

$$| \psi \rangle = \int_{-\infty}^{+\infty} \psi(x) |x \rangle dx = \int_{-\infty}^{+\infty} \langle x| \psi \rangle |x \rangle dx,$$

and we can also conclude that, by definition, \( \psi(x)= \langle x| \psi \rangle\).

We are now ready to reveal exactly how the velocity and the position of a particle can be combined in a single property. Firstly, it is more natural to not talk about the velocity \(v\) of the particle, but about its momentum \(p\). They are very similar, since they are related by the simple relation

$$p=mv,$$

where \(m\) is the mass of the particle. Now suppose we have a particle that has the property that it has momentum \(p\). We will write the state of the particle as \(|p\rangle\). As I told you, the position and the momentum are totally dependent properties, so we cannot add any further position information to it. This is just all there is!

What we can do, is ask how much the state \(|p\rangle\) has in common with state \(|x\rangle\). Now this is the moment when we will postulate something without deriving it, namely that that overlap is:

$$\langle x|p \rangle = e^{i x p / \hbar},$$

where \(\hbar\) is the (reduced) Planck constant and \(i^2=-1\) (see complex numbers).

So far, there really are a many aspects of quantum mechanics that I have skipped or talked very little about, but I wanted to focus on the fundamentals. It is surprising, also to me, how much we can actually already do with the extremely limited set of concepts we have developed so far. This must mean we are close to the truth! Next time, we will derive a basic model of the hydrogen atom already!

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