Relations between quantum states

Part 2 in the quest for the hydrogen molecule

We really have to get more exact.  The state \(|\rm{happy}\rangle\) seems a very fuzzy piece of information, there seem to be many ways to be happy, a whole continuum of them, and we require a state to describe a situation, not a property. It would be better if we could proceed with more terse and quantifiable information. We are in luck: nature absolutely loves terse and quantifiable information!

Let’s consider a particle. It has a few properties but really not that many. For instance, it has a position. Let us focus on the position in one dimension only, so that the particle has a position \(x\). That is really a well defined state. The particle can be at \(x=3\), meaning its state is

\(|x=3\rangle\).

Or it can be in state \(|x=0\rangle\), \(|x=7.5\rangle\), \(|x=-4.5\rangle\) etcetera.

Remember also that all states may be added. This means that we can also have this state:

\(|x=3\rangle + |x=7\rangle\)

Meaning the superposition of the pure state of the particle at \(x=3\) and the pure state of the particle at \(x=7\). As you can see, the number of states even a single particle can attain is absolutely huge. It seems there are many more states than we would have with ‘classical’ physics, where a particle can be at only one position. However, nature turns out to be very frugal with information, as we will see, which will make a big difference. Perhaps the world of quantum mechanics is even simpler than the ‘classical’ world, but in a rather counter-intuitive way. Let’s see how this plays out.

This means we can start to build a quantum theory with a particle state, and this is how we would normally proceed. But not so quick! – you may say. If a state really corresponds to a situation, what about the situation in the rest of the universe? Should this also be part of the state? A good point, and we will see where it will lead us.

Let’s suppose the universe is made of two particles. Let’s also suppose that their position is their only property. If one of them is at \(x=3\) and the other is at \(x=5\), then the state of the universe is:

\(|x=3  \cdot x=5\rangle\).

Wait! – you might say. Should we not write down which particle is where? If particle A is at \(|x=3\rangle\) and particle B is at \(|x=5\rangle\), then perhaps the state is really \(|x_A=3 \cdot x_B=5\rangle\)? Excellent question. I have stated before that nature loves terse information, so it will not surprise you it follows the more frugal design. If the universe has only two particles, and each particle has only one property, then the complete state of the universe is described by the fact that there is a particle at \(x=3\) and there is a particle at \(x=5\), a principle with far-reaching consequences, as we will see.

If we want to do physics, we will also have to let the universe evolve over time, so let’s guess how that might work. Naturally, we expect to add a property to the particles that they have a velocity, in addition to their position. This is the state of one particle at \(x=3\) with velocity \(v=1\):

\(|x=3, v=1\rangle\).

That looks quite reasonable, but it is wrong. The velocity and the position of a particle are not independent properties. In fact, they are one and the same property! Surely, nature saw again an opportunity to save a lot of information, but how does this work precisely? We will investigate it in the next installment.

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