Stepping into the quantum world

Part 2 of the quest for the hydrogen molecule.

There is no way around it: we are going to do quantum mechanics. So let’s start with an introduction to establish a starting point and get ourselves familiarized with the general ideas. After that, we are going to start calculating!

There are several ways you can start building a theory of quantum mechanics. Basically these are the main options:

  1. Schrödinger equation
  2. Hilbert space states
  3. Path integrals

This is also the order by which they were discovered, and of course it also became the order in which quantum mechanics is usually taught. Following the exact footsteps of previous physicists is a great way not to make progress, since you will build the same concepts and misunderstandings in your mind as the people before you. We will try to choose our own path, where possible.

So, let’s compare them.

  1. The normal procedure would be to start with the Schröding equation and then go calculate stuff. Which is fine, but it is also not very illuminating. Where does this equation come from? What does it tell us about the world, really? It does have the advantage that you can actually calculate things, so we will have to use it at some point. It is not much use when you go to more advanced topics  (relativistic quantum mechanics, quantum field theory).
  2. Hilbert space states are nice but a a little abstract in the beginning. They are good to develop first ideas and make you feel closer to “the truth”.
  3. Path integrals are very cool, and quite useful for keeping things simple (relatively speaking) when you go to more advanced physics. They are mathematically fuzzy and strange. They are not simple to calculate and a bit frightening. We can take a look later if it is possible to calculate the hydrogen atom with path integrals, because I don’t know really.

So let’s start with the second idea, and focus on quantum states. We will learn to reason from the abstract to the concrete and we will also learn that the world is a bit more… orthogonal (?) than what we think it is.

A state is a description of a situation. We will use this notation to write down a state: \(|\rm{state}\rangle\). Furthermore the states can be added and subtracted, and they can be multiplied by any number. Don’t worry if this feels a little abstract, it’s supposed to be that way. We don’t make any assumption about the underlying structure of such a state.

Let’s start with an example. If I am feeling happy, I could describe my state as


Since states can be multiplied with -1, this is another state:


Question: which of these states is happier? Answer: they are both equally happy. An unhappy state would be an entirely different one:


What if I am both happy and tired, how can I describe that? We could write it down like this:


where I have given the combined state a name “\(|\psi\rangle\)“. The sum of two ‘pure’ states is called a superpositon. Since states can be multiplied by a number, we can also have a superposition like this


The simple rule is: all states can be added. So that means we can also have the state:


Really? It looks confusing, so we must be doing quantum mechanics!

Next time, we are going to derive the Schröding equation, although it is generally agreed that this is not possible.