Demystifying Quantum Mechanics - Part I 
I've been struggling to understand QM for over 30 years. I don't mean struggling with the maths, I can just about do the maths, what I've always struggled with is why that maths works in the first place. I'm not quite there yet, but I recently had a road-to-Damascus moment while watching some YouTube lectures by Prof. James Binney of Oxford University. I'd like to share the result with you.

QM is usually taught in one of two ways. I'm going to call these two ways the "wavefunction approach" and the "abstract approach". When they are both applicable they're entirely equivalent.

The wavefunction approach introduces students to a wave that is associated with a quantum system and that has an equation to allow you to predict how the wave evolves over time. The reason it's taught this way is that students are normally already familiar with waves: sounds waves, electromagnetic waves, waves in membranes etc. The mathematics and the concepts are fairly familiar.

The abstract approach is often taught as the entry point for mathematicians or those more interested in theory, although even the wavefunction students have to move onto it eventually. It's in the abstract approach that the full power of the theory emerges, with the wavefunction approach just being a special case of it. The beauty of Prof Binney's lectures is that he uses the full abstract approach right from the start, but constantly returns to the wavefunction approach, making the equivalence of the two seem perfectly natural.

The abstract approach starts with a small set of assumptions and then builds the whole of QM on top of those assumptions. The assumptions are often presented as a generalisation of the wavefunction approach. This is the point where I always got stuck. The mathematical assumptions looked quite bizarre to me. It seemed almost perverse that nature would choose to express itself in such a strange way. Why would this obscure and abstract mathematics turn out to be the basis of the physical world? None of the textbooks even attempt to offer an explanation. Maybe they just think it's so obvious that only dullards like me would take 30 years for the penny to drop.

I'm going to illustrate how it works with a very simple example, nothing to do with atoms or the very small, just the rolling of a normal size, six sided die. When we roll a die the outcome is uncertain until it finally comes to rest with one value facing up. If it is a fair die then there is equal probability of getting each of the values 1 through 6. This bears a close analogy to what goes on in QM - things are uncertain until a measurement is made, at which point a specific value emerges. (This is only an analogy though, there are also some very significant differences.)

I'm going to encode the rolling of a die using the same mathemtics as QM. First we need to be able to encode the state of the die. If the die is at rest on a table then only one of six values is facing up. I'm going to encode this using a column of six numbers. If the die has the value "1" face up then I'll represent it by the following column of numbers.

0 a one in the first row and zeroes in all the others

If the die has the value "2" face up then I'll represent it by the following column of numbers.

0 a one in the second row and zeroes in all the others

And so on, a "1" in row n if the die shows the value n and "0" in all other rows. It can be a bit tiresome to write it out like that all the time, so I'll also have a shorthand notation.

|1> means the column of numbers with a one in the first row and zeroes in all the others.
|2> means the column of numbers with a one in the second row and zeroes in all the others.
|n> means the column of numbers with a one in row n and zeroes in all the others.

The next thing I need to do is to encode the act of measurement. I'm going to do this using a square block of numbers. This is probably going to look a bit daft. Unfortunately it will get worse - bear with me. The block of numbers looks like this.

1 0 0 0 0 0
0 2 0 0 0 0
0 0 3 0 0 0 the numbers 1 through 6 along the diagonal and zero everywhere else
0 0 0 4 0 0
0 0 0 0 5 0
0 0 0 0 0 6

I've only written this once and I'm already finding it tedious, so for quickness I'll just call this whole block of numbers "D". Anytime I use the capital letter D by itself, just imagine that big block of numbers in its place.

Now comes the tricky bit. I have to show you how mathematicians multiply blocks of numbers. If I want to multiply D into |2>, say, then I have to multiply each number in row one of D, by each number in the column |2>. This gives row 1 of the result. Then I have to multiply each number in row two of D, by each number in the column |2> to given row 2 of the result, and repeat for each row. The result is a new column of numbers.

1 0 0 0 0 0 0 | 1x0 + 0x1 + 0x0 + 0x0 + 0x0 + 0x0 0
0 2 0 0 0 0 1 | 0x0 + 2x1 + 0x0 + 0x0 + 0x0 + 0x0 2
0 0 3 0 0 0 0 | = 0x0 + 0x1 + 3x0 + 0x0 + 0x0 + 0x0 = 0 the original state times two
0 0 0 4 0 0 0 | 0x0 + 0x1 + 0x0 + 4x0 + 0x0 + 0x0 0
0 0 0 0 5 0 0 | 0x0 + 0x1 + 0x0 + 0x0 + 5x0 + 0x0 0
0 0 0 0 0 6 0 v 0x0 + 0x1 + 0x0 + 0x0 + 0x0 + 6x0 0

I know, it looks totally over the top and it doesn't seem to have achieved very much. I'm going to rewrite the above using my shorthand notation.

D|2> = 2|2>

This means exactly the same as the blocks of numbers above. All it says is that, if I apply the measurement operator D, to a die in the state showng a 2, (i.e. |2>), I get the same state back, |2>, but multiplied by the value it shows, 2. There's nothing deep or mysterious here, it's just how we've chosen to represent the die and its measurement, plus some rules for multiplying blocks of numbers. D acting on |2> returns the state |2> and the measured value, 2, because we've set the whole thing up to make it work that way.

The big block of numbers is called a "Linear Operator". The special columns of numbers, where the operator always returns a copy of the same column of numbers, is called an "eigenvector" and the special values, that multiply the eigenvectors, are called "eigenvalues". The assumptions of QM usually begin with a sentence something like this: "Measurements in QM are represented by Linear Operators. The possible outcomes of a measurement are the eigenvalues of those linear operators." And for 30 years I sat scratching my head going "Eh? WHY are the possible outcomes of a measurement the eigenvalues of those linear operators?" Believe it or not, it's as simple as the die example shows. Duh!

In QM, everything that we want to measure has its own measurement operator. For example, the energy of a system will have a measurement operator for measuring energy - let's call it E. This operates on states of definite energy |e> and returns the same states multiplied by the measured value, e:

E|e> = e|e>

This equation is so important that it has its own name. It's called the Time Independent Schrodinger Equation (TISE). A lot of QM involves figuring out what E looks like in a given situation and thus determining what values of e are valid. Unlike in classical mechanics, where the energy of a system can often take on any value, in QM values of e will often be discrete (or quantised, thus "Quantum" Mechanics).

A big block of numbers is called a "matrix" and when QM was first discovered by Heisenberg the theory was called Matrix Mechanics.

In our die example, |n> represented the possible face up states of the die. Suppose we want to measure the location of the die on the table instead. Whereas the face up value only had six possible values, the location has an infinite number of values, so we need an infinitely large column of numbers (which we obviously can't write down in its entirety). When |x> represents position it is called a "wavefunction" (because it often turns out to look wavy). When Schrodinger first did his version of QM he expressed everything as wavefunctions and it was known as Wave Mechanics.

If you're interested in learning more, especially if you're not too scared of a little bit of maths, I'd highly recommend Leonard Susskind's introductory lectures on the subject. If you feel a bit more ambitious, then James Binney's lectures will give you the full, uncensored and unexpurgated theory.

In Part II I'll show how we can use our columns of numbers to encode probabilities. This gives rise to something called "probability amplitudes" - another mystery that used to puzzle me for years. I'll also show you how states can be combined to make new states.