In Part I I started to show how you can take a very simple situation, such as the rolling of a normal, six sided die, and describe it using the mathematics of Quantum Mechanics (QM). I did this as follows.

The state of the die is represented by one of six columns of numbers. Each column is all zeros except for the number 1 in the row representing the face value of the die. You can either write down the columns explicitly, or use the shortcut notation |1>, |2>, |3>, ... I'm going to give these their proper names now and call them "Basis Kets."

A measurement was represented by a block of numbers with the numbers 1 through 6 on the diagonal and zeroes everywhere else. You can write the block down explicitly, or alternatively just use the capital letter D to represent it.

Using the rule "multiply rows into columns and add", D can act on basis kets to give the same basis ket back but multiplied by the value of the measurement:

This simple equation is essentially one of the basic assumptions of QM.

So far we haven't really achieved very much. The only states we have described are those where the die is lying still on the table showing a face up value. What about when the die is tumbling through the air and we have no idea what its final value will be? I'm going to represent this state of the die by a new column of numbers, this time with the number 0.41 in every row.

The number 0.41 is not chosen at random, it's an approximation to the reciprocal of the square root of 6, 1/sqrt(6). The reason for choosing this will become clear later. In shorthand I'll call this column |u> ("u" for uncertain). Notice that you can think of |u> as a sum over the basis kets, where each row of |u> is the sum of all the corresponding rows from the basis kets.

In fact, you can build an arbitrary column of numbers this way by adding together arbitrary multiples of basis kets:

We'll see how to interpret such arbitrary kets in a moment. Basis kets (i.e. |n>) and the totally uncertain ket (i.e. |u>) are both special cases of these arbitrary "kets".

We can compare kets to see how alike they are. Let's compare |1> to itself. We start by knocking it on its side: |1> goes to 1 0 0 0 0 0. Then we use the "multiply rows into columns and add" rule:

Unlike the case where D|n> produced a ket, multiplying a single row into a column gives just a single number. The number here tells us how alike |1> is to itself. The value we get is 1. If we compare |1> to |2>, we get:

The result is zero.

When the column is knocked on its side like that we represent it as <1| and call it a "bra". Multiplying a bra by the corresponding ket gives a number, just as in the examples above:

We shorten this to <1|1> = 1, <1|2> = 0. A bra and a ket together like this form a bra-ket. I suspect the inventor of this, Paul Dirac, meant it as a joke, but the terminology seems to have stuck. It also goes by some other names, the inner product, dot product, or overlap. I'll stick to the last term and call it the overlap. Kets that are identical have an overlap of 1 and kets that have nothing in common have an overlap of 0. Kets with some parts in common produce intermediate values. e.g. <1|u> = 0.41. Note that

We chose the value 0.41 precisely so that this would work. When the numbers in a ket have been chosen so that the sum of their squares is one, the ket is said to be "normalised". All kets in QM are assumed to be normalised and we will make the same assumption.

The overlap tells us how similar two kets are. Since a ket models the state of a random variable, the overlap is a measure of probability. As all kets are normalised, i.e. the sum of the square of their components is one, we'll find that we need to square the overlap to get probabilities. This is easiest shown with an example.

Suppose the die is tumbling though the air. We have no idea what the final face up value will be, i.e. it is in the state |u>, where all its components have the value 0.41. To find the probability that the final value will be 2, we take the overlap with |2> to see how similar they are, and square the result.

Which is 1/6, exactly as we would expect.

This leads to one of the other assumptions of QM, usually expressed something like this.

Of course this is an over-simplification. If this was all there was to QM you'd wonder what all the fuss was about. I've missed out a huge, and critical, complication. In the die example all the kets consist of a column of numbers. In QM, kets representing the state of a system likewise consist of columns of numbers, but alongside each number there is also a direction. It's as if each component is a little arrow. These arrows are called complex numbers and they're responsible for the wavy aspects of QM. Experiment shows that the resulting wavelengths correspond to the momentum of the system and the frequencies correspond to the energy of the system. I only have a very basic understanding of this aspect of QM. Obviously something that's changing rapidly has more energy than something that's not, but beyond that I really just take it as given.

The wavy aspects are responsible for the interference effects that we see in things like the double slit experiment. There is also a conserved quantity associated with the rotation of these arrows. The conserved quantity has a name that you may recognise, it's called electric charge.

In Part III I'll describe how some properties of a system can be mutually incompatible and show how this can be modelled using kets.

The state of the die is represented by one of six columns of numbers. Each column is all zeros except for the number 1 in the row representing the face value of the die. You can either write down the columns explicitly, or use the shortcut notation |1>, |2>, |3>, ... I'm going to give these their proper names now and call them "Basis Kets."

A measurement was represented by a block of numbers with the numbers 1 through 6 on the diagonal and zeroes everywhere else. You can write the block down explicitly, or alternatively just use the capital letter D to represent it.

Using the rule "multiply rows into columns and add", D can act on basis kets to give the same basis ket back but multiplied by the value of the measurement:

D|n> = n|n>

This simple equation is essentially one of the basic assumptions of QM.

So far we haven't really achieved very much. The only states we have described are those where the die is lying still on the table showing a face up value. What about when the die is tumbling through the air and we have no idea what its final value will be? I'm going to represent this state of the die by a new column of numbers, this time with the number 0.41 in every row.

0.41

0.41

0.41 0.41 in every row when the outcome could be any value

0.41

0.41

0.41

The number 0.41 is not chosen at random, it's an approximation to the reciprocal of the square root of 6, 1/sqrt(6). The reason for choosing this will become clear later. In shorthand I'll call this column |u> ("u" for uncertain). Notice that you can think of |u> as a sum over the basis kets, where each row of |u> is the sum of all the corresponding rows from the basis kets.

|u> = 0.41x|1> + 0.41x|2> + 0.41x|3> + 0.41x|4> + 0.41x|5> + 0.41x|6>

In fact, you can build an arbitrary column of numbers this way by adding together arbitrary multiples of basis kets:

2

4

0 = 2x|1> + 4x|2> + 0x|3> - 1x|4> + 9x|5> + 3x|6>

-1

9

3

We'll see how to interpret such arbitrary kets in a moment. Basis kets (i.e. |n>) and the totally uncertain ket (i.e. |u>) are both special cases of these arbitrary "kets".

We can compare kets to see how alike they are. Let's compare |1> to itself. We start by knocking it on its side: |1> goes to 1 0 0 0 0 0. Then we use the "multiply rows into columns and add" rule:

1 |

----------> 0 |

1 0 0 0 0 0 x 0 | = 1

0 |

0 |

0 V

Unlike the case where D|n> produced a ket, multiplying a single row into a column gives just a single number. The number here tells us how alike |1> is to itself. The value we get is 1. If we compare |1> to |2>, we get:

0 |

----------> 1 |

1 0 0 0 0 0 x 0 | = 0

0 |

0 |

0 V

The result is zero.

When the column is knocked on its side like that we represent it as <1| and call it a "bra". Multiplying a bra by the corresponding ket gives a number, just as in the examples above:

<1| x |1> = 1

<1| x |2> = 0.

We shorten this to <1|1> = 1, <1|2> = 0. A bra and a ket together like this form a bra-ket. I suspect the inventor of this, Paul Dirac, meant it as a joke, but the terminology seems to have stuck. It also goes by some other names, the inner product, dot product, or overlap. I'll stick to the last term and call it the overlap. Kets that are identical have an overlap of 1 and kets that have nothing in common have an overlap of 0. Kets with some parts in common produce intermediate values. e.g. <1|u> = 0.41. Note that

<u|u> = 1 (0.41 x 0.41 x 6 = 1)

We chose the value 0.41 precisely so that this would work. When the numbers in a ket have been chosen so that the sum of their squares is one, the ket is said to be "normalised". All kets in QM are assumed to be normalised and we will make the same assumption.

The overlap tells us how similar two kets are. Since a ket models the state of a random variable, the overlap is a measure of probability. As all kets are normalised, i.e. the sum of the square of their components is one, we'll find that we need to square the overlap to get probabilities. This is easiest shown with an example.

Suppose the die is tumbling though the air. We have no idea what the final face up value will be, i.e. it is in the state |u>, where all its components have the value 0.41. To find the probability that the final value will be 2, we take the overlap with |2> to see how similar they are, and square the result.

<2|u> x <2|u> = 0.41 x 0.41 = 0.16

Which is 1/6, exactly as we would expect.

This leads to one of the other assumptions of QM, usually expressed something like this.

**"If a system is in a state |s>, the probability of measuring a value x is given by the size of <x|s> squared."**This is often presented as being unique to QM, but as the example of the die illustrates, any random variable whose values are modelled as a space of normalised kets will exhibit this property. Some text books justify this better than others, but I haven't found any that explain it in the simple way I've described above, which I think is a shame.Of course this is an over-simplification. If this was all there was to QM you'd wonder what all the fuss was about. I've missed out a huge, and critical, complication. In the die example all the kets consist of a column of numbers. In QM, kets representing the state of a system likewise consist of columns of numbers, but alongside each number there is also a direction. It's as if each component is a little arrow. These arrows are called complex numbers and they're responsible for the wavy aspects of QM. Experiment shows that the resulting wavelengths correspond to the momentum of the system and the frequencies correspond to the energy of the system. I only have a very basic understanding of this aspect of QM. Obviously something that's changing rapidly has more energy than something that's not, but beyond that I really just take it as given.

The wavy aspects are responsible for the interference effects that we see in things like the double slit experiment. There is also a conserved quantity associated with the rotation of these arrows. The conserved quantity has a name that you may recognise, it's called electric charge.

In Part III I'll describe how some properties of a system can be mutually incompatible and show how this can be modelled using kets.