Bra-Ket Notation

Also called Dirac Notation.

Bra-Ket is a way of writing special vectors used in Quantum Physics that looks like this:

bra|ket

Here is a vector in 3 dimensions:

vector in 3d

We can write this as a column vector like this:

r =

Or we can write it as a "ket":

r =

But kets are special:

The values (a, b and c above) are complex numbers (they may be real numbers, imaginary numbers or a combination of both)
A ket is a quantum state
Kets can have any number of dimensions, including infinite dimensions!

The "bra" is similar, but the values are in a row, and each element is the complex conjugate of the ket's elements.

Example: This ket:

a =

2−3i

6+4i

3−i

Has this bra:

a =

2+3i

6−4i

3+i

The values are now in a row, and we also changed the sign (+ to −, and − to +) in the middle of each element.

In "matrix language", changing a ket into a bra (or bra into a ket) is a "conjugate transpose":

conjugate: 2−3i becomes 2+3i, etc...
transpose: rows swap with columns

Multiplying

Multiplying a bra a and ket b looks like this:

a|b

We use matrix multiplication, in particluar the dot product:

The "Dot Product" is where we multiply matching members, then sum up:

= 1×7 + 2×9 + 3×11 = 58

We match the 1st members (1 and 7), multiply them, likewise for the 2nd members (2 and 9) and the 3rd members (3 and 11), and finally sum them up.

In effect, the dot product "projects" one vector on to the other before multiplying the lengths:


		Like shining a light to see where the shadow lies

When the two vectors are at right angles the dot product is zero:

dot product shine light right angles
No shadow is cast!

Example:

vectors in 2d

a =

, and b =

So:

a|b =

= 1×0 + 0×1 = 0

This can be a simple test to see if vectors are orthogonal (the more general concept of "at right angles")

The dot product of a vector with itself is the length of the vector times the length of the vector. In other words it is length²:

dot product shine light same vector
Full shadow is cast!

Example:

c =

So:

c|c =

= 2×2 + 1×1 = 5

The dot product is 5

And we can also work out c's length to be √5

Example: What is the length of the vector [1, 2, −2, 5] ?

−2

= 1×1 + 2×2 + (−2)×(−2) + 5×5

= 1 + 4 + 4 + 25

= 34

The dot product is 34, so the vector's length is √34

Note: we can also use Pythagoras' Theorem to calculate its length:

√(1² + 2² + (−2)² + 5²) = √34

Basis

vector in 3d

We can separate the parts of a vector like this:

r =

= a

+ b

+ c

The vectors "1, 0, 0", "0, 1, 0" and "0, 0, 1" form the basis: the vectors that we measure things against.

In this case they are simple unit vectors, but any set of vectors can be used when they are independent of each other (being at right angles achieves this) and can together span every part of the space.

Matrix Rank has more details about linear dependence, span and more.

Orthonormal Basis

In most cases we want an orthonormal basis which is:

Orthogonal: each basis vector is at right angles to all others.
We can test it by making sure any pairing of basis vectors has a dot product a·b = 0
Normalized: each basis vector has length 1

Our simple example from above works nicely:

vectors in 2d
The vectors are at right angles,
and each vector has a length of 1

And this one also works:

vectors in 2d

Let's check it!

Is the dot product zero?

a·b = 1√2×1√2 + 1√2×−1√2

= 12 − 12 = 0

Is each length 1?

|a| = (1√2)² + (1√2)² = 12 + 12= 1

|b| = (1√2)² + (−1√2)² = 12 + 12= 1

So yes it is an orthonormal basis!

Schrödinger's Cat

dot product shine light same vector

A famous example is "Schrödinger's Cat": a thought experiment where a cat is in a box with a quantum-triggered container of gas. There is an equal chance of it being alive or dead (until we open the box).

It can be written like this:

cat = 1√2alive + 1√2dead

It says the state of the cat is in a superposition of the two states "alive" and "dead".

But why the 1√2 ?

First let us illustrate it like this:

vector in 3d

The basis is the two vectors alive and dead. The cat is shown in that probability space as a vector with equal components a and d.

Now let us normalize it!

Normalized

A normalized vector has a length of 1.

We know the dot product of a vector with itself is length², so a normalized vector has:

a|a = 1² = 1

Example: Normalizing the cat vector

If we assume a = d = 1 we get this:

cat|cat =

= 1×1 + 1×1 = 2

But it should be 1, right?

Let us try 1√2:

cat|cat =

1√2

= 12 + 12= 1

So a = d = 1√2, and we get:

cat = 1√2alive + 1√2dead

And it now has a length of 1

Probability

Let us try to find the probability by adding the component lengths a and d:

Probability = 1√2 + 1√2
= 2√2 = √2 ???

But that can't be right, probability can't be greater than 1

In fact we need to take the magnitude of each vector (shown using ||) and square it:

Probability = |1√2|² + |1√2|²
= 12 + 12 = 1 (yay!)

This is a general rule in Quantum Physics:

The probability equals the amplitude magnitude squared, in other words:

Probability = |Amplitude|²

The || means magnitude of a vector, not absolute value.

Naming Kets

Notice how we are free to use any word or symbol inside the ket. In some cases numbers are also used, but they are used as labels so don't try to do arithmetic with them.

Many Dimensions

superposition of dice

We can easily have many dimensions.

Imagine "Quantum Dice" that are in a superposition of 1, 2, 3, 4, 5 and 6.

The ket looks like this:

die =

For a fair die all elements (a, b, c, d, e, f) are equal, but your dice may be loaded!

Why?

Why do we do all this?

So we can "map" some real world case (usually one with probabilities) onto a well-defined mathematical basis. This then gives us the power to use all the math tools to study it.

Conclusion

The bra-ket notation is a simple way to refer to a vector with complex elements, any number of dimensions, that represents one state in a state space. The probability of any state equals the magnitude of its vector squared.