Bra-Ket Notation
Also called Dirac Notation.
Bra-Ket is a way of writing special vectors used in Quantum Physics:
bra|ket
It is a play on the word bracket. A bra on the left and a ket on the right combine to make a bra-ket.
Here's a vector in 3 dimensions:
We can write this as a column vector like this:
Or we can write it as a "ket":
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:
Has this bra:
Its values are 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, and so on...
- transpose: rows swap with columns
Read more at Matrix Types.
Multiplying
Multiplying a bra a and ket b looks like this:
a|b
We use matrix multiplication, in particular the dot product:
The "Dot Product" is where we multiply matching members, then sum up:
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 quantum physics the members are usually complex numbers. The bra is the complex conjugate of the ket (we flip the sign of the 'i' part). And then multiplying matching members and summing always gives us a real number and probabilities come out positive!
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:
No shadow is cast!
Example:
So:
This can be a simple test to see if vectors are orthogonal (the more general concept of "at right angles")
Example with Complex Numbers
Quantum states often use complex numbers. Here's how the inner product works:
Let's say:
Then the bra is the conjugate transpose:
(note the signs flipped on the i parts)
Now calculate ⟨ψ|ψ⟩:
= (1 − (−1)) + (4 − (−1))
= 2 + 5 = 7
The length squared is 7, a real, positive number, even with complex components!
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:
Full shadow is cast!
Example:
So:
The dot product is 5
And we can also work out c's length to be √5
Example: What's the length of the vector [1, 2, −2, 5] ?
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
We can separate the parts of a vector like this:
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:
The vectors are at right angles,
and each vector has a length of 1
And this one also works:
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
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's 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's illustrate it like this:
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's 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:
Example: Normalizing the cat vector
If we assume a = d = 1 we get this:
But it should be 1, right?
Let's try 1√2:
So a = d = 1√2, and we get:
cat = 1√2alive + 1√2dead
And it now has a length of 1
Probability
Let's 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 (shown using |...|) of each amplitude and square it:
Probability = |1√2|2 + |1√2|2
= 12 + 12 = 1 (yay!)
This is a general rule in Quantum Physics:
Probability = |Amplitude|2
Here, the bars |...| mean the absolute value (also called the modulus) of the complex number.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
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:
For a fair die all elements (a, b, c, d, e, f) are equal, but our 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.