Bra-ket notation (Dirac Notation)

#cs #physics/quantum #card

Why the Bra-ket notation?

Consider any one-dimensional wave function ψ(x) describing a quantum mechanical particle. The value of the wave function at point x1 is ψ(x1) , at point x2 is ψ(x2) , and so on.[1]

Then, we can represent all the function values as a list; we can then take this column vector ψ that lives in an abstract space and represent it with its elements as follows:

ψ=[ψ(x1)ψ(x2)ψ(x3)⋅⋅⋅]

Visualizing ψ in linear algebra

ψ(x1), ψ(x2) & ψ(x3) could be considered as three different coordinate axes. (You could consider infinitely many elements - but visualizing any more than 3 dimensions geometrically proves difficult.)

Now, ψ can be considered a state vector.

Since there are infinitely possible x values, there must also be infinitely many ψ(x) values. And if there are infinite many function values, then the space in which state vector ψ is infinite dimensional; and this abstract space, in which various quantum mechanical vectors live, is called Hilbert Space. Although this is an infinite dimensional vector space, it can also be finite dimensional space.

Hilbert Space with 2 dimensions

The spin up ψ↑ and spin down state ψ↓ which describe a single particle, (both) live in a two-dimensional Hilbert space, which means vectors like ψ↑ have only two components.

Wave function vs State-vector representation

To better distinguish the state-vector representation, of a quantum mechanical particle, from the wave-function representation, we can use the bra-ket notation.

Now, ψ can be represented as a state-vector with |ψ⟩ or ⟨ψ|, or as a wave-function with ψ(x)

Bra & Ket Vectors

There are two types of vectors in Dirac notation: the bra vector and the ket vector, named such because when put together they form a braket or an Inner Product.

If ψ is a column vector, then you can write it in Dirac notation as |ψ⟩, where the |⋅⟩ denotes that it's a unit column vector (or a ket vector). The vector adjoined to the ket vector is denoted as ψ† is called bra vector.

For a compact notation, we write the bra vector with a inverted arrow : ⟨ψ|.

Note

⟨ψ| or ψ† is obtained by applying entry-wise Complex Conjugation to the elements of the transpose of ψ. The bra-ket notation directly implies that ⟨ψ|ψ⟩ is the inner product of vector ψ with itself, which is by definition 1.

Info

More generally, if ψ and ϕ are quantum state vectors, then their inner product is ⟨ϕ|ψ⟩. This inner product implies that the probability of measuring the state ⟨ψ| to be |ϕ⟩ is |⟨ϕ|ψ⟩|2.

Scalar Product

If ψ and ϕ are quantum state vectors, we can represent the dot

#TODO Finish

Inner Product

Tensor/Outer Product

Projection Matrix

Representing the Hadamard operation with Dirac Notation

The following notation is often used to describe the states that result from applying the Hadamard gate to ⟨0| to ⟨1|. These states correspond to the unit vector in the +x and the −x directions on the Bloch Sphere:

12[11]=H|0⟩=|+⟩,12[1−1]=H|1⟩=|−⟩.

  1. Bra-Ket Notation and How to Use It (Alexander Fufaev) ↩︎