Dirac Notation

Kets (column vectors)

Before proceeding with the first postulate of quantum mechanics, we introduce the notation which quantum theorists use to perform their calculations: Dirac notation (after physicist Paul Dirac), also known as bra-ket notation.

Instead of denoting a vector $\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ in $\mathbb{R}^2$ or $\mathbb{C}^2$ by the notation $\vec{v}$, we instead write $|v\rangle$.

This is known as the ket (of bra-ket).

So:

$$|v\rangle = \begin{bmatrix}  v_1 \\  v_2  \end{bmatrix}  \quad \leftrightarrow \quad \vec{v} = \begin{bmatrix}  v_1 \\  v_2  \end{bmatrix}.$$

Do note that this is just notation. Nothing else has changed.

In fact, in quantum theory, physicists will often put many different symbols within the kets out of notational convenience! 

For instance, you might see things like:

$$\ket{\uparrow}, \ket{+}, \ket{\circlearrowright}, \ket{g}, \ket{e}$$

and more, depending on the context/physical application. So don''t be scared!

Bras (row vectors)

We know that the conjugate transpose of a vector $\vec{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ is given by $\vec{v}^\dagger = \begin{bmatrix} v_1^* & v_2^* \end{bmatrix}$.

That is, the conjugate transpose of a column vector takes the column vector and transposes it into a row vector, and also takes the complex conjugate of the entries.

In Dirac notation, we write $\vec{v}^\dagger$ as $\langle v|$.  This is known as the bra (of bra-ket).  

For example, 

$$\bra{0} =  \begin{bmatrix} 1 & 0 \end{bmatrix} \quad , \quad \bra{1} =  \begin{bmatrix} 0 & 1 \end{bmatrix}$$

In summary, for the conjugate transpose, we have: $\langle v| = (|v\rangle)^\dagger$ and $| v \rangle = (\langle v |)^\dagger$. And so, as we know, applying $\dagger$ twice returns the original vector.

Why bother?

You may be asking: Why was this invented? What’s the point?

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