Bras Test Directions and Kets Rebuild Them

The notation around bras and kets becomes much calmer once the shapes are visible.

A ket is a column-like vector. In finite dimensions, it is represented by a one-column matrix:

$$|x⟩.$$

A bra is a row-like linear functional. In coordinates, it is represented by a one-row matrix:

$$⟨y|.$$

So when a bra sits next to a ket,

$$⟨y||x⟩,$$

it is not a symbolic merge. It is function application: the bra $⟨y|$ takes the ket $|x⟩$ as input and returns a scalar.

That scalar is written as the inner product

$$⟨y|x⟩$$

A bra eats a ket

In finite dimensions, think of

$$|x⟩=\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right],⟨y|=\left[\begin{array}{ccc}\overline{y_1}& \cdots & \overline{y_n}\end{array}\right].$$

Then

$$⟨y||x⟩=\left[\begin{array}{ccc}\overline{y_1}& \cdots & \overline{y_n}\end{array}\right]\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]=\sum _{i=1}^n\overline{y_i}{x}_i.$$

The result has no remaining vector shape. The row and the column contract into one scalar.

A ket followed by a bra makes an operator

Now reverse the order:

$$|x⟩⟨y|.$$

This does not contract to a scalar. A column times a row makes a matrix-like object, namely an outer product. It acts on an input $|z⟩$ by associativity:

$$(|x⟩⟨y|)|z⟩=|x⟩(⟨y||z⟩)=⟨y\mid z⟩|x⟩.$$

The same fact in coordinates

Let $x,y,z\in {\mathbb{C}}^n$. The outer product $|x⟩⟨y|$ is the matrix with entries

$$(|x⟩⟨y|{)}_{ij}=x_i\overline{y_j}.$$

Applying this matrix to $z$ gives, in coordinate $i$,

$$\begin{array}{rl}((|x⟩⟨y|)|z⟩{)}_i& =\sum _{j=1}^n{x}_i\overline{y_j}{z}_j\\ & =x_i\sum _{j=1}^n\overline{y_j}{z}_j\\ & =x_i⟨y\mid z⟩.\end{array}$$

The factor ${\sum }_j\overline{y_j}{z}_j$ is the scalar test $⟨y\mid z⟩$. After the test, every coordinate of the output is that same scalar times the corresponding coordinate of $x$.

So $|x⟩⟨y|$ is a rank-one operator with two steps:

1. test how much $|z⟩$ points in the $|y⟩$ direction;
2. rebuild that scalar amount in the $|x⟩$ direction.

The order matters:

expression	shape	result
$⟨y||x⟩$	row times column	scalar
$|x⟩⟨y|$	column times row	operator / matrix

Projection tests and rebuilds in the same direction

The projection

$$P_i=|e_i⟩⟨e_i|$$

uses the same direction twice. Applied to $|\psi ⟩$,

$$\begin{array}{rl}{P}_i|\psi ⟩& =(|e_i⟩⟨e_i|)|\psi ⟩\\ & =|e_i⟩(⟨e_i||\psi ⟩)\\ & =⟨e_i\mid \psi ⟩|e_i⟩.\end{array}$$

First, $⟨e_i|$ tests how much of $|\psi ⟩$ lies in the $|e_i⟩$ direction. Then $|e_i⟩$ rebuilds exactly that component along the same direction.

If

$$|\psi ⟩=\sum _j{\alpha }_j|e_j⟩$$

in an orthonormal basis, then

$$⟨e_i\mid \psi ⟩={\alpha }_i$$

and therefore

$$P_i|\psi ⟩={\alpha }_i|e_i⟩.$$

Everything except the $i$-th component is removed.

Linearity tells you where scalars go

In Dirac notation, the ket side is linear:

$$⟨y\mid (\alpha x+\beta z)⟩=\alpha ⟨y\mid x⟩+\beta ⟨y\mid z⟩.$$

The bra side is conjugate-linear over complex numbers:

$$⟨\alpha y+\beta z\mid x⟩=\overline{\alpha }⟨y\mid x⟩+\overline{\beta }⟨z\mid x⟩.$$

This is why bras contain complex conjugates in coordinates:

$$|y⟩=\left[\begin{array}{c}{y}_1\\ ⋮\\ y_n\end{array}\right]⟹⟨y|=\left[\begin{array}{ccc}\overline{y_1}& \cdots & \overline{y_n}\end{array}\right].$$

For outer products, the resulting operator is linear in its input:

$$\begin{array}{rl}(|x⟩⟨y|)(\alpha |z_1⟩+\beta |z_2⟩)& =|x⟩⟨y\mid (\alpha z_1+\beta z_2)⟩\\ & =\alpha ⟨y\mid z_1⟩|x⟩+\beta ⟨y\mid z_2⟩|x⟩.\end{array}$$

So the outer product is an operator built from an inner-product test followed by vector construction.

The rule of thumb

Read Dirac expressions by shape and order.

• bra-ket: $⟨y||x⟩=⟨y\mid x⟩$ is a scalar.
• ket-bra: $|x⟩⟨y|$ is an operator.
• projection: $|e⟩⟨e|$ tests along $e$ and rebuilds along $e$.

The bra asks a question: “how much points in this direction?”

The ket provides a direction to rebuild the answer.