Dirac’s notation gives every state two faces. The ket is the state as a column of amplitudes. The bra is its partner, the row of conjugated amplitudes, and a bra meeting a ket makes a single number. The notation is built so that the brackets close exactly when a number comes out, and it makes most of the algebra of quantum mechanics write itself.

A ket is a question’s answer written down; a bra is a question that can be asked of any answer. Put the question next to the answer and you get a number back: how much of what the bra asks about the ket contains.

In a basis, a ket is a column of complex components and its bra is the row of their complex conjugates, the conjugate transpose, written $\bra{\psi}=\ket{\psi}^{\dagger}$. A bra followed by a ket is a bracket, $\braket{\phi|\psi}$: the row times the column, a single complex number.

• In components, $\braket{\phi|\psi}=b_{1}^{*}c_{1}+b_{2}^{*}c_{2}$ when has components and has .

To turn a ket into its bra, write its components as a row and conjugate each one.

The bracket of a ket with its own bra is the sum of the moduli squared of its components: always real and positive for a nonzero vector.

A bracket multiplies the conjugated components of the bra by the components of the ket, entry by entry, and adds.

Reading a bracket the other way round gives the complex conjugate of the original number.

Great work! You now know how to:

• turn a ket into its bra by conjugate transposition
• compute a bracket as a row times a column
• use the rules for conjugating coefficients and swapping a bracket
• tell a bracket, which is a number, from a ket-bra, which is not