1.5: Bra-ket Notation

In this section we detail the notation that we will use to describe a quantum state, \(|\psi\rangle\). This notation is due to Dirac and, while it takes some time to get used to, is incredibly convenient.

Inner Products

We saw earlier that all of our quantum states live inside a Hilbert space. A Hilbert space is a special kind of vector space that, in addition to all the usual rules with vector spaces, is also endowed with an inner product. And an inner product is a way of taking two states (vectors in the Hilbert space) and getting a number out. For instance, define

\[|\psi\rangle=\sum_{k} a_{k}|k\rangle\]

where the kets \(|k\rangle\) form a basis, so are orthogonal. If we instead write this state as a column vector,

\[
|\psi\rangle=\left(\begin{array}{c}
a_{0} \\
a_{1} \\
\vdots \\
a_{N-1}
\end{array}\right)
\]

Then the inner product of \(|\psi\rangle\) with itself is

\[
\langle\psi, \psi\rangle=\left(\begin{array}{cccc}
a_{0}^{*} & a_{1}^{*} & \cdots & a_{N_{1}}^{*}
\end{array}\right) \cdot\left(\begin{array}{c}
a_{0} \\
a_{1} \\
\vdots \\
a_{N-1}
\end{array}\right)=\sum_{k=0}^{N-1} a_{k}^{*} a_{k}=\sum_{k=0}^{N-1}\left|a_{k}\right|^{2}
\]

The complex conjugation step is important so that when we take the inner product of a vector with itself we get a real number which we can associate
with a length. Dirac noticed that there could be an easier way to write this by defining an object, called a "bra," that is the conjugate-transpose of a ket,

\[\langle\psi|=| \psi\rangle^{\dagger}=\sum_{k} a_{k}^{*}\langle k|\]

This object acts on a ket to give a number, as long as we remember the rule,

\[\langle j|| k\rangle \equiv\langle j \mid k\rangle=\delta_{j k}\]

Now we can write the inner product of \(|\psi\rangle\) with itself as

\[
\begin{align}
\langle\psi \mid \psi\rangle & =\left(\sum_{j} a_{j}^{*}\langle j|\right)\left(\sum_{k} a_{k}|k\rangle\right) \\ \notag \\
& =\sum_{j, k} a_{j}^{*} a_{k}\langle j \mid k\rangle \\ \notag \\
& =\sum_{j, k} a_{j}^{*} a_{k} \delta_{j k} \\ \notag \\
& =\sum_{k}\left|a_{k}\right|^{2}
\end{align}
\]

Now we can use the same tools to write the inner product of any two states, \(|\psi\rangle\) and \(|\phi\rangle\), where

\[|\phi\rangle=\sum_{k} b_{k}|k\rangle\]

Their inner product is,

\[\langle\psi \mid \phi\rangle=\sum_{j, k} a_{j}^{*} b_{k}\langle j \mid k\rangle=\sum_{k} a_{k}^{*} b_{k}\]

Notice that there is no reason for the inner product of two states to be real (unless they are the same state), and that

\[\langle\psi \mid \phi\rangle=\langle\phi \mid \psi\rangle^{*} \in \mathbb{C}\]

In this way, a bra vector may be considered as a "functional." We feed it a ket, and it spits out a complex number.

The Dual Space

We mentioned above that a bra vector is a functional on the Hilbert space. In fact, the set of all bra vectors forms what is known as the dual space. This space is the set of all linear functionals that can act on the Hilbert space.