In Dirac notation, a vector $\psi$ is denoted by the symbol $|\psi\rangle$, which is called a ket.

We know that there is always a dual vector $f_{\psi}$ corresponding to any vector $\psi$, i.e. the unique dual vector $f_{\psi}$ such that $f_{\psi}(\phi) = (\psi,\phi)$ for any vector $\phi$. In Dirac notation, this dual vector is denoted $\langle \psi |$ and is called a bra.

Much sophomoric humor is possible over this name, which we shall do our best to avoid here.

In Dirac notation, the inner product $(\phi,\psi)$ is denoted $\langle \phi | \psi \rangle$ and is called a bra-ket. This makes it clear that the inner product $\langle \phi | \psi \rangle = f_{\phi}(\psi)$ can be thought of as a dual vector acting on a vector, or a bra acting on a ket.

To gain familiarity with the notation, here are some examples of things we have already learned, written in Dirac notation.

• The correspondence between vectors and dual vectors: \begin{align*} & |\psi\rangle & \qquad \leftrightarrow \qquad & \langle \psi | & \\ & |a\psi \rangle = a|\psi \rangle & \qquad \leftrightarrow \qquad & \langle a \psi | = a^* \langle \psi | & \\ & a|\psi \rangle + b|\phi \rangle & \qquad \leftrightarrow \qquad & a^* \langle \psi | + b^* \langle \phi | & \end{align*}
• Properties of the scalar product: \begin{align*} \langle \phi | \psi \rangle & = \langle \psi | \phi \rangle^* \\ \langle \phi | a \psi_1 + b \psi_2 \rangle & = a \langle \phi | \psi_1 \rangle + b \langle \phi | \psi_2 \rangle \\ \langle a \phi_1 + b\phi_2 | \psi \rangle & = a^* \langle \phi_1 |\psi \rangle + b^* \langle \phi_2 | \psi \rangle \end{align*}
• The Cauchy-Schwartz Inequality: \[|\langle \phi | \psi \rangle |^2 \leq \langle \psi | \psi \rangle \langle \phi | \phi \rangle \]
• The Triangle Inequality: \[\sqrt{\langle \psi + \phi | \psi + \phi \rangle} \leq \sqrt{\langle \psi | \psi \rangle } + \sqrt{\langle \phi | \phi \rangle} \]

Even when we are dealing with a very concrete Hilbert space like $\mathbb{R}^n$ or $\mathbb{C}^n$, the vector $|\psi \rangle$ should not be thought of as identical with a column vector \[\left ( \begin{array}{c} b_1 \\ b_2 \\ \vdots \\ b_n \end{array} \right ).\] Instead, the column vector is a representation of the vector $|\psi \rangle$ in a specific basis, in this case the standard basis $|e_1 \rangle, |e_2 \rangle, \cdots , |e_n \rangle$. The vector $|\psi \rangle$ itself is basis independent, and has different components when represented in different bases.

To understand the relationship between representations in different bases, consider an orthonormal basis $|e_1 \rangle, |e_2 \rangle, \cdots $ (which may have countably infinite dimension). We can always write $|\psi \rangle$ in this basis as \[|\psi \rangle = \sum_j b_j |e_j\rangle,\] for some components $b_j$.

We can find the components as follows. \[\langle e_k | \psi \rangle = \sum_j b_j \langle e_k | e_j \rangle = \sum_j b_j \delta_{jk} = b_k.\] The components are found by taking inner products with the basis elements, so, in fact, we can just write \[|\psi \rangle = \sum_j |e_j \rangle\langle e_j | \psi \rangle.\] The convention of writing the inner products to the right of the basis vectors in this representation is chosen for good reasons, which will be discussed in the next module.

Now suppose we have a different basis $|f_1\rangle, |f_2\rangle, \cdots$. In this basis we will have \[|\psi \rangle = \sum_j c_j |f_j\rangle,\] where the components $c_j$ are different from the comppnents $b_j$ in the $|e_j\rangle$ basis. In the $|f_j\rangle$ basis, we will have $c_j = \langle f_j | \psi \rangle$ and so we also have \[|\psi \rangle = \sum_j |f_j \rangle \langle f_j | \psi \rangle. \]

We can convert between the two representations as follows. \begin{align*} b_j & = \langle e_j | \psi \rangle \\ & = \langle e_j | \left ( \sum_k |f_k \rangle \langle f_k | \psi \rangle \right ) \\ & = \sum_k \langle e_j | f_k \rangle \langle f_k | \psi \rangle \\ & = \sum_k \langle e_j | f_k \rangle c_k. \end{align*} Knowing the inner products $\langle e_j | f_k \rangle$ is what allows us to convert between representations in two different orthonormal bases.

We can do the same thing in a Hilbert space with a continuous basis, essentially by replacing the sums by integrals, but before doing so it is helpful to properly introduce the Dirac delta function.

The Dirac delta function $\delta(x)$ is not really a function. It is defined to be the mathematical object such that, for any function $f(x)$, \[\int_{-\infty}^{+\infty} \delta(x) f(x)\,\mathrm{d}x = f(0),\] i.e. when integrated it “picks out” the value of $f(x)$ at $x=0$. The Dirac delta function is an example of what is called a distribution in mathematics.

From the definition, we can also show that \[\int_{-\infty}^{+\infty} \delta(x-x') f(x) \, \mathrm{d}x = f(x').\] To show this, we make the substitution $x'' = x - x'$. This gives $\mathrm{d}x'' = \mathrm{d}x$ and $x = x' + x''$. Hence, we have \[\int_{-\infty}^{+\infty} \delta(x-x') f(x) \, \mathrm{d}x = \int_{-\infty}^{+\infty} \delta(x'') f(x'+x'') \, \mathrm{d}x''. \] By the definition of $\delta(x'')$, this integral picks out the value of $f(x'+x'')$ at $x'' = 0$, so we have \[\int_{-\infty}^{+\infty} \delta(x-x') f(x) \, \mathrm{d}x = f(x'+0) = f(x'). \]

We have already seen that a wavefunction can be represented in terms of position as $\psi(x)$ or momentum as $\phi(p)$. These are two different functions, related by the Fourier transform. This is just like representing the vector $|\psi \rangle$ in two different bases $|e_j \rangle$ or $|f_j\rangle$ in the discrete case. As in that casse, we want to develop a formalism in which $|\psi\rangle$ is an abstract, basis indepenent, vector and $\psi(x)$ and $\phi(p)$ are just representations of it in two different bases.

To achieve this, we introduce position “vectors” $|x\rangle$ that have “inner product” \[\langle x' | x\rangle = \delta(x-x').\] The scare quotes are because $\delta(x-x')$ is not really a function, so the value of $\langle x' | x\rangle$ is not a scalar, and hence not an inner product. As good physicists, we will proceed as if these were well defined vectors and inner products and hope everything will be OK. The mathematicians in the audience may wish to note that this can be made rigorous in the theory of rigged Hilbert space.

We can then write the position representation of the vector $|\psi\rangle$ as \[|\psi \rangle = \int_{-\infty}^{+\infty} \psi(x) |x\rangle \,\mathrm{d}x,\] where we are thinking of the values of $\psi(x)$ at different values of $x$ as components of a vector in the basis $|x\rangle$, just like the components $b_j$ in the decomposition \[|\psi \rangle = \sum_j b_j |e_j\rangle,\] for the discrete case.

As in the discrete case, we can retrieve the components by taking inner products. \begin{align*} \langle x | \psi \rangle & = \int_{-\infty}^{+\infty} \psi(x') \langle x | x' \rangle \, \mathrm{d}x' \\ & = \int_{-\infty}^{+\infty} \psi(x')\delta (x'-x)\,\mathrm{d}x' \\ & = \psi(x), \end{align*} and hence we can write \[|\psi \rangle = \int_{-\infty}^{+\infty} |x\rangle\langle x |\psi \rangle \,\mathrm{d}x.\]

We can do the same thing for the momentum representation. We introduce momentum “basis vectors” $|p\rangle$ with inner products \[\langle p' | p\rangle = \delta(p-p'),\] and write the momentum representation of the vector $|\psi\rangle$ as \[|\psi \rangle = \int_{-\infty}^{+\infty} \phi(p) |p\rangle \,\mathrm{d}p.\] We will then have $\phi(p) = \langle p | \psi \rangle$ and \[|\psi \rangle = \int_{-\infty}^{+\infty} |p\rangle\langle p |\psi \rangle \,\mathrm{d}p.\]

To convert between the two basis representations, we will use \[\psi(x) = \langle x | \psi \rangle = \int_{-\infty}^{+\infty} \langle x | p \rangle \langle p |\psi \rangle \,\mathrm{d}p = \int_{-\infty}^{+\infty} \langle x | p \rangle \phi(p) \,\mathrm{d}p,\] and \[\phi(p) = \langle p | \psi \rangle = \int_{-\infty}^{+\infty} \langle p | x \rangle \langle x |\psi \rangle \,\mathrm{d}x = \int_{-\infty}^{+\infty} \langle p | x \rangle \psi(x) \,\mathrm{d}x.\]

To do this, we need to know the “inner product” $\langle x| p \rangle$, which we will derive in a future lecture. (It should be no surprise that it is related to the Fourier transform.)

For now, the point is that we have abstract, basis independent, symbols like $|\psi \rangle$, $\langle \phi |$ and $\langle \phi | \psi \rangle$, which we can use in Hilbert spaces that have discrete or continuous bases, or a combination of both, and which give us a uniform and consistent notation.