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Consider a continuously infinite dimensional Hilbert space, i.e. it has a basis $\ket{\chi_k}$ labelled by a continuous index $k$. We call such a basis orthonormal if \[\braket{\chi_{k'}}{\chi_k} = \delta(k-k').\]

Note that this means that \[\braket{\chi_k}{\chi_k} = \delta(0),\] and by the integral representation of the Dirac delta function this is \[\delta(0) = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\D k\, e^{i0} = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\D k\,=\infty,\] so, unlike in the discrete case, the basis vectors are not normalized, $\|\chi_k\| \neq 1$. Since their norms are actually infinite, we cannot normalize them by dividing by a constant either. It is just a fact that there are no normalized continuous bases, so we adopt this convention for orthonormality instead.

The completeness condition (Dirac notaty) in a continuous basis becomes \[\hat{I} = \int_{-\infty}^{+\infty} \D k \, \proj{\chi_k},\] and we can use this to expand any ket $\ket{\psi}$ in the $\ket{\chi_k}$ basis as \[\ket{\psi} = \hat{I}\ket{\psi} = \int_{-\infty}^{+\infty} \D k\,\ket{\chi_k}\braket{\chi_k}{\psi} = \int_{-\infty}^{+\infty}\D \k\, f(k)\ket{\chi_k},\] where $f(k) = \braket{\chi_k}{\psi}$ are the components of $\ket{\psi}$ in the $\ket{\chi_k}$ basis, with the main difference to what we have seen before being that $f(k)$ is now a function of a continuous variable rather than a discrete set of components. We can think of it as a “column vector” with a continuous index.

As in the discrete case, we can similarly represent a bra in a particular basis using the Dirac notaty: \begin{align*} \bra{\psi} & = \bra{\psi}\hat{I} \\ & = \int_{-\infty}^{+\infty} \D k\,\braket{\psi}{\chi_k}\bra{\chi_k} \\ & = \int_{-\infty}^{+\infty} \D k\,\braket{\chi_k}{\psi}^*\bra{\chi_k} \\ & = \int_{-\infty}^{+\infty}\D \k\, f(k)^*\bra{\chi_k}, \end{align*} where we can think of $f^*(k)$ as the components of a “row vector” with a continuous index.

As another example, we can calculate an inner product in a particular basis via: \begin{align*} \braket{\phi}{\psi} & = \sand{\phi}{\hat{I}}{\psi} \\ & = \int_{-\infty}^{+\infty} \D k\, \braket{\phi}{\chi_k} \braket{\chi_k}{\psi} \\ & = \int_{-\infty}^{+\infty} \D k\, \braket{\chi_k}{\phi}^* \braket{\chi_k}{\psi} \\ & = \int_{-\infty}^{+\infty} \D k\, g^*(k) f(k), \end{align*} where $g(k)$ are the components of $\ket{\phi}$ in the $\ket{\chi_k}$ basis, i.e. $\ket{\phi} = \int_{-\infty}^{+\infty} \D k\, g(k) \ket{\chi_k}$.

Finally, we can represent an operator in the $\ket{\chi_k}$ basis via: \begin{align*} \hat{A} & = \hat{I}\hat{A}\hat{I} \\ & = \int_{-\infty}^{+\infty} \D k\, \int_{-\infty}^{+\infty} \D k'\, \ket{\chi_k}\sand{\chi_k}{\hat{A}}{\chi_{k'}} \bra{\chi_{k'}} \\ & = \int_{-\infty}^{+\infty} \D k\, \int_{-\infty}^{+\infty} \D k'\, \sand{\chi_k}{\hat{A}}{\chi_{k'}} \ketbra{\chi_k}{\chi_{k'}} \\ & = \int_{-\infty}^{+\infty} \D k\, \int_{-\infty}^{+\infty} \D k'\, A_{kk'} \ketbra{\chi_k}{\chi_{k'}}, \end{align*} where we think of $A_{kk'} = \sand{\chi_k}{\hat{A}}{\chi_{k'}}$ as the components of a “matrix” with a continuous infinity of rows and columns.

2.vi.1 The Position Representation