A note on conventions

Is it ok to define kets in this way?

Some may object to the following syntax:

psi = ket ( exp(-x**2) )

Did we just loose some generality by imposing a coordinate representation on the ket? You may have seen the awkward notation sometimes used to caution against this:

\[\begin{equation} \vert \rangle "=" \Psi(\mathbf{x}) \end{equation}\]

The above expression seems to imply some relationship between kets in Hilbertspace and scalar functions on \(\mathbb{R}^3\) which is not fully captured by our notation, and will most definently confuse and irritate most people seeing it for the first time.

Does the BraketLab-syntax erronously confine us to work in a coordinate representation by equating abstract vectors in Hilbert-space to functions in \(\mathbb{R}^3\)? Not really.

Instead, the ket-class in Braketlab has to be instanciated from a suitable starting point, which in this case happens to be a function in coordinate space.

Note that there is a difference between the assignment operator = common to most modern code-languages (like the Python-case above) and mathematical equality. The mathematical equivalent to the BraketLab syntax is more comparable to the following

\[\begin{equation} \vert \Psi \rangle := ket ( e^{-\mathbf{x}^2} \rangle \forall x \in \mathbb{R}^3 ), \end{equation}\]

where \(ket\) turns whatever you put into it into a vector in Hilbert space. Fair enough, the internal representation of the ket is still unfortunately in coordinate space. Future versions of BraketLab could perhaps be able to automatically transform between representations, but for now this task is left for the user.

Now, the following is true: a wavefunction \(\Psi\) is composed by the expansion coefficients in coordinate space \(\{ \delta(\mathbf{x}-\mathbf{x}')\} \forall x' \in \mathbb{R}\) of a ket (or state vector) \(\vert \mathscr{S} \rangle\) in Hilbert space. Thus,

\[\begin{equation} \Psi(\mathbf{x}) = \langle \mathbf{x} \vert \mathscr{S} \rangle := \int_{\mathbb{R}^3} \delta(\mathbf{x}-\mathbf{x}') \mathscr{S}(\mathbf{x}') d\mathbf{x'} \end{equation}\]

The ket \(\vert \mathscr{S} \rangle\) can be projected onto any basis (or be represented in any space), but that does not mean we are not allowed to instanciate it from a position space representation. Note that BraketLab is general, so you may just as well define

psi = ket ( exp(-p**2) ),

and take p to signify a momentum representation.

To state it explicitly: if the definition above holds, i.e.

\[\begin{equation} \langle \mathbf{x} \vert \mathscr{S} \rangle := \int_{\mathbb{R}^3} \delta(\mathbf{x}-\mathbf{x}') \mathscr{S}(\mathbf{x}') d\mathbf{x'}, \end{equation}\]

then it does not seem so unreasonable to adopt the following convention:

\[\begin{equation} \vert \mathscr{S} \rangle := \vert \mathscr{S}(\mathbf{x}') \rangle, \end{equation}\]

where a ket is defined from its (in this case) coordinate representation.