# [plt-scheme] Currying and physics

It seems to me that operators actually provide a more direct analog to
currying than the (classical) fields discussed in that note. In
elementary quantum mechanics, a particle is represented by a wave
function (a complex valuied function of time and position). Observable
quantities (or just observables) correspond to (linear) operators on
the space of so-called wave functions. For example, in one dimension,
position corresponds to i h bar (the imaginary unit times Planck's
constant divided 2 pi) times differentiation with respect to x). In
LaTeX, that's
P = i\hbar \frac{\partial}{\partial x}
So, if psi (the letter traditionally used to represent wave functions)
is x^2, then Px is 2i \hbar x (never mind the fact that it isn't
square integrable). So, if you think of the probability density for
position being function of both the observable (in this case,
position) and the quantum state, you take the first input variable
(the observable) and generate a function (or, as some people like to
say, functional) that can be applied to to the wave function to give
you a new function (this time of the interval over which you are
integrating), then you take the integral (another function!) to get
the expected position. Without that last step, you get yet another
function the norm of which is the probability density of position.
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