[plt-scheme] Currying and physics
----- Original Message -----
From: Gregory Woodhouse
To: PLT List
Sent: Thursday, January 01, 2009 8:33 PM
Subject: [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
You probably mean (linear) momentum. Position can be represented by an operator (function, functional) Ψ -> xΨ. The probability to find the particle at position x at time t is:
integral over x of Ψ*(x,t) x Ψ(x,t) divided by
the integral over x of Ψ*(x,t) Ψ(x,t),
where Ψ* is the complex conjugate of Ψ.
In this case momentum is represented by the function Ψ -> (iħ/2π)(δΨ/δx).
You could have choosen Ψ(x)=1/(1+x^2) as a function with finite norm,
or in three dimensions 1/(1+x^2+y^2+z^2)
In practice wave functions are often represented by time independent vectors (called kets) in a Hilbert space.
Which functions Ψ are to be included in this space is determined by the law of conservation of energy. In quantum mechanics this law says: HΨ=EΨ, where H is the so called Hamiltonian (an operator representing energy) and E a real number. The equation must be solved for both Ψ and E (and the solution usually consists of an infinite number of pairs Ψ and E) By using the symmetry properties of the system being studied, many parts of the integrals can be simplified to summations.
Jos
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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