A Neighborhood of Infinity: Some parallels between classical and quantum mechanics

Introduction
This isn't really a blog post. More of something I wanted to interject in a discussion on Google plus but wouldn't fit in the text box.

I've always had trouble with the way the Legendre transform is introduced in classical mechanics. I know I'm not the only one. Many mathematicians and physicists have recognised that it seems to be plucked out of a hat like a rabbit and have even written papers to address this issue. But however much an author attempts to make it seem natural, it still looks like a rabbit to me.

So I have to ask myself, what would make me feel comfortable with the Legendre transform?

The Legendre transform is an analogue of the Fourier transform that uses a different semiring to the usual. I wrote briefly about this many years ago. So if we could write classical mechanics in a form that is analogous to another problem where I'd use a Fourier transform, I'd be happier. This is my attempt to do that.

When I wrote about Fourier transforms a little while back the intention was to immediately follow it with an analogous article about Legendre transforms. Unfortunately that's been postponed so I'm going to just assume you know that Legendre transforms can be used to compute inf-convolutions. I'll state clearly what that means below, but I won't show any detail on the analogy with Fourier transforms.

Free classical particles
Let's work in one dimension with a particle of mass $m$ whose position at time $t$ is $x(t)$ . The kinetic energy of this particle is given by $T=\frac{1}{2}m\dot{x}^2$ . Its Lagrangian is therefore $L=\frac{1}{2}m\dot{x}^2-V(x)$ .

The action of our particle for the time from $t_0$ to $t_1$ is therefore

$\displaystyle{\int_{t_0}^{t_1}(\frac{1}{2}m\dot{x}^2-V(x))dt}$

The particle motion is that which minimises the action.

Suppose the position of the particle at time $t_0$ is $x_0$ and the position at time $t_1$ is $x_1$ . Then write $\psi(t_0,t_1,x_0,x_1)$ for the action minimising path from $x_0$ to $x_1$ . So

$\displaystyle{\psi(t_0,x_0;t_1,x_1) = \min_x\int_{t_0}^{t_1}(\frac{1}{2}m\dot{x}^2-V(x))dt}$

where we're minimising over all paths $x$ such that $x(t_i)=x_i$ .

Now suppose our system evolves from time $t_0$ to $t_2$ . We can consider this to be two stages, one from $t_0$ to $t_1$ followed by one from $t_1$ to $t_2$ . Let $\phi$ be the minimised action analogous to $\psi$ for the period $t_1$ to $t_2$ . The action from $t_0$ to $t_2$ is the sum of the actions for the two subperiods. So the minimum total action for the period $t_0$ to $t_2$ is given by

$\displaystyle{\psi(t_0,x_0;t_2,x_2) = \min_{x_1}(\psi(t_0,x_0;t_1,x_1)+\psi(t_1,x_1;t_2,x_2))}$

Let me simply that a little. I'll use $\psi(t,x)$ where I previously used $\psi(t_0,x_0;t,x)$ and $\phi(x_1,x_2)$ for $\psi(t_1,x_1;t_2,x_2)$ . So that last equation becomes:

$\displaystyle{\psi(t_2,x_2)=\min_{x_1}(\psi(t_1,x_1)+\phi(x_2-x_1))}$

Now suppose $\phi$ is translation-independent in the sense that $\phi(x+s,x'+s)=\phi(x,x')$ . So we can write $\phi(x_1,x_2)=\phi(x_2-x_1)$ . Then the minimum total action is given by

$\displaystyle{\psi(t_2,x_2)=\min_{x_1}(\psi(t_1,x_1)+\phi(x_2-x_1))}$

Infimal convolution is defined by

$\displaystyle{(f\odot g)(x) = \min_s(f(s)+g(x-s))}$

so the minimum we seek is

$\displaystyle{\psi(t_2,x_2) = (\psi(t_1,\cdot)\odot\phi))(x_2)}$

So now it's natural to use the Legendre transform. We have the inf-convolution theorem:

$(f\odot g)^\ast=f^\ast+g^\ast$

where $f^\ast$ is the Legendre transform of $f$ given by

$\displaystyle{f^\ast(p) = \sup_x(px-f(x))}$

and so $\psi^\ast(t_2,p) = \psi^\ast(t_1,p)+\phi^\ast(p)$ (where we use $\ast$ to represent Legendre transform with respect to the spatial variable).

Let's consider the case where from $t_1$ onwards the particle motion is free, so $V=0$ . In this case we clearly have translation-invariance and so the time evolution is given by repeated inf-convolution with $\phi$ and in the "Legendre domain" this is nothing other than repeated addition of $\phi^\ast$ .

Let's take a look at $\phi$ . We know that if a particle travels freely from $x_1$ to $x_2$ over the period from $t_1$ to $t_2$ then it must have followed the minimum action path and we know, from basic mechanics, this is the path with constant velocity. So

$\displaystyle{T = \frac{1}{2}m(x_2-x_1)^2/(t_2-t_1)^2}$

and hence the action is given by

$\displaystyle{\phi(s) = \frac{1}{2}ms^2/(t_2-t_1)}$

So the time evolution of $\psi$ is given by repeated inf-convolution with a quadratic function. The time evolution of $\psi^\ast$ is therefore given by repeated addition of the Legendre transform of a quadratic function. It's not hard to prove that the Legendre transform of a quadratic function is also quadratic. In fact:

$\displaystyle{\phi^\ast(p) = \frac{1}{8}mp^2(t_2-t_1)^2}$

Addition is easier to work with than inf-convolution so if we wish to understand the time evolution of the action function it's natural to work with this Legendre transformed function.

So that's it for classical mechanics in this post. I've tried to look at the evolution of a classical system in a way that makes the Legendre transform natural.

Free quantum particles
Now I want to take a look at the evolution of a free quantum particle to show how similar it is to what I wrote above. In this case we have the Schrödinger equation

$\displaystyle{i\hbar\frac{\partial}{\partial t}\psi(t,x) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x,t)+V\psi(t,x)}$

Let's suppose that from time $t_1$ onwards the particle is free so $V=0$ . Then we have

$\displaystyle{i\hbar\frac{\partial}{\partial t}\psi(t,x) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(t,x)$

Now let's take the Fourier transform in the spatial variable. We get:

$\displaystyle{i\hbar\frac{\partial}{\partial t}\hat{\psi}(t,k) = -\frac{\hbar^2}{2m}(ik)^2\hat{\psi}(t,k)$

$\displaystyle{\hat{\psi}(t,k) = \exp(\frac{i\hbar k^2(t-t_1)}{2m})\hat{\psi}(t_1,k)$

We can write this as

$\displaystyle{\hat{\psi}(t,k) = \hat{\phi}(k)\hat{\psi}(t_1,k)$

where

$\phi(x) = \exp(\frac{ix^2}{2\hbar m})$

So the time evolution of the free quantum particle is given by repeated convolution with a Gaussian function which in the Fourier domain is repeated multiplication by a Gaussian. The classical section above is nothing but a tropical version of this section.

Conclusion
I doubt I've said anything original here. Classical mechanics is well known to be the limit of quantum mechanics as $\hbar\rightarrow 0$ and it's well known that in this limit we find that occurrences of the semiring $(\mathbb{R},+,\times)$ are replaced by the semiring $(\mathbb{R},\min,+)$ . But I've never seen an article that attempts to describe classical mechanics in terms of repeated inf-convolution even though this is close to Hamilton's formulation and I've never seen an article that shows the parallel with the Schrödinger equation in this way. I'm hoping someone will now be able to say to me "I've seen that before" and post a relevant link below.

Note
I'm not sure how the above applies for a non-trivial potential $V$ . I wrote this little Schrödinger equation solver a while back. As might be expected, it's inconvenient to use the Fourier domain to deal with the part of the evolution due to $V$ . In order to simulate a time step of $dt$ the code simulates $dt/2$ in the Fourier domain assuming the particle is free and then spends $dt/2$ solving for the $V$ -dependent part in the spatial domain. So even in the presence of non-trivial $V$ it can still be useful to work with a Fourier transform. Almost the same iteration could be used to numerically compute the action for the classical case.

1 comment:

John BaezSaturday, 21 January, 2012
Great blog post! I feel pretty sure this material is known, since there's a long tradition of 'idempotent analysis' in Russia which seeks to treat classical mechanics using linear algebra over the tropical semirig. I've provided a short list of references here. I'm not sure they contain what you want, but they should give a reasonably good picture of the state of the art.

Saturday, January 21, 2012

Some parallels between classical and quantum mechanics

1 comment: