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$\displaystyle i \hbar \partial_t |\psi \rangle = H |\psi\rangle$

is the fundamental equation of motion for (non-relativistic) quantum mechanics, modeling both one-particle systems and ${N}$-particle systems for ${N>1}$. Remarkably, despite being a linear equation, solutions ${|\psi\rangle}$ to this equation can be governed by a non-linear equation in the large particle limit ${N \rightarrow \infty}$. In particular, when modeling a Bose-Einstein condensate with a suitably scaled interaction potential ${V}$ in the large particle limit, the solution can be governed by the cubic nonlinear Schrödinger equation

$\displaystyle i \partial_t \phi = \Delta \phi + \lambda |\phi|^2 \phi. \ \ \ \ \ (1)$

I recently attended a talk by Natasa Pavlovic on the rigorous derivation of this type of limiting behaviour, which was initiated by the pioneering work of Hepp and Spohn, and has now attracted a vast recent literature. The rigorous details here are rather sophisticated; but the heuristic explanation of the phenomenon is fairly simple, and actually rather pretty in my opinion, involving the foundational quantum mechanics of ${N}$-particle systems. I am recording this heuristic derivation here, partly for my own benefit, but perhaps it will be of interest to some readers.

This discussion will be purely formal, in the sense that (important) analytic issues such as differentiability, existence and uniqueness, etc. will be largely ignored.

My penultimate article for my PCM series is a very short one, on “Hamiltonians“. The PCM has a number of short articles to define terms which occur frequently in the longer articles, but are not substantive enough topics by themselves to warrant a full-length treatment. One of these is the term “Hamiltonian”, which is used in all the standard types of physical mechanics (classical or quantum, microscopic or statistical) to describe the total energy of a system. It is a remarkable feature of the laws of physics that this single object (which is a scalar-valued function in classical physics, and a self-adjoint operator in quantum mechanics) suffices to describe the entire dynamics of a system, although from a mathematical perspective it is not always easy to read off all the analytic aspects of this dynamics just from the form of the Hamiltonian.

In mathematics, Hamiltonians of course arise in the equations of mathematical physics (such as Hamilton’s equations of motion, or Schrödinger’s equations of motion), but also show up in symplectic geometry (as a special case of a moment map) and in microlocal analysis.

For this post, I would also like to highlight an article of my good friend Andrew Granville on one of my own favorite topics, “Analytic number theory“, focusing in particular on the classical problem of understanding the distribution of the primes, via such analytic tools as zeta functions and L-functions, sieve theory, and the circle method.

This post is derived from an interesting conversation I had several years ago with my friend Jason Newquist on trying to find some intuitive analogies for the non-classical nature of quantum mechanics. It occurred to me that this type of informal, rambling discussion might actually be rather suited to the blog medium, so here goes nothing…

Quantum mechanics has a number of weird consequences, but here we are focusing on three (inter-related) ones:

1. Objects can behave both like particles (with definite position and a continuum of states) and waves (with indefinite position and (in confined situations) quantised states);
2. The equations that govern quantum mechanics are deterministic, but the standard interpretation of the solutions of these equations is probabilistic; and
3. If instead one applies the laws of quantum mechanics literally at the macroscopic scale, then the universe itself must split into the superposition of many distinct “worlds”.

In trying to come up with a classical conceptual model in which to capture these non-classical phenomena, we eventually hit upon using the idea of using computer games as an analogy. The exact choice of game is not terribly important, but let us pick Tomb Raider – a popular game from about ten years ago (back when I had the leisure to play these things), in which the heroine, Lara Croft, explores various tombs and dungeons, solving puzzles and dodging traps, in order to achieve some objective. It is quite common for Lara to die in the game, for instance by failing to evade one of the traps. (I should warn that this analogy will be rather violent on certain computer-generated characters.)