Physicists at Imperial College London and the advertising guys at Hi Rezz are working together to bring this poster to you at two West London locations.  Our poster is written in the international language of mathematics, which if you don't speak don't worry. We think of our sign in motorway blue as a chunk of blue sky, a moment for pause, a street-side poem. Our hope is to open a window to share one of our greatest and most iconic equations, one that has enabled much of the modern world.  And we want to communicate a little flavour of our world, our approach, fascination and enthusiasm with you. We warmly encourage you to get in touch and ask us a question about physics or the scientific life and we will do our best to answer.

How the Mona Lisa is to artists, this equation is to physicists - beautiful, instantly recognisable, incredibly important and yet enigmatically hiding something from us.

What is this?

What a dazzling little troupe of dancing characters these are!  This strange combination of symbols is an important equation in physics.  It is one of our greatest - of vast application and accuracy; and also humbly, pleasingly and elegantly compact.  Just as we have equations that may be familiar to you describing for example the motion of tennis balls or aeroplanes, this equation describes the movements and interactions of the world at the smallest scales: by this we mean the atom itself or the little pieces of matter that make it, the structure of molecules or light scattering from the atom.  This is the Schrödinger Equation; it is the basic equation of quantum theory.  

What is an equation?

An equation describes relations between various quantities in nature and allows us to predict results... if we know ‘this and this’, then we can calculate a related outcome.   For example if you travel at 60 miles per hour for 2 hours, you know you will have traveled 120 miles.   If you stop to think for one moment you’d be right to consider it an amazing thing that aspects of nature can be described as mathematical relations. The ability to make a mathematical hypothesis and hold it up against nature to test its fit is at the very core of physics and a great part of the delight!  A good equation is like a wonderful tool: a trusty spanner, or a well loved bicycle.  It's also like a magical gift box revealing to us more about nature than we previously knew.  And to some physicists it will have a beauty of its own that words struggle to describe - there is a delicate dance, a subtle interplay between its components.  

What does this equation mean?

At very small scales light and matter behave in beautiful and surprising ways that can be difficult to describe in everyday terms. The most important feature is the wave function Ψ which describes (in probabilities) all manner of properties: the energy, location, speed and momentum to name a few of a microscopic particle of matter or light.  The left hand side describes how the wave function and hence these physical quantities change with time and relates it to the energy of the system on the right hand side which drives this change.  So it is a dynamical equation for infinitesimally small systems.

Out of this equation come an almost endless supply of wondrous gifts: some portraits to show us what the basic building blocks of nature look like, story-books that explain how they get their characters and why they behave the way they do, crystal balls for predicting their futures.

How did it come about and what does it enable?

Though it bears the name of one man, Erwin Schrödinger, the formulation of this equation was born from a great international effort.  Over 90 years ago people around the world were trying to explain particular important and striking patterns they had discovered in Nature: like the organisation of the elements in the periodic table and the stunning and mysterious absorption lines encoded in the light from stars.  This equation brilliantly answered such vital questions.  It also opens the door to our imaginations and has allowed us to create previously unthinkable things: the fibre optic internet backbone, lasers, X-ray machines, electron microscopes, GPS, solid state electronics, nuclear reactors, computers and solar panels.   It has made possible the design of much of our modern world.  We have tested it a million times over and it has never failed.

Where are we going?

Despite the equation’s immense robustness, accuracy and utility, there are still deep questions about its interpretation that consume entire careers.   A physicist will always hope to find circumstances that reveal flaws in an equation or even smash it to smithereens.  This is not a destructive urge - in this way she may reveal a fresh layer of understanding and see the world in a new and even more accurate way.

The Schrödinger equation describes how quantum states, which are complex vectors in a Hilbert space, evolve. The physical content enters in the choice of correct Hilbert space and the specific Hamiltonian for the problem, and probabilities of measurement outcomes are extracted once the state is known - basically via the 2-norm. Note that the equation is linear in time so it is easy to write a formal solution by exponentiation of the (Hermitian) Hamiltonian operator that appears on the right hand side. This is clearly a unitary operator, thus it preserves the 2-norm of the state vector (which amounts to conservation of total probability) and so Schrödinger evolution corresponds to a smooth rigid rotation of the state vector. Unfortunately the dimension of the Hilbert space is typically exponentially large in both the number of systems and the degrees of freedom being investigated, so in practise the solution is almost always approximated.