At this point it s best to suspend your intuition about what it really means to say that a particle

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where   is acceleration,   is force and   is mass. This wonderfully straightforward, yet subtle law allows car rental in cape town you to describe motion of all kinds and so it can, in theory at least, answer pretty much any question a physicist might want to ask about the world.
Or can it? When people first started considering the world at the smallest scales, for example electrons orbiting the nucleus of an atom, they realised that things get very weird indeed and that Newton s laws no longer apply. To describe this tiny world you need quantum car rental in cape town mechanics , a theory developed at the beginning of the twentieth car rental in cape town century. The core equation of this theory, car rental in cape town the analogue of Newton s second law, is called  Schrödinger car rental in cape town s equation .
In classical mechanics we describe a state of a physical system using position and momentum, explains Nazim Bouatta, car rental in cape town a theoretical physicist at the University of Cambridge . For example, if you've car rental in cape town got a table full of moving car rental in cape town billiard balls and you know the position and the momentum (that's the mass times the velocity) of each ball at some time  , then you know all there is to know about the system at that time  : where everything is, where everything is going and how fast. The kind of question we then ask is: if we know the initial conditions of a system, that is, we know the system at time   what is the dynamical evolution of this system? And we use Newton's car rental in cape town second law for that. In quantum mechanics we ask the same question, but the answer is tricky because position car rental in cape town and momentum are no longer the right variables to describe [the system].
The problem is that the objects quantum mechanics tries to describe don t always behave like tiny little car rental in cape town billiard balls. Sometimes it is better to think of them as waves. Take the example of light. Newton, car rental in cape town apart from his work on gravity, was also interested in optics, says Bouatta. According to Newton, light was described by particles. But then, after the work of many scientists, including the theoretical understanding provided by  James Clerk Maxwell , we discovered that light was described by waves.
But in 1905 Einstein realised that the wave picture wasn t entirely correct either. To explain the  photoelectric effect (see the  Plus  article  Light s identity crisis ) you need to think of a beam of light as a stream of particles, which Einstein dubbed  photons . The number of photons is proportional to the intensity of the light, and the energy  E  of each photon is proportional to its frequency  f :
Here,    is Planck s constant , an incredibly small number named after the physicist  Max Planck  who had already guessed this formula car rental in cape town in 1900 in his work on  black body radiation . So we were facing the situation that sometimes the correct way of describing light was as waves and sometimes it was as particles, says Bouatta.
The double slit experiment: The top picture shows the interference pattern created by waves passing though car rental in cape town the slits, the middle picture shows what you d expect to see when particles are fired through the slits, and the bottom picture shows what actually happens when you fire particles such as electrons through the slits: you get the interference pattern car rental in cape town you expect from waves, but the electrons are registered as arriving as particles.
Einstein s result linked in with the age-old endeavour, started in the 17th century by  Christiaan Huygens  and explored again in the 19th century by  William Hamilton : to unify the physics of optics (which was all about waves) and mechanics (which was all about particles). Inspired by the schizophrenic behaviour of light the young French physicist Louis de Broglie  took a dramatic step in this journey: he postulated that not only light, but also matter suffered from the so-called  wave-particle duality . The tiny building blocks of matter, such as electrons, also behave like particles car rental in cape town in some situations and like waves in others.
De Broglie s idea, which he announced in the 1920s, wasn t based on experimental evidence, car rental in cape town rather it sprung from theoretical considerations inspired by Einstein s theory of relativity. But experimental evidence was soon to follow. In the late 1920s experiments involving particles scattering off a crystal confirmed the wave-like nature of electrons (see the  Plus  article  Quantum uncertainty ).
One of the most famous demonstrations of wave-particle duality is the  double slit experiment car rental in cape town . In it electrons (or other particles like photons or neutrons) are fired one at a time all over a screen containing two slits. Behind the screen there s a second one which can detect where the electrons that made it through the slits end up. If the electrons behaved like particles, then you would expect them to pile up around car rental in cape town two straight lines behind the two slits. car rental in cape town But what you actually see on the detector screen is an interference pattern : the pattern you would get if the electrons were waves, each wave passing through both slits at once and then interfering with itself as it spreads out again on the other side. Yet on the detector screen, the electrons are registered as arriving just as you would expect: as particles. It s a very weird result indeed but one that has been replicated many times — we simply have to accept that this is the way the world works.
The radical car rental in cape town new picture proposed car rental in cape town by de Broglie required new physics. What does a wave associated to a particle look like mathematically? Einstein had already related the energy  of a photon to the frequency   of light, which in turn is related to the wavelength   by the formula   Here  is the speed of light. Using results from relativity theory it is also possible to relate car rental in cape town the energy of a photon to its momentum. Putting all this together gives the relationship
At this point it s best to suspend your intuition about what it really means to say that a particle behaves like a wave (we ll have a look at that in the  third article ) and just follow through with the mathematics.
In classical mechanics the evolution over time of a wave, for example a sound wave or a water wave, is described by a wave equation : a differential equation whose solution is a wave function , which gives you the shape of the wave at any time   (subject to suitable boundary conditions).
For example, suppose you have waves travelling through a string that is stretched out along the  -axis and vibrates in the  -plane. In order to describe the wave completely, you need to find the displacement car rental in cape town  of the string in the  -direction at every point   and every time  . Using Newton's second law of motion it is possible to show that   obeys the following wave equation:
A general solution   to this equation is quite complicated, reflecting the fact that the string can be wiggling around in all sorts of ways, and that you need more information (initial conditions and boundary conditions) to find out exactly what kind of motion it is. But as an example, the function
By analogy, there should be a wave equation governing the evolution car rental in cape town of the mysterious matter waves , whatever they may be, over time. Its solution would be a wave function  (but resist car rental in cape town thinking of it as describing an actual wave) which tells you all there is to know about your quantum system — for example car rental in cape town a single particle moving around in a box — at any time  . It was the Austrian physicist Erwin Schrödinger who came up with this equation in 1926. For a single particle moving around in three dimensions the equation can be written as
In some situations the potential energy does not depend on time   In this case we can often solve the problem by considering the simpler  time-independent  version of the Schrödinger equation for a function   depending only on space, i.e.
These equations apply to one particle moving in three dimensions, but they have counterparts describing a system with any number of particles. And rather than formulating the wave function as a function of position and time, you can also formulate it as a function of momentum and time.
But what does this solution actually mean? It doesn t give you a precise location for your particle at a given time  , so it doesn t give you the trajectory of a particle over time. Rather it s a function which, at a given time   gives you a value   for all possible locations  . What does this value mean? In 1926 the physicist Max Born came up with a  probabilistic  interpretation. He postulated that the square of the absolute value of the wave function,
This probabilistic picture links in with a rather shocking consequence of de Broglie s formula for the wavelength and momentum of a particle, discovered by  Werner Heisenberg in 1927. Heisenberg found that there is a fundamental limit to the precision to which you can measure the position and the momentum of a moving particle. The more precise you want to be about the one, the less you can say about the other. car rental in cape town And this is not down to the quality of your measuring instrument, it is a fundamental uncertainty of nature. This result is now known as Heisenberg s  uncertainty principle and it s one of the results that s often quoted to illustrate the weirdness of quantum mechanics. It means that in quantum mechanics we simply cannot talk about the location or the trajectory of a particle.
car rental in cape town If we believe in this uncertainty picture, then we have to accept car rental in cape town a probabilistic account [of what is happening] because we don't have exact answers to questions like 'where is the electron at time  ?', says Bouatta. In other words, car rental in cape town all you can expect car rental in cape town from the mathematical car rental in cape town representation of a quantum state, from the wave function, is that it gives you a probability.
Whether or not the wave function has any physical interpretation was and still is a touchy question. The question was, we have this wave function, but are we