In 1905 Albert Einstein presented two equations for energy. The first was:
E = hf
for the energy of electromagnetic waves (later to become known as photons). Here h is Planck’s constant and f = the frequency of the photon.
And the second formula was:
E = mc2
where m is the mass of any particle or object, and c is the speed of light.
In 1925 Louis de Broglie wondered why these two forms of energy had to be considered different. Could they not be combined? The only way they could be combined is if the entities being described (electrons and such) had both a particle and wave nature—as photons do.
From this de Broglie surmised that all particles have a complementary wavelike aspect, with a wavelength λ (lambda). The formula he derived for the wavelength is:
λ = h/mv
where again m = mass and v = the velocity at which the particle (or composite of particles) is moving.
So the larger the mass and higher the velocity for a particle, then the shorter its wavelength. This implies that the wavelengths for large objects (like cats or humans) are so small as to never be observable (even with high-tech instrumentation). But the wavelengths for tiny objects (like electrons and other sub-atomic particles) are long enough to be observed by our instruments: and so they have been (via diffraction and interference experiments).
In 1926 Irwin Schrödinger inserted de Broglie’s wave-like representation of particles into the conservation of energy equation (total energy = kinetic energy plus potential energy) and from this derived an equation to describe their behavior— which has become known as the Schrödinger wave equation.
A wave equation is an example of an ‘equation of motion’ which, as the name suggests, can be used to predict the motion of an object. In this case the object is a wave. In other words, if we know the amplitude and velocity of the wave at a given time and place, we can project forward (or backward) and predict the amplitude and velocity of the wave at some other time and place. For example, if you drop a pebble into a pond it makes a wave of a given height (amplitude) which will decrease with time as the wave spreads. Knowing the rate at which the wave spreads and loses amplitude, we can predict what it will look like in ten seconds, twenty seconds and so on. Or conversely, we can look at the circular wave pattern at a given time and run the whole thing in reverse to re-create the original pulse created by the pebble.
This so-called ‘classical’ wave (aka ‘wave function’ when expressed mathematically) differs from the ‘quantum’ waves described by the Schrödinger equation, in that the height of the peak (amplitude) and depth of trough in the ‘Schrödinger wave function’ correlates to the probability of finding the particle at that location at a particular time. In this sense the wave amplitude is a ‘probability amplitude’ which can be positive (a peak) or negative (a trough). So in the case where two such waves interfere they can add constructively or destructively (whereby they would get larger or tend to cancel each other out). To get the actual probability of finding the particle at a given location at a given time from the Schrödinger equation, we have to square the probability amplitude: and the square is always a positive number that ranges from zero to one. The fact that it is a kind of probability wave, also implies that the electron (or photon as the case may be) is not actually present at any location described by the wave function, until we make a measurement with some kind of electromagnetic device. In other words the individual micro-particle is in a fuzzy wave-like spread-out state prior to interacting with the macro-world via a measurement, and only collapses to look like an individual particle at a specific location after the interaction. So, when we make that measurement we find that it is located at one of the possible locations outlined by the wave function. And if we were to repeat the experiment hundreds of times, the locations we found would follow the pattern prescribed by the wave function.
To better understand a ‘probability wave’, imagine that you have a friend who you know is at a large fair-ground and you want to find her as quickly as possible. On that fair-ground there are animal cages in which there is zero chance of finding your friend; and large open areas with no attractions, in which there is some chance of finding your friend; and there are also attractions around which there is a higher probability of finding her. From this you could construct a probability distribution, and then look in the more high probability places to find her. A plot of the probability distribution would look like an air photograph of the fair-ground, with clusters of people around the high probability sites, less people in the lower probability sites, and zero people in the animal cages. This kind of distribution is what is represented by the solution to the Schrödinger wave equation (i.e. the wave function) when applied to a specific situation. The famous ‘double slit experiment’ shows a probability distribution for photons that have been directed through two closely placed narrow slits, which differs from the pattern we get for one slit or three slits; or a round pinhole vs. a square one.
From a practical perspective, the Schrödinger equation can be used to predict the behavior of electrons (and other particles) in various situations: such as when bonded to an atom, or trapped in a box. This level of understanding has allowed us to manipulate electrons in materials (like semi-conductors), which in turn allow us to manipulate the flow of electricity into complex patterns: which then allows us to turn inanimate objects into devices on which we can watch movies and talk to our friends on the other side of the world.