The first great stride that the champions of QM made towards understand particles-waves was to make a new definition for subatomic particles. The new definition stated that particles were a tendency to exist. The next great step came when they started to attack the problem from a mathematical point of view. They found that not only can a particle be quantized, but a wave can be quantized as well. A quantized wave is better known as a standing wave. A standing wave is the kind of wave that happens when something oscillates at just the right rate in order for nodes to form. The figure below depicts various standing waves. If you have ever played with a slinky this effect is common. It is when the waves troughs and crest simply trade places with out moving down the slinky.

Following the ideas of standing waves a QM named Erwin Schrodinger developed the idea that electrons were actually standing waves. Schrodinger studied the mechanics of standing waves in a wave tank. Unlike the waves above Schrodinger's were three dimensional. These standing waves had many different shapes that are depicted below. After a great deal of studying he developed a wave equation that describes the propagation on standing waves with respect to time and initial conditions. It was at the same time that another QM named Wolfgang Pauli showed that no two electrons can exist exactly the same state. This idea is called the Pauli exclusion principle. The exclusion principle explains a plethora of natural phenomena including why some star form neutron stars and others form white dwarfs. More important than that, it allowed Schrodinger to perfect his wave equation. When the exclusion principle is accounted for, the wave equation is a great model for electrons. For example, the wave equation predicts that there can only be two electrons in the lowest energy shell because their are only two standing wave forms that exist at this energy level. It was the wave equation that helped develop the quantum numbers for describing electrons that chemists use today.

The state of an electron is described by its four quantum numbers. The 1st number describes the energy level of the electron. Another way to think of this is the size of the shell. These numbers usually range from 1 to 7. The 2nd number describes the shape of the shell. Their are currently four shapes: (s, p, d, f). The 3rd number describes which axis (x, y, or z) that orbital is centered about. The fourth number describes the spin of the electron. It can be either +1\2 or -1/2. For example in helium there are two electrons. One electron is (1, s, n/a,+1/2) the other is (1, s, n/a, -1/2). Their is a blank in the d number because the s means that the shell is sphere shaped and a sphere is symmetric about all 3 axis. The wave equation received its final touch when it accounted for the two-slit experiment

In the advanced two slit experiment if an individual photon is fired at the target it seems to go through both holes, interfere with itself, and falls into one of the stripes at random. This randomness is governed by probability. Although, it is currently not possible to predict the trajectory of an individual photon, it is possible to count the number of photons in each interference band. Once this is done it is now possible to make a statistical model governed by probability.

For example, say 100 photons travel through the two slit experiment and end up in a pattern.

30 photons are found in the middle band. 20 photons are found in the band left of center and in the band right of center. 15 photons are found to be scattered among various bands left of the band that is left of center. Another 15 photons are found to be scattered among various bands right of the band that is right of center. With this information it is possible to make a statistical model. It can be said that thirty percent of the time a photon traveling through the two slit apparatus will go to the center band. Thirty out of one-hundred is thirty percent. A percentage can also be assigned to each of the individual bands.

A percentage is simply a measure of probability and it is the amount probability that explains why the middle stripe is brighter than the others. There is a higher probability that a photon will go to the middle band as opposed to the other stripes beside it. QMs thought about all of this and theorized that since a particle was the tendency to exist maybe the wave component of the particle-wave duality was a wave of probability to govern the particle's state of existence. A standing wave of probability would explain the quantized aspect of a photon and how light can travel through a vacuum. It was probability that was waving and not some strange ether.

When all these ideas were brought together and put into mathematical form the QMs had what is now known as Schrodinger's wave equation. The wave equation is named after Erwin Schrodinger because he was the physicists who developed the mathematics behind it. The wave equation is one of the cornerstones of quantum mechanics. The wave function models a probably wave, which governs the probability of a corresponding particle's tendency to exist in certain states. The wave function works like this. When the proper initial conditions for a particle(s) are accounted for the wave function generates a multi-dimensional probability wave which governs the probability of the particle(s) existing in certain states. The wave function can deal with more than 3-dimensional waves. For example, if two particles are being modeled, the wave function is 6-D. It has three dimensions for each particle. When a measurement is made on the system it causes the wave function to collapse and the particle(s) to exist in one of the probable real positions. This is known as the quantum jump. If thousand of measurements are made a pattern will emerge which matches the up the statistical pattern predicted by the wave function. The wave equation is essentially a socket wrench for a QM. It is one of his most useful and versatile tools. With it he is able to find the probability of a particle existing in a certain place at a certain time.

The probability wave model has had terrific success explaining natural phenomena with amazing accuracy. It also can be used to explain why an atom is stable. All the models till now had either waves or particles orbiting the nucleus. Bohr with his model showed that the atom was stable because the electrons can only exist in certain quantized levels. These levels are like stair steps. In Bohr's model the electrons could only exist at certain levels of energy. Bohr used this model to explain the bright line spectrums, but he never said why electrons exist at quantized energy levels. If these levels were stable it would be easy to explain why atoms are relatively stable. If the ground state is the lowest level that an electron can exist on then an electron could not crash into the nucleus. These levels are stable because the probability wave determines that these are the only probable places for an electron to exist. The probability wave explains why electrons have quantized energy levels and why atoms are stable.

The wave equation has some remarkable consequences for quantum mechanics. For one, it changes quantum mechanics from a science of absolute laws to one governed by probability. It was this that caused Einstein to become disgusted with quantum mechanics and state 'I do not believe in a God who plays dice.' Another question brought up by this interpretation was the importance of the observer and quantum uncertainty.