Werner Heisenberg (1901-1976) decided to solve the problem. He decided to start with what can be measured experimentally - that is the spectral lines. He created a theory called the matrix mechanics. Thanks to calculating matrixes (some mathematical expressions) the same results as from the Bohr theory could be achieved. And that was without creating a vision of any strange orbits of electrons. The Heisenberg matrixes explained even more facts than the Bohr theory. Unfortunately, as is usually goes with the theories using complicated mathematics it was very difficult and that is why not much liked by physicians.

In 1925 Schrodinger (being thirty eight) (1887-1961) tried to study the problem. Interested in the wave description of the electron suggested by de Broglie he decided to make a theory using this description. He created an equation in which the electron is treated just like it was a wave - the wave of matter. The equation is called Schrodinger equation. - the wave function is described by the Schrodinger equation. The function describes all the features of the electron which we know or which we can measure. The is variable in time and space. According to Schrodinger the electrons were to be just the waves of matter, and it's corpuscular form was to be only a illusion. The Schrodinger equation is the second power of the wave function () described by probability density function of that wave of matter. In the hydrogen atom the waves of matter were dense in the places where the orbits described by the Bohr theory were situated. Solving the equation one could calculate the radiuses of that orbits directly.

    What more, thanks to the model suggested by Schrodinger the radiuses of orbits and the placement of the spectral lines of many other than hydrogen chemical elements can be calculated.
    The Schrodinger equation described well not only the electrons. It is used also for description of many other particles such as protons, neutrons all the whole atom.
    Besides the Schrodinger theory is simplest than the Heisenberg's matrix mechanics although they are equivalent
    But the interpretation of the Schrodinger wave was wrong. The experimental facts pointed that the electron can't be a wave of matter. For example it couldn't explain the observed fact described below:
    Let's imagine that we have a wire net which we connect with a battery of a known voltage. Now, if we would be shooting the net with the electrons which have the higher energy than the energy of such construed barrier the electrons should penetrated without any problem. But if the electrons are of the lower energy than the energy of the barrier they shouldn't penetrate it. They all should be impeded and then repulsed - just like there were reflected. That's how it should be. But the experimental facts didn't confirm that. Even if the energy is higher than of the bombarding electrons some part of the electrons will penetrate the barrier anyway (the rest will be reflected ). According to the Schrodinger equation some part of the wave of electron goes through the barrier and some part is reflected. But in such situation there would be something strange happening to the electron - some part of the electron would be penetrating and some would be reflected. No such think as the parts of electrons occurred.
    How can it be explained?

In 1926 Max Born noticed that the second power of the wave function ( 2) described by Schrodinger is just a description of probability of the electron being situated in a given place. The function is time and space dependent and if 2achieves a big value in a given place there is also a high probability of the electron being find in that place. But where 2is small the probability is also small. In places where 2is equal 0 the electron can not be situated.

    We can predict the situation of the electron only with the determined probability. For example in the atom of hydrogen the probability of finding an electron in the ground state in a sphere of a radius equal 10^-8 cm around the nucleus is equal about 80%.
    So the phenomena described before can be now easily explained. A part of the wave goes through the barrier and thanks to it there is some probability of finding an electron behind the barrier. So the electron can with some probability go through the barrier of the potential. For example shooting the net of a hundred electrons we get eighty five of them reflected and fifteen go through.
    Summarising Born noticed when measured the electrons behave like they were particles and in other cases their probability distribution in space is in accordance with the probability coming from the Schrodinger equation. So the ²is the description of the probable positions of the electron.
    The laws of the probability are also used for describing the electrons changing the energy states. If the electron placed on a given orbit can "jump" on more than one lower orbit then there is some probability of taking each of them.
    Thanks to the Schroedinger equation and the Born's interpretation of it the behaviour of atoms of different chemical elements and their bounding into chemical compounds was explained. Such great systems for researching microstructures like the electron and the proton microscopes where constructed.
    In 1927 Paul Dirac developed a formula very well describing the electron wave. The equation explained and predicted many facts connected with the electron. The only problem was that there were two solutions. That was the suggestion of the subsistence of the particle identical to the electron but of a positive charge.
    The particle was discovered in 1932 by Carl Anderson. It was called the positron and was the first discovered antiparticle (the particle of anti-matter).
    Also in 1927 Heisenberg gave indeterminacy principle. It says that there are pairs of the quantities that appear in atomic physics which can not be known simultaneously with a big accuracy. For example it is not possible to defined exactly both the position and the momentum of electron at the same time. The indeterminacy quotient of those two quantities is has got to be equal or bigger than the Planck constant (h):

            (1)

            (2)