The concept of the dual nature of the photon was extended to other particles such as electrons by DeBroglie whose equation: mv = h /l (mass x velocity = Plank's constant /wavelength ) is like Plank's for photons, i.e. E = hn ( or energy = Plank's constant x frequency ).

All we have to do is express the wavelength l as velocity, v / frequency n, and rearrange the equation so that the left hand side is mv², which is the kinetic energy, and the right hand side becomes hn and we have Plank's equation.

The form of DeBroglie's equation emphasises the dual nature of the electron and photon since the left hand side contains classical mechanical terms, mass and velocity, easily associated with a particle, while the right hand side is all about waves. Your expression of an "electron wave" sums it up. Is it a particle or a wave ? Answer: both according to the results of its behaviour.

Let's look at why electrons behave as they seem to in atoms. The simple explanation which skates over a lot of detail is that when particles behave like waves and find themselves in situations where the waves are constrained by certain they become "standing waves" and only certain wave-lengths or energy levels are allowed. This is the case of electrons under the influence of the oppositely charged atomic nucleus. Some of these energy levels correspond with the "shells" of electrons surrounding the nucleus. So it is not so much the "balance of kinetic and potential energy" which explains the existence of shells as the wave-like behaviour of electrons. Let's explore this in more detail.

When particles are as small as electrons, we can not observe their behaviour without affecting it because the means we have at our disposal for observation are similar in size and energy as the particles we want to observe. This means that we cannot observe the precise behaviour of an electron without changing the behaviour by interacting with it when trying to observe it. Heisenburg first pointed this "Uncertainty Principle" out. Imagine trying to detect a ball moving on a pool table by firing another ball across the table. When a collision occurs we might know the position of the collision but we do not know the speed of the target ball or its direction after the collision. We can however, repeat the observation enough times in enough positions to establish what the probability of the target ball being at any position on the table is.

The mathematical functions which describe the probability of finding electrons at various positions near the atomic nucleus satisfy differential equations which are analogous to those describing the amplitude of a wave. As we have recognised, it is a characteristic of waves, that under certain circumstances, standing waves can be established where the amplitudes are fixed in space. A simple example is a vibrating string whose amplitude is greatest in the middle of the string. In the case of electrons, the amplitudes of the waves correspond with the probabilities of finding electrons.

If we were to describe the motion of electrons in atoms in classical terms these terms such as potential energy and kinetic energy would have a continuum of allowed values. The blend of particle and wave behaviour exhibited by electrons in atoms results in only some values being allowed by the standing-wave characteristics. The potential energy, for example, is dependent on the distance of the electron from the nucleus and so the distance, ( the average distance, since we are not sure where the electron is ) has certain "quantised" values. In this case, we use the letter "n" to indicate the quantum level for this characteristic, the "principal quantum number". The lower the number, the lower the energy of the electron, and the closer the electron is, on average, to the nucleus.

For kinetic energy, we use quantised values for angular momentum called the "azimuthal quantum number". This number "l" can have values from 0 to (n-1).

To describe the observed behaviour of electrons more completely, we have to take another factor into account. The motion of the electron around the nucleus produces a magnetic moment. This enables electrons with the same potential energy ( principal quantum number, n ) and kinetic energy ( azimuthal quantum number, l ) to be distinguished from each other according to how this magnetic moment behaves if a magnetic field is present. The wave nature of the electron again dictates that this characteristic is quantised. The magnetic moment vector assumes a number of orientations relative to the applied field ranging from fully aligned in one direction, where the magnetic quantum number, m_l is equal to the azimuthal quantum number, to 180 degrees opposite. This quantum number, m_l , therefore has values between l and -l.

These three quantum numbers dictate the shapes and positions of the probability functions which we call orbitals rather than the classical orbits we would use for particles. Finally, because the electron displays behaviour as if it were a spinning charged particle, it appears to have an intrinsic magnetic moment and the magnetic vector can be either aligned in the direction of an applied field or opposite to it. This "spin" is governed by a quantum number, m_s.

At last we come to the Pauli exclusion principle, which has mathematical origins but is simply stated as a rule which dictates that no two electrons have the same set of quantum numbers. In other words, no more than two electrons ( distinguished by their spin ) can occupy the same orbital. This rule is accepted because of its success in matching the conclusions drawn from it to what we observe. Our picture of the electronic structure of atoms and molecules depend on the exclusion principle and is the basis on our understanding how chemical behaviour is dictated by electronic behaviour.

With our four quantum numbers and the Pauli exclusion principle, we can build up pictures of the electrons in their orbitals. As we increase the positive charge on the nucleus from 1 for hydrogen we add more electrons each of which occupies the lowest-energy orbitals available. The "shells" of electrons are the regions occupied by groups of electrons with the same principal quantum number.

To go any further requires diagrams, tables and graphs. This question sent me rushing back to some of my old text books, "Valance" by C.A.Coulson ( Oxford University Press ) and "Physical Chemistry" by Moelwyn-Hughes ( Pergamon Press ), both pretty ancient. For something more up-to date, very visual but light on explanation try www.orbitals.com. For something in between try: http://antoine.fsu.umd.edu/chem/senese/101/electrons/index.shtml