Electronic Structure of Atoms. The Periodic Table

The occurrence of discrete energy states of a bound electron in an atom and the co-existence of wave and particle properties in a free electron, can be accounted for on the basis of wave mechanics (Schrodinger, 1926). Observation of the diffraction patterns produced when electrons of known energy encounter crystals of known atomic spacing shows that the wave length, A, associated with an electron of velocity, v, is given by 

Wave and particle properties can then be reconciled by the following argument. 

The need to associate a wave packet with a inovung particle has another important consequence. In the one-dimensional case of plane waves representing a particle moving in tlie, v-direction, the extension of the wave packet (/Jx, say) is determined by tlie range of wavelengths admitted in the packet and this in turn, by the momentum range ay). It can then be shown (Heisenberg, 1927) that 

Any disturbance W [e,g, the displacement of an elastic string or the height of a water wave) which is propagated along the r-direction with velocity u satisfies a simple partial differential equation, the wave equation  

Here W depends on both position and time, W More generally 

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