Wavefunctions and Operators The Schrodinger Equation

It would be hopeless to try to describe the complex motions of electrons and nuclei as a function of coordinates in space and time. The only useful concept is that of distribution or probability of presence. This distribution is represented by the wavefunction xjj, jf, z) such that xp1 gives the 

The lowest frequency standing wave has no node (t/ ) and is called the fundamental5 (frequency). Higher frequency modes are harmonics5 which have increasing numbers of nodes. 

These concepts of the classical physics of standing waves have important implications in photophysics, in particular for the understanding of orbital symmetries and laser light emission. In the case of a standing wave the propagation velocity does not exist, and the important relationship defines the wavelengths as a function of the distance between the boundaries and the number of nodes in the wavefunction 

In eqn. 2.7 the number n is a quantum number5. It is in fact related to the number of nodes in the wavefunction and must in this case be a positive integer (n = 1, 2, etc.). This would apply to a wave which follows one direction only. Since real space is three-dimensional a standing wave must be defined by three quantum numbers. The motions of electrons around nuclei are essentially circular, so that the use of polar coordinates is preferable and the three quantum numbers are  

While it is obvious that the principal quantum number n must be a positive integer (it is impossible to have less than zero nodes in a wavefunction), the values of the other quantum numbers can be negative when they are defined in polar coordinates. The allowed values of the various quantum numbers are 

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