Spherical Harmonics

Todays poem is about Spherical Harmonics, just as any function in rectangular coordinates can be expressed as a Fourier Series in Trigonometric functions and in Cylindrical coordinates we can expand any function in terms of Bessel Functions. So in spherical coordinates it is possible to express any function as a series in Spherical Harmonics. Naturally then this expansion  has been applied to spherical waves, the calculation of the fields outside a sphere which leads to a series of Polynomials called Legendre polynomials. One of the most exciting application is to the calculation of the angular dependence of the wave function of the hydrogen atom. It is often thought that because the "waves" are quantised that this is unique to quantum systems however this is mistaken. Any spherical wave be it classical or quantum is quantised in the sense that depending on the size of the sphere only a certain set pf standing waves are available. 

Of course the biggest misconception about the so called wavefunction of the hydrogen atom is the fact that many people think they are really waves. Well as I have argued many times in quantum mechanics the so called waves are really probability density functions. Despite the misleading pictures of orbitals of the hydrogen atom the electron is not spread out over all space as they would seem to imply. Like any probability density function they just represent the probability that an electron of a certain energy will be seen at a certain place with a given probability. The surfaces represent the boundary at which the electron is likely to be found upto 95% of the time. But an electron is essentially a small point like particle with a definite mass, intrinsic spin and charge. Quantum mechanics does not change this picture. Also it should be remembered that the probability density functions are three dimensional functions so the electron is not in an orbit at all. But can be anywhere within that three dimensional region. Calling these three dimensional probability density functions orbitals is just so misleading. Matters are not helped by energy level diagrams showing the electron jumping from one energy level to another. But it must be remembered that these are energy level  diagrams not positional diagrams. An electron does not jump from one energy level to another. It just changes energy and with each energy level the form of the probability density function changes. 

Anyway having said that Spherical Harmonics just like Bessel Functions are truly amazing functions. There are a whole load of other 'Special Functions' associated with the differential equations of mathematical physics, but anyone who masters the mathematical properties of Spherical Harmonics and Bessel functions will be well placed to understand a lot of physics. 

Finally it is of interest to note that the Spectrum of the Cosmic Microwave Background radiation has been expressed in terms of Spherical Harmonics. Also via the Spherical Harmonic addition theorem there is a close link between Spherical Harmonics and Bessel Functions another example of the interconnectedness of mathematical ideas. Here is my tribute to them 

                                                        Spherical Harmonics

 

                                        Many uses we have found,

                                    For your functions so profound.

                                    The fields inside a sphere,

                                    Have become very clear.

 

                                    You describe the spectrum,

                                    Of an atoms angular momentum.

                                    And the background radiation,

                                    That comes from natures creation. 

        Probability density funtions of an electron outside a hydrogen atom