The integral of sin x,y,z at t is equal to the energy concerns of the system an equivalence given by
a is a constant such as the speed of light=c not to be confused with the C in the equation below which simply denotes any constant, n is any number as is x.
Maths gibberish representing the area of the line under the graph in terms of sin aka an antiderivative or integral
Differential showing the rate of change of the line of a graph which is consequently a derivative is undoing what and an integral did. Hence you get things like s=ut+1/2at^2
Which is clearly the equation for speed, and if you you used the equation
Which you may recognise as quadratic from long boring maths equations at school, it really is not that complicated.
Hence this bastard:
Newton merely used integrals to explain why speed=distance over time, acceleration is a rate of change proportional to speed at time t represented by a=x/m^-2 and so on. Which is why the metre is defined by the speed of light in a vacuum because it is a constant. Likewise time is defined by space by space-time.
You can't model a quantum system with gravitation using Newtonian mechanics and get a correspondence to reality like general relativity which is non euclidian classical system, at the small scale though matter can't be analogised like that..?
As I opined on another thread the Planck model of planets orbiting a sun you are taught in high school is meaningless, although simple enough to let the novice understand.
lambda is the periodicity and it works well to model water and air, but energy doesn't come in nice convenient classical packets, it comes in quanta, and quanta are definite propositions of absorption and emission which can be empirically verified as can the forces hence be modelled by analogy but with some serious maths caveats: we have to use an imaginary axis called i to represent x,y,z at t, aka as Minkowski metric, and we have to introduce a constant that agrees with the proportions of energy found in the test apparatus in spectroscopy experiments, we also have to renormalise the integral so there are realistic degrees of freedom and things can't magically appear on the other side of the universe we hence have the limits - and positive infinity which denotes the waves extent and energy which unlike Newtons classical calculus is never 0 because energy is never nothing, an assymptotic limit of energy hence cannot have no energy although constructive and destructive interference can produce zero points on the graph the energy of the system is still clearly not nothing when two phases destructively interfere.
This more clearly shows the relation of sin to a circle if you moved the bottom phase under the top you can clearly c it forms a circle. And if you put one graph over the top of the other you would see a series of circles.
If we plug the energy concerns of say Deuterium an isotope of hydrogen into the Dirac equation what we get exactly pictures the resultant spectra of dueterium, which is not to say the maths is denoting reality, that is a philosophical concern, merely that when the experiment is done the results agree with the model.
h bar is the reduced plancks constant which is needed to introduce the energy of a system in terms that whilst not classical are analogous to a wave with uncertainty equal to the measurements uncertainty at least.
Which lead to copenhagen and hence to anti matter experiments, and the standard model with Higgs.
I lost the plot around here, so if you even read the source and made it this far kudos.Hydrogen atom densities and dissociation degrees were derived from the absolute intensities of three hydrogen Balmer lines. Due to the high fraction of molecules, direct atom excitation and dissociative excitation must both be taken into account in the interpretation. The hydrogen atom density nH is then related to the measured photon emission coefficient by
with the effective emission coefficients Xem. The molecular density was derived from Dalton's law using the molecule partial pressure and the gas temperature. The molecular pressure was obtained from the known mixture, subtracting electron and ion pressures as well as the atom pressure--all small corrections--by iteration. Knowledge of Te is relatively important for evaluating equation (11). Contributions to the Balmer lines from dissociative recombination or mutual H - /H + recombination are negligible (below 5%) in the considered plasmas. Quenching processes by hydrogen molecules also influence the upper states of the Balmer lines only very little. Estimates on the basis of  show that the maximum correction is of the order of 5%, which has been taken into account in the results. Depending on the neutral hydrogen densities in these plasmas, both rate coefficients must be corrected for the optical thickness of the Lyman lines
I had to look this up I must admit... :S
Wow they does like their maths gibberish. Even I had to think twice before understanding this even remotely. :S
You can at least see where the electron volt values come from, well if your brain hasn't leaked out of your ears, which is a consequence of maths...
I am reminded of the difficulty of reunifying space kittens with buttered toast in the resultant imaginary plane of wtf.