Breaking down reality(the advent of quantum mechanics)
For decades people have been trying to unravel the mysteries of the universe and in doing so they gave way to mathematical descriptions of nature which are more consistent say, than writing down the entire bible and not understanding which god to mention, mathematics is a tool we use to describe the nature around us so understanding what kind of math works and what doesn’t and logically driving out a hypothesis from all that scramble is what theoretical physicists do
For example describe gravity , this has only one correct logic
Relativity
Or Newtonian physics
which is correct, relativity, why, the mathematics here is describing a lot more things than the Newtonian equations of gravity, this is called mathematical consistency, newton stated the effect not the cause of gravity which we will examine in later articles.
Newton gave us a simple day-to-day understanding of physics using his equations of motion, the most famous being f=ma, but as humanity shifted and expanded its ideas in search of a brighter horizon these things were given lesser and lesser importance because of the weird rules obeyed by quantum mechanics, ellipses round the sun, parabolas on the run which used to describe day to day motion of objects were now being replaced by probability and a lot of nonsensical intuition which we call quantum mechanics
You might know from basic high school physics the expression for a moving wave is
y(x,t)=A+cos(kx-wt)
Tedious, no sweat we will break it down, the last term is the shape of the wave oscillating with a definite period and the cos factor is the shape, from w=2pi/t we get wt=2pi stating the periodicity of the wave, for more, check out my upcoming article on calculus, an intuitive approach for physics, not to mention that calculus is the most basic mathematics you will need to understand quantum physics or rather any kind of physics
For quantum systems the same idea is given a more complicated expression, for , it was observed that particles and waves are but the same thing, it all kickstarted with the photoelectric effect where Einstein gave the idea of photons the quantized version of light, according to the equation e=hv, where e is the energy of one photon to knock an electron, makes sense, cause waves cannot hit anything we need particles and that too with a high frequency not wavelength which is, color, for example, take a red bottle and a green bottle which hits your eyes most, green, cause it has more frequency so if you hit the electron with a green photon it would get ejected much more easily than with a red one
The expression of an electron spiraling around the atom is given by a circling (standing) wave
Ae^ik-wt
This is called the wavefunction, which tells us the probability of an electron,s presence somewhere within the atom, e and i are constants that we don't need to deal with just now
Schrodinger, gave it a deep thought and decided to introduce a relation for the energy and momentum of a particle E=p²-m², where c is the speed of light, for simplification we have taken c equals 1 thus the change in notation from which we get the Schrodinger equation for motion which is
Where psi is the wave function of the electron,,
It can also be extended to fields where the latter is denoted by phi
This all kickstarted when the physics prophet Einstein introduced wave-particle duality saying
Particles and waves are the same things, he noticed that when light falls on metals, electrons can be knocked out of it, this all seems weird but he said, no idiots, forget that light was a wave, instead treat it as a particle, photon, the more frequency, the more electrons does it kick out, not the wavelength which says it to be a wave, later de Broglie put forward a relation for the same relating the wavelength to the total energy and momentum of the electron, the more the momentum, the less the wavelength and vice versa, if it moves faster the wave will get smaller and shorter indicating a decreased wavelength and the slower, the more the amplitude and the wavelength. But this was baffling, how do we know where the electron is if it is a wave,
Schrodinger gave it a deep thought and introduced the relation
Where k(r,t) is the wavefunction of the electron and e is the energy denoted by the first term, it is just relating the energy of an electron to its Hamiltonian or the total energy it has, it gives the hint as to how fast and with how much energy is the electron going with the change in the wavefunction, later using the same relation Klein, and Gordon found out that the wavefunction could be replaced with a field interaction, specifically the electromagnetic field for photons!
A photon is a bunch of oscillations up and down in an electromagnetic field and it is described using the proca equation, specifically duFuv=m²Av
Where Fuv is the electromagnetic stress tensor indicating the components of the field in various directions and the mass term is the mass of the photon(nonexistent but still we use it)so the rate of change of the electromagnetic field is equal to the energy of the photon because A is the potential energy of the system remembering that B=gradA, from the maxwell equations
What next, well dirac was sucking his pen at university of cambridge working on a new theory in QM when suddenly he got it and jumped off his chair immediately giving birth to the dirac equation which is
ih du-mcpsi=0
Here ih is the energy of the electron and another term is required which i forgot to add, it is c(a.p) so the equation fully reads
Bmc²psi +c(a.p)psi=Epsi
Where psi is the electron wavefunction and the first term is the RELATIVISTIC energy of the electron, and a and b are called the Dirac matrices determining the spin of the electrons, now spin is another very complex thing to understand but we will tackle it step by step
If you spin your colorful water bottle or even a girl while dancing(haha lol) you will notice that their sides change, nothing is the same, and their backs are differently colored, but in the case of particles they don’t have any distinct features to give them away they stay the same so it feels like nothing changed they are not colorful or black and white to see the difference in the front and back sides(more on that later) , they are not actually spinning but it is just a term given to describe this weird phenomenon, so the Dirac equation describes the relativistic(obeying relativity, more on that next ) behavior of the spinning electron in the atom while schrodinger and klein gordon equations describe their positions and energies ….
We are done for now
adios!
