## Quantum Mechanics for the rest of us

This is my humble experiment to collect in one place the reasons of why Quantum mechanics came up, all the WHY's and HOW's a layman person can ask. These series of articles is targeted towards the people which are not afraid of some high school mathematics, either. You don't have to decipher the formulas to understand most of the content, but that is the added benefit I'm hopping to pursue. Also in roll-up boxes I'll will try to come up with the full or partial derivation of the laws we encounter, or at least you should expect some general explanation of the derivation. Again you don't have to even look at those.
My other motivation for writing this was because there are many articles on the internet for parts of how QM come to be and explanations of different phenomena that caused it, but I have not seen one which covers it end to end with historical perspective in mind.
Keep also in mind that this is work in progress and if some part seem a little bit of sketchy I will fill more info over time.
I will be also very greatful to people who can point me to some crucial parts I missed OR got wrong once I put the Comments section.

There are two main approaches, that I've seen both in books and tutorials when it comes to explaining Quantum mechanics.
The first one is the wishy-washy ala-magic approach in which authors are scared the hell from even using a basic formula to give some insight into the matter, the second approach is to jump into bracket notation, derivatives, integrals, differential equations, complex vectors and all the high math.
The third approach which I will try to use here is somewhat of a middle ground, as picturesque explanation as possible, but at the same time using some high school math to illustrate the concepts. Don't get me wrong there are some articles which dip their toes and use this third method, but just to a point, most of the time they expect the reader to be fluent in Physics. I hope to be able to present the whole "fact and figures" in this one place, so you don't have to jump around.
So as additional side project I will try over time to build a library of short introductions to some basic terminology that is often used when we are talking physics, things like momentum, energy and so on, so you don't have to jump to other links to read about it. Of course it will go without saying that I will give you links at the Reference section where you can find in many cases much more elaborate and detailed discussion on those topics if you are further interested.
The goal I have in mind is that you can get basic understanding just from these couple of articles and then this will ignite your interest to dig deeper. I hope to give you the fundamentals to understand the ideas and concepts used in QM and Physics in general. Then I have my selfish goal to document in one place the things I was able to understand, so I can be able to refer to it later on and also expand on it.
I'm trying to combine bits and pieces I was able to extract from different places and then combine them in a coherent enough way, so that you can grasp what is QM all about ? At the same time I hope to capture the historical context so articles read more like a story, than pure physics text.
Don't expect to learn how to do QM calculations and such, because in all of the cases I'm using simplified scenarios not real life ones which will require QM-math, which as you probably guessed I don't understand very much anyway.
Again you will have to know high school math... pleaase.
So..... lets begin...

It is the beginning of the 20th century Newton mechanics is all the rage, Maxwell electromagnetic theory seems to be able to explain the non mechanics phenomena .... little by little the structure of matter is revealed... tadaaaa ... it is atomic ... Scientists think they could explain atoms behavior by regarding them as point-objects under the spell of mechanical laws in the well behaved, ordered clock-work Universe.
In 1900 Lord Kelvin (yes that Kelvin, the unit we use to measure absolute temperature) in address to an assemblage of physicists at the British Association for the advancement of Science announces, "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement."
Little did he know of the explosion of new physics that was about to be unleashed, shattering the long held beliefs. Most of it happened trying to dispel several unexplained and fascinating phenomena namely black body radiation, photo electric effect, wavy behavior of light ..., but let us first start with something so ephemeral that has baffled us for so long...
Somebody long ago had said :) "Let there be light ..."
It may be surprising for you at first, but the history of light theories played major role for the advent of QM.

### What is light ?

In different ages there were different opinion what light is.
Plato thought there were three streams and we can see because of the interaction between them. Most of the Greeks thought light emanate from the eyes and that is why when we look in any direction we see the objects we are looking at, because we shine on them with our eyes like super-hero's from today's comics book . This didn't explain how we can see in the dark, but hey ... we can't have everything. Another school the Pythagorean's thought that every luminous object emanated light in all directions. As with everything with the ancient Greeks there were proliferation of views an ideas.
But once "the chicken was out of the egg" what do we do with it, the explanations raised more questions than they answered.
What is light made of ? Is it material or immaterial ? How fast it is moving ? How exactly does it move trough space ?
The rise of the 16-17 century science started to rely not only on logical reasoning as the Greeks did, but on experiment too, the ultimate arbiter of all physics. Over time two main candidate theories become popular which explained some of the behavior of light.
The first was that light is made of corpuscles with Isaac Newton as its chief champion. Imagine throwing a stone, the way it fly's and how it bounces, we can easily associate such picture with the real light the only thing we need is that those "stones" are very minuscule indeed. if we have a myriad of them we can easy visualize how light can go in straight lines, create edge shadows, the phenomena of spectra even more with the colossus of Newton and his mechanics behind it it was easy to explain this mechanically looking behavior.
The chief opponent of particle theory was Huygens supported wave theory, which become prevalent sometime after corpuscle theory could not explain many newly discovered phenomena. It explained how different beam of lights can go through each other, which particle theory could not explain, but the problem was if the light was made of waves how does it propagate trough space. From everyday experience we know that waves need medium though which they can propagate. So scientists had to fabricate a medium, namely the artifice of all penetrating all encompassing luminiferous ether. (the problem with the aether was that it must be extremely rigid because light travel so fast and at the same time very transparent because there is no drag i.e. strong yet soft). Now with the ether spread everywhere we can visualize light behaving as a wave in a pound of water. This view overtook the Newton view and was a predominant theory until the beginning of the 20th century before the advent of Quantum mechanics
Further proof was Maxwell electromagnetic theory which was wave theory and showed that light is electromagnetic in nature. At the beginning electro-magnetism needed yet another encompassing "tentacle"-like ether i.e. "electromagnetic ether". But it slowly gave up ground to "field"-like theories which did not needed ether any more, thus ever more conclusively proving that light is a wave.
The only thing which was missing was experimental proof of those electromagnetic waves, which came in 1887 with the famous experiment of Hertz, that undeniably proved the wave nature of Maxwell electromagnetic waves, hence light. Different colors of light were nothing more but electromagnetic wave which pulsate with different frequency.
So the wave theory overtook Newton pulsating corpuscles, but not for long .....

Black body is idealized object that absorb all radiation falling on it and then radiates it in a perfect manner. Every object in ideal world works as a black body : stars, metals, gases .... everything made of matter.
Our tale starts with principle postulated by Kirchhoff "that the intensity of radiation from a black body is dependent only upon the wavelength of the radiation and the temperature of the radiating body, a relationship worth while investigation".
Then in 19th century scientists calculated that a black body in thermal equilibrium should radiate energy in a way so that the higher the frequency the higher the energy radiated (because the shorter the wave length the number of the possible waves increases). The problem with that deduction was that it did not agree with experimental data. The formula they come up, based on classical theories, so called Rayleigh-Jeans law (black line in the diagram below) :
 B(T) = ((8*pi*f^2)/c^3)*k*T B - spectral radiation f - is the frequency of the emitted radiation c - is the speed of light k - Boltzman constant T - absolute temperature
It is referred in scientific history as the "ultra-violet catastrophe", if this proposition were true that meant left on its own devices any object will radiate its total energy in a burst of ultraviolet radiation and this as we know does not happen in reality.
You probably know this already, but anyway it is good place to remind you, that wavelength and frequency are reciprocal i.e. the higher the frequency the shorter the wave length and vs. versa. This is consequence of their definition i.e.: f = c/lambda or in other words lambda = c/f. Here c is the velocity of the wave, in our specific case the speed of light. But if you look at the experimental data in the diagram you will see it somewhat agrees with the Rayleigh law in longer wavelength (i.e. lower frequencies), but as we go towards shorter wavelength radiation grows exponentially to infinity very fast
What we see in reality is that radiation does not increase to infinity as the frequency goes up, but instead after a maximum at specific point it drops back to a lower level.
OK so far so good, what? then happens in the high frequency range, if Rayleght was not right. Austrian physicist Wien assumed that light was composed of particles for which the same laws held as for gas molecules, based on that assumption he derived similar law in 1896, which agreed well with experimental data but only in high frequencies. So there was a dilemma Rayleght law worked in one end of the spectrum, Wein law in the other end of the spectrum. What are we to do ?
Max Plank was first to solve the problem in a satisfactory manner, even that he did not liked his own explanation, for the result pointed out that energy is emitted in bundles, not like Maxwell theory said smoothly as waves.

Enough talk let see how the Plank equation looks like :
 Plank law : B(T) = ((2*h*f^3)/c^2) * (1/(e^((h*f)/(k*T)) - 1)) B - spectral radiation f - is the frequency of the emitted radiation c - is the speed of light e - is the Euler's number : constant 2.7182... h - is Plank constant : 6.626.. * 10^-34 Joules * sec k - Boltzman constant T - absolute temperature
The variable things as we said already are the frequency and the temperature, all the other parameters are constants.
If we simplify Plank's law to its bare bones i.e. strip the constants it would look something like this : B ~ f^3/e^(f/T) then we can see that the emitted energy is dependent mainly on two variables i.e. frequency and temperature in the non-linear fashion like shown in the image.
Just for the sake of it let's see the overall shape of bare-bone equation : wolframalpha.com : f^3 * 1/e^(f/1000), from 0 to 10000
You can personally attest for the behavior shown in the diagram ... if you heat a material it will first glow red, then as the temperature grows it will start glowing in white and finally at very high temperature in blue ... Look at the graph above, the 'bumps' (the peaks are where the most radiation is generated) it just happen to be in the visual spectrum (interesting that all flora and fauna evolved to take advantage of this phenomena), now follow the different temperatures and you will see why this happens :), the light where the most energy is radiated shifts to the left, higher frequencies. Like in the photoelectric effect as we shall see in a minute it looked like as if the energy is radiated piecemeal rather than in a waves like manner. Plank showed that lump of matter could be represented as a myriad of jiggling particles, the more the particles jiggle the warmer the object. To make his calculation work he had to assume that the energy is not absorbed in smooth way like a wave, but in bundles.
Even that Plank succumbed to the conclusion that the energy absorbed and radiated in bundles, he still imagined that it was transported as wave, Maxwell theory said that very clearly indeed. We would have to wait for Einstein in 1905 to forcefully prove that transportation is also in bundles.
*Stefan-Boltzman law : P = sigma * A * e * T^4 *Wien's displacement lambda_max = 2.898 / T Examples : Human body : T = 310K => 9.35 micro-meters Molten iron : T = 1810K => 1600 nm Sun : T = 5800 K => 500 nm

 Planks law derivation click to toggle> [+] To derive the law scientists normally used the so called equpatriation theorem i.e. that the energy states of atoms in thermal equilibrium have the same average energy associated with each degree of freedom of their motion namely e = 1/2k*T (electromagnetic waves have 2 degree of freedom per mode, so the formula becomes just k*T). That is why the kT shows up in all formulas. Back to the derivation, in essence they represented a black body as a cavity object, then when the body is in thermal equilibrium calculate the number of possible waves encapsulated in unit of volume and knowing that and the average energy per wave we just mentioned calculate the total energy with statistical methods. Plank on the other hand added one crucible deduction to the calculation. In the theory before him as you may expect the waves are bounded by the physical limits of the body, Plank added one more boundary condition namely the quantization of energy i.e. now the possible waves are not only constrained by the physical limits of the body but also energy is limited to assume only specific values. This "double whammy" constraints, made calculations agree with experiments. And that as we shall see later is one of the very ideas that will help us understand QM. Side note: For those familiar with Cantor infinities, the difference between Plank-energies and all-the-possible energies, seems very much as the difference between integer-infinity and real-number-infinity Once I figure a way to explain derivation of Plank law in more simple way, I will post it here. Until then use this link : !fm

### Photo electric effect

When a light is shone on a metal a small current flow through it, because light is giving energy to the atoms of the metal. The interesting thing is that not all colors of light affect the metal in the same way. Even very bright red light will not generate current, but a dim blue light will.
This comes to show that blue light should transport more energy than red light. But if we follow the logic from our normal day to day experience or Classical mechanics it would seem that the brighter the light, we should expect the bigger the wave (amplitude) i.e. more energy. Consider big world phenomena...f.e. waves hitting the coast cliffs.. the higher (amplitude) the wave have, the more splashing (releasing energy) we get. Just so you know, wave energy in classical physics is dependent on amplitude in the following way E ~ A^2 * v, where v is velocity of the wave.
But as we said in the example above that is not the case, it seems that the energy content of light is dependent on the frequency (color) rather than brightness (amplitude).
Another anomaly which later prove to be the same photo electric effect was what Herz noticed when he did his experiments. When ultraviolet light (higher frequency than blue visible light, look at the em-waves diagram) shone on his apparatus the sparks came out more easily. According to Maxwell theory if the intensity of light was increased then the speed of the electrons that was knocked out should have increased, but instead their speed was the same, what changed instead was that the number of the knocked out electrons increased. When the frequency was increased then the speed of the electrons increased. Wave theory again could not explain what was happening.
What Einstein though can explain this phenomena is if instead of thinking of light as wave we quantize the Energy and carry it by using particles (he called them photons). This quantization works in such a small sizes that in our everyday life for all intents and purposes the behavior of normal matter still shows as analog and uninterrupted. The cumulative averaging effect blurs the quantization in the big world of sand speckles and dust. Just for comparison sake in one qubic cm of air, there quintillions of atoms i.e ~ 3 * 10^19. To give you perspective this number is after billion and then trillion, and then, wait for it... sextillion..and then finally quintillion.
The formula that Einstein was able to come up (which followed on the insights of Max Plank) was :
E=h*f

h - Planck constant 6.6*10^-34 J*s
f - is the frequency of light that the photon is transferring


shows that energy quantization is dependent on the frequency not on amplitude as it is in the big world of specks and planets.
To get a grasp of the concept you should think of every elementary particle as a bullet shooting on a wall, the heavier(higher frequency) the bullet the bigger the jolt the faster the knocked out pieces will fly off the wall. But if we have a wave to wash on the wall, we have to wait a millennia for the waves to erode pieces of it. Even a single bullet will take chunks of the wall, exactly what happens when a single photon with the right frequency hit the object. But we need very intensive wave to make any damage.
Where Plank was just assuming that energy was absorbed and released in quanta's, Einstein insisted it was carried in bundles too, rather than waves.

 Photo electric effect click to toggle> [+] The Russian physicist Stoletov, discovered that the energy of the emitted electron was in a linear correlation with the applied radiation frequency. On the right is a sample diagram for sodium metal : What this meant was that according to the Plank hypotesis to disrupt the electron bond from the metal a minum energy is required : e = h*v_0 Generally this energy is represented by the letter W and is called work function. The resudual energy i.e. the energy of the incoming photon minus the work function h*v - h*v_0 then is transformed to kinetic energy E_k=1/2*m*v^2 of the departing electron. h*v - h*v_0 = 1/2*m*v^2 let's use W : h*v = W + 1/2*m*v^2 Here the left side represent the energy of the radiation the right side the energy of the released electron plus the work-function the energy absorbed by the material. In other words this relation is a formulation of the principle of conservation of energy and neatly explains the transfer of energy from the photon to the absorbing material and knocked out electron.

### Compton scattering

If a wave reaches obstacle we get the effect of diffraction like shown in the image on the right. Characteristic of diffraction as we know from ordinary experience is that the wavelength/frequency of the diffracted waves does not change. Look closely at the picture again.
So it was a surprise when Compton found there is case for which this does not hold true. What he did was to bombard paraffin wax with X-rays, he found out that the rays scattered sideways and backwards had wavelength slightly greater than the original rays.
But as we saw already wave theory says that the wavelength should stay the same. Compton recognized that, yes wave theory may not be able to explain the phenomena, but the results were intelligible if the photon hypothesis was applied instead.
In paraffine-wax electrons are pretty loosely bonded with their atoms, we can assume them to be almost free electrons floating around, when Compton bombarded them with high energy X-rays (~10 000 times stronger than visible light) the electrons are expelled from the paraffin-atoms.
Now according to the laws of conservation of energy the total energy have to stay the same, and the rebounding photon should have lost some energy to the scattered electron. But as we know the energy of the photon is E = h*f, less energy means lower frequency i.e. longer wavelength. Which perfectly matches the Compton observation.
One more thing if instead of paraffin-wax we use gas in a cloud chamber the recoiled electron can be observed and there is trace of the scattered photon. And after calculations were made they agreed with experiment, which proves that photons behave like billiard balls, rather than like waves. The theory shows that if the photon is deflected at right angle i.e. 90 degrees the change of the wavelength would be equal to h/(m*c) where m is the mass of the electron. The value acquired if we do the calculation is called Compton wavelength and is numerically equal to 2.42 * 10^-10 cm.

 Compton scattering derivation click to toggle> [+] In a nutshell we derive the equation using the principle of conservation of momentum to find equation representing the deflected electron-momentum (1). Then we use the principle of conservation of energy to find another representation of the same momentum (2). Then we "equate" them, substitute in de Brogile wave equation, so that we get in the final formula relation of wavelengths instead of momentum. We are onto getting some equation which involves wavelengths not momentum i.e. finding the difference of the photon wavelength before and after the collision i.e. the Compton equation. Lets do some math : p_1 - momentum of the photon before collision p_2 - momentum of the photon after collision p_e - momentum of electron after collision We know that momentum is p = m*v, also E = m*c^2, thus : E = m*c*c = p*c \ \ \ \ rArr \ p = E/c Now that we know this and the other Einstein eq for photon energy E=h*f, then photon momenta is : p_1 = E_1/c = (h*f_1)/c = h / lambda_1 also the momentum of the deflected photon will be : p_2 = E_2/c = h / lambda_2 if vec p_e is the momentum of the recoiled electron then according to the law of conservation of momentum : vec p_1 = vec p_2 + vec p_e \ \ \ rArr \ vec p_e = vec p_1 - vec p_2 We raise both sides to the power of 2 to get the scalar quantity instead of a vector i.e: (vec p_e)^2 = (vec p_1 - vec p_2)^2 p_e^2 = p_1^2 + p_2^2 - 2 * (vec p_1 @ vec p_2) dot product of two vectors is vec p_1 * vec p_2 = |p_1| * |p_2| * cos theta, so we get : (1) p_e = p_1^2 + p_2^2 - 2 * p_1 * p_2 * cos theta We got our momentum trough conservation of momentum, now lets find another one this time using the principle of conservation of energy. The energy of electron before collision is E_0 = m*c^2, after the collision E_f-final, E_e = p_e * c : E_f^2 = E_0^2 + E_e^2 = E_0^2 + p_e^2 * c^2 Then according to the law of conservation of energy : E photon before + E electron rest = E electron after + E photon after i.e. p_1*c + E_0 = p_2 * c + sqrt( E_0^2 + p_e^2 * c^2) moving p_2*c on the left side, we get : p_1*c + E_0 - p_2 * c = sqrt( E_0^2 + p_e^2 * c^2 ) c(p_1-p_2) + E_0 = sqrt( E_0^2 + p_e^2 * c^2 ) then squaring : c^2*(p_1-p_2)^2 + E_0^2 + 2*c*E_0(p_1-p_2) = E_0^2 + p_e^2 * c^2 We substract E_0^2 from both sides and then express p_e : p_e^2 = (c^2*(p_1-p_2)^2+2*c*E_0(p_1-p_2) ) / c^2 so finally : (2) p_e^2 = p_1^2 + p_2^2 - 2*p_1*p_2 + ( (2*E_0*(p_1-p_2))/c)  If we "equate" p_e using (1) and (2) : -2*p_1*p_2 + ( (2*E_0*(p_1-p_2))/c) = -2*p_1*p_2*cos theta get -2 out : -2 * (p_1*p_2 + ( (E_0*(p_1-p_2))/c)) = -2*p_1*p_2*cos theta divide by -2 : p_1*p_2 + ( (E_0*(p_1-p_2))/c) = p_1*p_2*cos theta substract p_1*p_2 from both sides, then we get : (E_0*(p_1-p_2))/c = p_1*p_2*(1-cos theta) Multiplying by (h*c)/(p_1*p_2*E_0), using lambda=h/p we get Compton equation : lambda_2 - lambda_1 = ((h*c)/E_0)*(1-cos theta) E=mc^2 , so : lambda_2 - lambda_1 = (h/(mc))*(1-cos theta) lambda_2 - lambda_1 = h/(mc)(1-cos theta) h/mc = lambda_compton = 2.4 * 10^-10 Large effect-deflection if lambda >> lambda_compton Small effect if lambda << lambda_compton High freq behave like particle, low freq like wave i.e. no scattering

### Wave like behavior

Then again there were the other type of experiments with light. They showed that light behaves like a wave ... Light (electromagnetic radiation in general and later the atom constituents) seem to exhibits wave like qualities, if we made our conclusions observing phenomenas like refraction, diffraction, Doppler effect and interference.
We could summarize this wavy behavior using the famous two slit experiment done by Thomas Young (left image).
If we shone light trough one slit, what we will see is a image with one maximum. This is what we would normally expect (right image top result).
Now if we shone a light trough two slits instead of getting an image with two maximums, which would be what we may expect if the light is corpuscular or if we made a deduction from our one-slit experiment, but instead we get a pattern resembling wave interference like the one shown on the image on right. And interference is something we are familiar in everyday life..think of a waves in a pond, if you drop two stones away from each other, the wave generated by them interfere with each other i.e. cancel in some places and double in other.
We know already from Einstein that light is made of photons, if we are to agree with him this means we will be able to slow down the radiation sending less and less photons ... until we can send just one at a time like the one the image right-down.
Ok, experimenters did that...and after gazillion of particles later what they got on their detector screen was again interference pattern. Crazy stuff !!!.. what is happening here !?
Light going out like particles and then absorbed like waves. Does our photons crossed via the two slits simultaneously and then interfered with themselves or did all the photons made a pact before they left the photon-gun and coordinate their behavior over time to make this interference happen ?.
Let's then be even "cutier" and put a detector on the slits so we can see where does the photon goes trough and catch him with the "pants down". To our surprise what happens in this case is that we get two light-maximums on the detector screen !!? Something we expected in our original double slit experiment. For some reason at the moment we decided to detect the photons they started behaving like photons instead of like waves, weird !! What we are to conclude from this, it is like the light is behaving like whatever we want it to be.
You wont probably have to make the mental jump now, but ponder the following statement for a while : "In general if the measuring device is of the smaller size than the wavelength we get wavy behavior, if we measure with device bigger that the wavelength we get particle behavior". And this is logical if our measuring device is bigger than what we measure, then it will look to us as a blob..

Let's summarize what we have learned so far.
Phenomenas and effects which support particle theory :
• light goes in straight lines
• photo electric effect
• Compton scattering
• cloud chamber tracking
• Scattering of light
• Reflection (better explained by particle theory)
Phenomenas and effects which support wave theory :
• Interference
• Diffraction -
• Refraction - change of direction of light beam passing trough different mediums.
• Polarization of light
• Light rays can cross each other
So after we discussed all those phenomena and went trough the different theories over time one thing continued top pop up. Once light is wave, once it is particle, once it is wave, once it is particle ...... what is it ? How can we explain this duality in sensible way.
It just happens that this whole mismatch could be straightened up if we grasp two very simple concepts. This whole preliminary talk I was preparing you for what is to come.
Ok, enough talking lets get with it.
Wavy-particle behavior can be explained by, tadaaa... :
• Standing waves
• Boundary conditions
What are those things ?

### Standing waves

There are two main types of waves, namely transverse and longitudinal. waves(todo)
Of those we are interested in the transverse waves which further subdivide to traveling and standing waves, we would discuss the latter one only.
Traveling wave is for example a wave in the sea, or a wave generated by a rock dropped in a pond.
A standing wave on the other hand is a wave on which both ends are pinned/anchored to stationary boundary, like a waves generated by the strings of a musical instruments. Check the example of standing wave on the right. It looks like the wave is frozen in time, oscillating in place.
A thing I probably don't have to remind you is that we measure the wavelength of a wave from crest to crest. Now lets take a more thoughtful approach and see what type of standing waves are possible in the case where both ends are anchored. Check the image below, does something strike you as odd !
Standing waves of this kind can have only specific wavelengths, and those wavelengths have to be a fraction of length of the boundary on which the string is pinned and even more importantly not whatever fractions, but whole-number fractions only.
 Let's try and present this in more mathematical way. The wavelength (lambda) is dependent on the total-length (L) in the following way where n is a whole-number : L = 1/2 n lambda One Length can contain n-half-waves. from this we can express the wavelength as : lambda = (2*L)/n and if we do quick calculation on several values of n : lambda_1 = 2L, lambda_2 = L, lambda_3 = (2L)/3, lambda_4 = L/2 ... etc As we can see we can have infinitely many of them, but only specific wavelengths are allowed.

Taking this into account and what we have been talking about all this time, does some nagging thought start tickling your mind ?
Let me remind you again : Waves, quantas, bundles ! Something ?
We know wavelength and frequency are reciprocal ? and Energy in particle world is very much connected to frequency ? are we getting there ... still nothing ?
Think.. think.. standing waves ... only specific wavelengths/frequencies possible....

#### !! EUREKA !!

Standing waves are WAVES and also can wiggle only at specific frequencies i.e. the ENERGY can be only specific values i.e. the energy of such waves will be QUANTIZED.

We got it !!! We found our goldilock : BOUNDARY RESTRICTED STANDING WAVE

Now if we could just use this potent explanatory mechanism to build a theory around it to explain the phenomenas and allow us to calculate scenarios which match with experiment.

### Final remarks

If there is one thing or should I see two that I want you to remember from this whole exercise, that will be the concepts of standing waves and boundary condition. In my opinion those are the two intuitive ideas on which you can base your understanding of wave-particle duality of matter. When those two work hand in hand you get both quantization and waviness in very natural manner.
The other moral from this story is that we should look at wave-particle duality of light not as opposite sides of a coin, but as a two complementary properties of light. Once you bend your thinking to think of them as complementary rather than opposite everything goes into place.

### What's next ?

In the next part we will concentrate more on the matter side of things i.e. structure of the Atom, how mathematical abstractions helped both clarify and mystify the explanations and also some historical perspective on how the theories and ideas evolved and intersected each other. More importantly :
• The early Bohr atom theory
• De Brogile insight on the waviness of matter
• Heisenberg breakthrough with his matrix mechanics and then his uncertainty principle
• Shredinger wave equation and later refinements
• Pauli exclusion principle
... and much more..

### Quantum Mechanics references

Here are Interenet and book references, from which I shamlessly "borrowed" ideas to build my little article ;) I don't know about you, but normally when I can't grasp something from a book or article, what most of the times I will do is to find another one and read it, then probably another one... and generally at some point my brain plays a magic trick on me and everything clicks.
It happened so numerous times that not only I'm not surprised anymore but I always employ this technique on complex topics I want to learn and can't grasp immideately.
So here they are :