Quantum Theory Made Easy [2]

the year was 1924 for over ten years physicists had been struggling with the emerging picture of the world of atoms contradictions between theory and experiment made the previous decade or so a dramatic one for physics as if to deliberately fan the flames the French nobleman Louie de Blois published a doctoral dissertation that only made matters worse his ideas challenged not only conventional established physics but common sense itself dubrow believed that nature contained symmetries inherent to the laws of physics so if light could behave like a wave or a particle then maybe matter could - this idea would set the stage for the insanity of quantum mechanics in the early 19th century physicists discovered a way to identify different elements simply by examining patterns in light when a gas is heated up it emits light if that light then gets diffracted it produces aspects from containing all of its constituent wavelengths instead of producing the usual spectrum that we're all familiar with however the outgoing light produces a series of sharp isolated parallel lines that represent only a few discrete pieces of the electromagnetic spectrum this is called an emission line spectrum the bars of light are called spectral lines and each one corresponds to a definite wavelength and frequency curiously every element has a unique emission spectrum the same isotope of hydrogen always yields this particular pattern of spectral lines and no other element in the universe ever includes this pattern in its spectrum while heated gases will emit very specific wavelengths of light cool gases will absorb very specific wavelengths of light it's the exact opposite of what happens with emission spectra when light is passed through a cold gas and is subsequently diffracted the spectrum of light that emerges will contain sharp isolated parallel dark spots that represent only a few discrete pieces of the spectrum once again the locations of these dark spots which are called absorption lines are unique to each individual element additionally when the emission and absorption spectra of the same element are superimposed the entire electromagnetic spectrum is reproduced perfectly an elements emission lines are located in the exact same places as it's absorption lines because all elements have unique spectra emission and absorption lines can be used to identify any gas in the universe they can be thought of as fingerprints of elements and they can be used to determine among other things an astronomical objects chemical composition for example this is the absorption spectrum of the Sun by comparing it with the absorption spectra of other elements one can easily determine that this is composed principally of hydrogen and helium in the early 20th century it was still unclear as to what makes an element exhibit a certain spectrum what did these lines represent and why did they appear in these locations by 1910 the existence of atoms had finally been established and the best model describing them had been developed by the English physicist JJ Thompson who was already famous for having discovered electrons and measuring their charge to mass ratio the previous year the american physicist robert millikan had performed the first measurements of the electrons charge which combined with Thomson's charge to mass ratio revealed that the electron was an incredibly light particle far lighter than the lightest atoms this and other discoveries led physicists to conclude that almost all of the mass of an electron comes from its positively charged constituents so Thompson hypothesized that atoms resembled Christmas pudding a traditional British dessert the raisins in the pudding represented the negatively charged electrons while the pudding surrounding the raisins represented the atoms positive charge the exact nature of the pudding itself was a big mystery all that the model said was that the pudding was saturated with positive charge and that the electrons were somehow embedded inside this was known as the plum pudding model this model had tremendous appeal because it was able to account for atomic spectra if the atom collided with another atom as in a heated gas it would be like plucking the strings of a guitar the strings start vibrating and in doing so transfer energy to the air in the form of sound waves each string vibrates with a different frequency and therefore releases unique frequencies of sound when an electron starts shaking because it has charged it releases light instead of sound waves and that light has the frequency that the electron is shaking with Thompson believed that each electron in an element could shake with only specific unique frequencies which he called natural oscillations and so could release only specific frequencies of light this would explain emission spectra if an atom were illuminated with light of many frequencies like when white light is shined onto a cold gas each electron would selectively absorb only light whose frequency matched that of the electrons natural oscillation this would explain absorption spectra and if as Thompson believed each element had electrons with unique natural oscillations then the consistency and uniqueness of emission and absorption spectra were easily explained along with the fact that they complement each other what determined an atoms fingerprint he reasoned was the natural oscillations of the electrons though the details of the plum-pudding models predictions were not particularly successful it at least provided a starting point for our understanding of atomic structure however in 1911 ernest rutherford one of the greatest experimentalists in history published a paper that utterly demolished the plum pudding model he had overseen an experiment carried out by Geiger and Marsden in which alpha particles which are positively charged particles emitted from radioactive materials were fired at gold foil in an effort to test Thomson's model according to the plum pudding hypothesis once the positively charged alpha particles entered the gold foil and interacted with the atoms that comprise it the particles would pass through the pudding and experience minor deflections due to the interaction between the positive charges the electrons would not play any role because of how little mass they'd had compared to the tremendous momentum of the incoming alpha particles the negatively charged electrons would not alter the course of the positively charged alpha particles any more than a swarm of gnats would alter the course of a speeding bullet any deflections in the experiment would be due to the positive charge within the atoms and these deflections were expected to be minor because of the wide distribution of positive charge Rutherford and his colleagues were astounded at the results of the experiment when alpha particles were fired through the foil the detector registered that most of them experienced no deflection whatsoever a few of the particles did as predicted end up slightly deflected but the most shocking result was that several of the alpha particles were deflected quite severely in fact some of them came right back having been deflected by a full 180 degrees Rutherford later famously said that this was the most incredible thing that had ever happened to him in his life akin to firing a cannon ball at a piece of tissue paper only to watch it come back and hit him the results of the gold foil experiment yielded the following insights first since most of the alpha particles passed through the atoms without experiencing any deflection atoms are comprised primarily of empty space rather than of a positively charged pudding secondly since some of the particles ended up rebounding they must have encountered resistance from a highly concentrated positive charge contrary to the wide distribution that Thompson had envisioned taken together this suggested that the atom was comprised primarily of empty space which contains negatively charged electrons that surrounded a tightly packed nucleus of positive charge based on the distribution of angles at which the alpha particle scattered Rutherford was able to calculate the diameter of the nucleus to be about a hundred trillionth of a meter by the end of the decade he discovered the positively charged constituents of atomic nuclei which he named protons once the results were published the Rutherford model replaced the Thompson model as the most dominant framework for atomic structure but there were some serious problems an electron is negatively charged and a proton is positively charged so being within such close proximity should force the far less mass of electron to crash against the proton upon realizing this Rutherford made the proposition that the electron orbited about the nucleus the same way that planets orbit around the Sun but this came with catastrophic consequences recall that when an electric charge is accelerated it will produce electromagnetic waves well circular motion is a type of acceleration even if that motion has constant speed therefore an electron orbiting about a nucleus must be releasing light at all times this does not happen even if it did the fact that orbiting electrons must continuously emit light would still make the entire configuration unstable since the corresponding loss in energy would force the electron to jump to lower and lower orbits the end result would be an electron that spirals and crashes into the nucleus and this would take a fraction of a second worst of all as electrons spiraled into the nucleus the continuous change in their angular frequencies would cause the frequencies of the corresponding electromagnetic waves to continuously change which would result in the emission of the entire electromagnetic spectrum rather than the discrete spectral lines that the plum pudding model had so neatly explained in short the Rutherford model was a disaster it got good results for the distribution of charges but it produced an unstable atom that also failed to reproduce atomic spectra in 1913 the Danish physicist Niels Bohr proposed a major revision of Rutherford's atom that would resolve these problems Bohr's model as it would come to be called represented the first radical departure from classical ideas about matter he proposed that atomic spectra can arise only if electrons were allowed to occupy very specific orbits or energy levels within an atom in other words the distance from the nucleus at which an electron can orbit and by extension the amount of energy in the atom is quantized an electron can occupy either of these energy levels but it cannot occupy any point in between the strangest part of this model is that when an electron moves from one orbit to another it does so instantaneously and without crossing the gap in between the electron simply disappears and reappears somewhere else Bohr's reasoning went like this since the spectra of elements can take on certain specific frequencies than by the planck einstein relation they must also take on certain specific energies and because these energies depend on the distance of the electron from the nucleus electrons can only occupy very specific energy levels with no electrons between the allowed ones since different elements have different spectra they likewise must have different energy levels so in summary the reason that all of the atoms of a given element will have the same spectra is because they all have the same set of energy levels but different elements have different sets of energy levels and so their spectra will likewise be different the actual mechanism behind emission and absorption involves the transition of an electron between one energy level and another when an atom gets excited that is when it absorbs energies via collisions or via the absorption of photons elec Jones will jump to higher energy levels by an amount that corresponds to how much energy was transferred when a photon strikes an atom it needs to have just the right amount of energy in order to make the electron jump to a higher orbit but what determines this right amount of energy the answer is actually quite simple it's the difference in energy between the levels the energy of the absorbed photon which has plunks constant times the photons frequency is equal to the difference in energy between the initial and final orbits the units of energy dealt with at these scales are electron volts let's refer to them as Evy suppose there's an atom with energy levels equal to 7 Evie and 4 Evie and that the electron is occupying the lower orbit the difference in energy between these levels is 3 Evie if a yellow photon which has an energy of about 2 Evie strikes the atom it will not get absorbed because it doesn't have enough energy to push the electron into the higher orbit instead it'll just scatter off of the atom but when white light which contains all of the frequencies of light is shined on to the atom it will absorb the piece of light that is needed to force the jump to a higher energy level when that light is subsequently diffracted which again allows all of its constituent wavelengths to be displayed the entire spectrum is reproduced except for a tiny narrow band in the bluish violet region this is where the photon with 3 V's the amount of energy needed for the atom to transition into a higher state was located it has been absorbed by the atom emission is exactly the opposite and equivalent process an excited atom that is one that is absorbed a photon or perhaps has undergone many collisions within a hot gas will release a photon and jump to a lower energy level this generally occurs within a fraction of the second of the atom being excited the equation describing the properties of the outgoing photon is the same as before the photons energy equals the difference in energy between the initial orbit and the final orbit since atoms have a finite amount of energy there is a minimum orbit that can be occupied electrons cannot keep jumping down forever or else they wouldn't definitely be releasing energy in the form of photons the lowest possible energy denoted n equals one is called the ground state and equals to represents the first excited state N equals three represents the second excited state and on and on it goes because the atoms and a hot gas are excited they tend to jump to lower energy levels and in doing so they release photons whose energies correspond to the difference in energy of the orbits these orbits have energy is depending on their distance from the nucleus so there are specific differences in energy between the orbits themselves because of this the outgoing photons likewise have specific characteristic energies and so by the Planck Einstein relation they also have specific characteristic frequencies those frequencies of light are the only ones that get released and they constitute emission spectra and because emission and absorption are opposite and equal processes they yield opposite and equal spectra the Bohr model for the most part is intuitively easy to grasp if you're full of energy you'll go out and run a marathon after the marathon when your energy has been expended you'll lie down and relax when an electron is at a higher orbit it's running a marathon when it occupies the ground state it's relaxing if you give the electron some energy it'll get back up and running again take that energy away and the electron goes back to sleep once this idea was developed it became natural to ask where energy levels actually real was Borel Eon to something when he postulated that quantization applied to matter or was he merely crafting a convenient fiction to explain atomic spectra in an ad hoc manner as it turned out he was completely correct the year after Bohr created his model the physicists James Franck and Gustav Hertz put his energy level hypothesis to the test here's a simplified version of their experimental setup electrons were fired with an adjustable amount of energy into a chamber full of mercury gas at the opposite end of the chamber was a means to detect the final energies of the electrons in a controlled setup the electrons would be fired straight at the detector without passing through anything as a result their initial energies would be equal to their final energies let's say the initial energy of each electron was 10 evey since they encounter no resistance their final energies are also 10 evey now let's place a barrier in the middle which requires a minimum of 80 V in order to pass this is analogous to a tollbooth on a highway where you have to pay in order to proceed an electron that initially has 10 evey of energy would ultimately have to evey since it transferred a TV to the barrier if an electron of 7 TeV is fired into the chamber the detector will not register it since it didn't pass through the barrier with all of this in mind let's look at the results of the Franck Hertz experiment were instead of a barrier the chamber contained mercury gas the initial energies of these electrons being adjustable were the independent factor the final energies would depend on how the electrons interacted with the mercury gas here's the graph showing what happened even though the axes are in volts and currents the principle is still the same as with our simplified version of the experiment at first electrons have very low initial energies which are slowly being cranked up to 5 V's though it can't be directly seen from this graph the initial energies of the electrons are at first roughly equal to their final energies this indicates that when the electrons pass through the mercury gas they're losing almost no energy when they scatter against the mercury atoms they just bounce around with their initial energies until they hit the detector which registers an almost identical amount of energy to what they'd started with the result is almost the same as what one would get if the mercury weren't there at all but then when the electrons get to 4.9 evey there is a sudden drop in the amount of energy registered by the detector as the initial energy slowly increase so to the final energies until once again they peak and sharply drop to another Valley this one's slightly higher than the previous one the process then repeats itself what's going on here why are these sudden drops happening when the electrons have energy is lower than 4.9 evey they're barely interacting with the atoms and so barely lose any energy but once they reach 4.9 TV they transfer their energy to the mercury atoms which then get kicked into their first excited States the incoming electrons now having completely lost their energies to the mercury atoms are not being detected the only electrons being detected at this point are the ones that completely missed the mercury atoms as their energies continue to be cranked up the electrons are capable of both exciting the mercury atoms and registering with a detector the energies of these detected electrons are their initial energies minus the four point nine EVs that they gave up to the mercury atoms then at the second peak the electrons detected energies drop again which of course is due to them kicking the mercury atoms into their second excited States the stripping the electrons of all their energy again the only electrons being registered here are the ones that missed the mercury entirely and since their initial energies are higher than those of the previous Valley their final energies are likewise higher this is why this valley is slightly elevated above this one and then the process repeats itself shortly after the drops in energy occurred the mercury emitted electromagnetic radiation whose frequencies Franck and Hertz measured it turned out that the frequencies of these photons turned out to be exactly what Bohr's model said they should be the first peak took place at 4.9 evey which by the plunk Einstein relation corresponds to a frequency of about one point one eight five times ten to the fifteen Hertz this was the same frequency as those of the emitted photons after the first energy drop which confirmed that the mercury atoms did indeed release photons as they returned to the ground state the frequencies of all other emitted photons corresponded exactly to what Bohr's model predicted moreover these frequencies matched those found on Mercury's emission spectrum and when the experiment was repeated with neon as well as other gases the Bohr model only continued to succeed when these results were presented to Einstein some time later he is said to over marked it's so lovely that it makes you cry Bohr had emerged triumphant the quantization of energy levels was confirmed by the Franck Hertz experiment and so the Bohr atom replaced the Rutherford everyone involved got Nobel prizes for this now that Bohr had determined the relationship between atomic spectra and energy levels he turned his attention to making the atom stable and mathematically predicting its energy levels he began with the simplest case the hydrogen atom Bohr's research in 1913 was very preliminary so he made some crude derivations and assumptions when developing his model as a result Bohr was only partially successful he was missing an important piece of the puzzle he was missing the duality of matter though Bohr had utilized the particle nature of electromagnetic waves in his model it would not be until 1922 that Arthur Compton would definitively confirm the duality of light once he did physicists accepted duality into the mainstream and two years later Louie de Blois published his revolutionary observations matter was quantized and light was quantized light was dualistic and in every instance where the particle nature of light emerged quantization in the form of Planck's constant made an appearance so what if the same thing was happening with matter what if the presence of quantization scaled by Planck's constant was likewise accompanied by dualistic behavior and matter what if particles could act like waves as it turns out they can a few years after de broad public his hypothesis diffraction which is explicitly a wave behavior was accidentally observed in electrons many years later the famous double slit experiment was performed the setup involved a beam of electrons being fired at a barrier with two narrow slits particles would pass through these slits and impose their shapes onto the detector behind the barrier waves on the other hand would interfere some parts of the waves would combine to form larger peaks and troughs while other parts of the waves would cancel each other out in doing so they would produce a distinct pattern on the detector called an interference pattern the areas with the most hits indicate regions where the waves combine empty areas represent the regions where the waves destroy each other when electrons were fired through the double slit an interference pattern emerged indicating that they behave like waves even when one electron was fired at a time the interference pattern still emerged thus making the duality of matter indisputable a very curious thing happens when the experiment is modified with detectors which measured the trajectories of the electrons in order to determine which slit they might have gone through once the detectors are placed the electrons no longer produce an interference pattern they go back to behaving like particles and when the detectors are modified in such a manner that makes their measurements less reliable analogously to modifying a firearm such that its sites are slightly off-center some of the electrons behave like particles while others behave like waves the results of these modified double slit experiments suggested a number of possible interpretations with far-reaching consequences none of which will be considered until the final video of the series as was stated in the previous video before one can ask what quantum mechanics means one must first understand what quantum theory says for the present purposes it is sufficient to note that the double slit experiment demonstrates that electrons can have either wave properties or particle properties all matter has wavelengths you have a wavelength your car has a wavelength the earth has a wavelength everything exhibits wave behavior matter waves or debris waves have wavelengths that conform to this equation lambda equals H over P the dubrow wavelength equals Planck's constant divided by momentum once again the quantum of action makes an appearance and once again it is the reason we don't witness the strangeness of quantum mechanics in our everyday lives after all when you walk through a doorway you don't diffract but why not if you're a wave then shaking hands with someone should cause the both of you to combine some of your body parts and erase others why doesn't this happen the reason is that in order for the wave behavior of matter to be noticeable the debris wavelength needs to be comparable to the size of the system being considered to illustrate this point consider a grain of sand that weighs one-half of one billionth of a kilogram suppose that it falls with a terminal speed of about half a meter per second the debris wavelength for this grain of sand is Planck's constant divided by momentum momentum is mass times velocity so plugging everything in we find that the sands dubrow wavelength is two point six five times ten to the negative 24 meters this is a trillionth of the size of an atom's nucleus and is therefore absolutely negligible even at subatomic scales since the mass of an object is inversely related to its dubrow wavelength more massive objects like human beings will have even smaller wavelengths and thus even more negligible wave properties this is why you don't diffract when you walk through a doorway and thank goodness that you don't an electron on the other hand has very noticeable wave effects and this is because it's debry wavelength is actually quite large plugging in the mass of an electron moving at 1500 kilometers per second the resulting debris wavelength is about the size of a typical atom as a result the wave properties of electrons are extremely important at atomic scales armed with the duality of matter it became possible to reformulate Bohr's model in terms of debris waves when working on his model board had discovered that angular momentum was quantized he came to this conclusion via an extremely convoluted derivation but when equipped with the duality of matter the process can be vastly simplified if the electron occupies a certain specific distance from the nucleus that corresponds to the energy level and if the electron can be thought of as a wave rather than as a particle then the electron becomes a closed standing wave that surrounds the nucleus at the center in order for this wave to neatly close in on itself the circumference of the orbit that it occupies must be some integer multiple of the electrons wavelength and is called the principal quantum number it and other quantum numbers will be discussed in greater detail in a later video for now however suffice it to say that the principal quantum number determines the energy levels of the atom and that it can only take on integer values it can never be an irrational number replacing lambda with the equation for the debris wavelength and then rearranging this expression emerges recall that H divided by 2 pi is such a common term in quantum physics that the letter H bar was invented to express it the end result is this expression being equal to some integer times H bar in classical physics this expression represents the angular momentum of a circle our orbit so in the end the angular momentum of the electron is some integer times H bar in other words angular momentum is quantized which is what Bohr had discovered the previous decade the importance of the quantization of angular momentum is that it confirmed Bohr suspicions that electrons simply disappear from one orbit and reappear in another without crossing the gap in between this is because of the fact that only specific distances from the nucleus can be occupied by electrons which follows naturally and inevitably from angular Momentum's quantization this makes it necessary for electrons to undergo quantum leaps in order to change energy levels in fact the quantization of angular momentum makes it possible within the Bohr model to determine the orbital radii occupied by the electrons and the velocities with which they travel with regard to the orbital radius all of these terms are either fundamental or mathematical constants which can be combined into a single constant term called the Bohr radius denoted by a knot it is the radius of the ground-state orbit within a hydrogen atom since the radius of higher energy levels is the square of the principal quantum number times the Bohr radius the permitted energy levels have these values the ground state is a knot the first excited state is for a knot the second excited state is 9 a knot the third excited state is 16 a knot and so on and so forth finally with the orbital radii and velocities available to physicists it became possible to derive a general equation that predicted the energies of the hydrogen atoms orbits and by extension the wavelengths of its emission and absorption lines the result is this the Rydberg equation which was named after the Swedish physicist who did pioneering work on hydrogen spectrum the inverse of the spectral photons wavelength is equal to this constant which is about 11 million inverse meters times this expression where NL is the principal quantum number of the lower orbit and and U is the principal quantum number of the upper orbit remember the principal quantum numbers are just sequential integers the ground state is 1 the first excited state is 2 and so on let's put this equation to the test by determining the wavelengths of the photons that the atom spits out when electrons jump down to the second energy level the jump from 3 to 2 yields a photon whose wavelength is 656 nanometers the jump from 4 to 2 yields of photon whose wavelength is 486 nanometers the jump from 5 to 2 yields a photon whose wavelength is 434 nanometers and the jump from 6 to 2 yields a photon whose wavelength is 410 nanometers notice that the larger the jump between orbits the smaller the photons wavelength and thus the greater the photons frequency and by extension energy this is exactly what one would expect amazingly the emission spectrum of hydrogen yields exactly these wavelengths which the Rydberg equation was just used to reproduce these particular spectral lines which constitute the visible portion of hydrogen spectrum are called the Balmer series and all of the other spectral lines are reproduced to just as perfectly by the Rydberg equation the Bohr model disappeared to be wildly successful and combined with dubrow's insights the atom now became a positively charged nucleus surrounded by negatively charged waving electrons but as was often the case there were a few problems first of all the model utterly failed when it attempted to reproduce the spectra of atoms with more than one electron to search for a general principle that would allow physicists to predict and explain the motions of electrons within atoms would have to continue additionally it combined classical and quantum ideas in a somewhat disorganized mesh with dubrow's hypothesis further steps could be taken away from the classical paradigm but it still wasn't enough it was still a classically orbiting electron that just happened to have a wavelength further the model provided no insight as to what happens during a transition between energy levels and it predicted that the ground state of hydrogen would contain a magnetic dipole moment which doesn't happen meanwhile some very important questions were being raised about dubrow's wave chiefly among them what exactly is it that's waving this was a deeply perplexing issue because an electron is essentially a point how does a point wave clearly matter did have the property of waves electrons interfere and diffract so they must be waves and if they weren't we wouldn't have electron microscopes electron microscopes have far more resolution that normal microscopes do because electrons can be made to have wavelengths that are hundreds of thousands of times shorter than those of visible light which allows for far greater resolution and obviously if electrons have wavelengths than they must be waves yet at the same time electrons were particles since they could scatter waves cannot scatter particles cannot diffract and yet electrons can do both what was going on so by the mid-1920s there were a number of serious questions being thrown around and quantum mechanics appeared to be a disorganized mess of problematic models troubling challenges to conventional physics paradigms and no general mathematical framework to unite everything it was however starting to become clear that in order to get an accurate picture of how subatomic particles like electrons behave one first needs an equation that describes them entirely in terms of their wave properties atomic theory would have to be revised yet again in order to accommodate the behaviors of objects with large debris wavelengths but in the meantime the questions persisted what did the debris wave actually represent what was it describing what about the duality of light and matter are they part of a more general underlying principle of nature what about the maths how does one make predictions about the evolution of a quantum system over time physicists believed that once these questions were answered there would be no more mysteries they thought that as soon as the disparate physical frameworks and contradictory experiments would be resolved that once a general theory of quantum mechanics would develop under some unified physical principles that quantum mechanics surely would no longer be strange they were half correct when these questions finally did get answered and when the foundations of quantum mechanics were at long last developed quantum theory stopped being strange and started to become completely insane [Music] you

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Keywords: Quantum mechanics, quantum, quantum theory, quantum theory made easy, king crocoduck, science, duality, double slit experiment, de Broglie, matter waves, animation, quantum physics, skepticism, bohr, physics

Id: FlIrgE5T_g0

Length: 35min 25sec (2125 seconds)

Published: Tue Dec 01 2015