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            What is Quantum Physics?


      1. The word "quantum" comes from the Latin word for "how much" (as does "quantity"). Something that is "quantized", like the energy of Planck's harmonic oscillators, can only take specific values. For example, in most countries money is effectively quantized, with the "quantum of money" being the lowest-value coin in circulation. "Mechanics" is the branch of science that deals with the action of forces on objects, and so "quantum mechanics" is the form of mechanics that deals with objects for which particular properties are quantized.
      2. Actually there can be intensity-dependent effects, but at light intensities achievable with non-laser sources these effects are unobservable.
      3. Einstein's photoelectric effect equation can be derived and explained without requiring the concept of "photons". That is, the electromagnetic radiation can be treated as a classical electromagnetic wave, as long as the electrons in the material are treated by the laws of quantum mechanics. The results are quantitatively correct for thermal light sources (the sun, incandescent lamps, etc.) both for the rate of electron emission as well as their angular distribution. For more on this point, see.
      4. The classical model of the atom is called the planetary model or the Rutherford model, after Ernest Rutherford who proposed it in 1911, based on the Geiger-Marsden gold foil experiment which first demonstrated the existence of the nucleus.



      Quantum physics is a branch of science that deals with discrete, indivisible units of energy called quanta as described by the Quantum Theory. There are five main ideas represented in Quantum Theory:

      1. Energy is not continuous, but comes in small but discrete units. 
      2. The elementary particles behave both like particles and like waves. 
      3. The movement of these particles is inherently random. 
      4. It is physically impossible to know both the position and the momentum of a particle at the same time. The more precisely one is known, the less precise the measurement of the other is.
      5. The atomic world is nothing like the world we live in. 


      While at a glance this may seem like just another strange theory, it contains many clues as to the fundamental nature of the universe and is more important then even relativity in the grand scheme of things (if any one thing at that level could be said to be more important then anything else). Furthermore, it describes the nature of the universe as being much different then the world we see. As Niels Bohr said, "Anyone who is not shocked by quantum theory has not understood it." 


      Particle/Wave Duality

      Particle/wave duality is perhaps the easiest way to get aquatinted with quantum theory because it shows, in a few simple experiments, how different the atomic world is from our world.

      First let's set up a generic situation to avoid repetition. In the center of the experiment is a wall with two slits in it. To the right we have a detector. What exactly the detector is varies from experiment to experiment, but it's purpose stays the same: detect how many of whatever we are sending through the experiment reaches each point. To the left of the wall we have the originating point of whatever it is we are going to send through the experiment. That's the experiment: send something through two slits and see what happens. For simplicity, assume that nothing bounces off of the walls in funny patterns to mess up the experiment.


      Generic Experiment Setup

      First try the experiment with bullets. Place a gun at the originating point and use a sandbar as the detector. First try covering one slit and see what happens. You get more bullets near the center of the slit and less as you get further away. When you cover the other slit, you see the same thing with respect to the other slit. Now open both slits. You get the sum of the result of opening each slit.  The most bullets are found in the middle of the two slits with less being found the further you get from the center.

      Bullets Experiment

      Well, that was fun. Let's try it on something more interesting: water waves. Place a wave generator at the originating point and detect using a wave detector that measures the height of the waves that pass. Try it with one slit closed. You see a result just like that of the bullets. With the other slit closed the result is the same. Now try it with both slits open. Instead of getting the sum of the results of each slit being open, you see a wavy pattern ; in the center there is a wave greater then the sum of what appeared there each time only one slit was open. Next to that large wave was a wave much smaller then what appeared there during either of the two single slit runs. Then the pattern repeats; large wave, though not nearly as large as the center one, then small wave. This makes sense; in some places the waves reinforced each other creating a larger wave, in other places they canceled out. In the center there was the most overlap, and therefore the largest wave. In mathematical terms, instead of the resulting intensity being the sum of the squares of the heights of the waves, it is the square of the sum.

      While the result was different from the bullets, there is still nothing unusual about it; everyone has seen this effect when the waves from two stones that are dropped into a lake in different places overlap. The difference between this experiment and the previous one is easily explained by saying that while the bullets each went through only one slit, the waves each went through both slits and were thus able to interfere with themselves.

      Now try the experiment with electrons. Recall that electrons are negatively charged particles that make up the outer layers of the atom. Certainly they could only go through one slit at a time, so their pattern should look like that of the bullets, right? Let's find out. (NOTE: to actually perform this exact experiment would take detectors more advanced then any on earth at this time. However, the experiments have been done with neutron beams and the results were the same as those presented here. A slightly different experiment was done to show that electrons would behave the same way . For reasons of familiarity, we speak of electrons here instead of neutrons.) Place an electron gun at the originating point and an electron detector in the detector place. First try opening only one slit, then just the other. The results are just like those of the bullets and the waves. Now open both slits. The result is just like the waves!

      There must be some explanation. After all, an electron couldn't go through both slits. Instead of a continuous stream of electrons, let's turn the electron gun down so that at any one time only one electron is in the experiment. Now the electrons won't be able to cause trouble since there is no one else to interfere with. The result should now look like the bullets. But it doesn't!  It would seem that the electrons do go through both slits.

      This is indeed a strange occurrence; we should watch them ourselves to make sure that this is indeed what is happening. So, we put a light behind the wall so that we can see a flash from the slit that the electron went through, or a flash from both slits if it went through both. Try the experiment again. As each electron passes through, there is a flash in only one of the two slits. So they do only go through one slit! But something else has happened too: the result now looks like the result of the bullets experiment!! 

      Electron Experiment with Light

      Obviously the light is causing problems. Perhaps if we turned down the intensity of the light, we would be able to see them without disturbing them. When we try this, we notice first that the flashes we see are the same size. Also, some electrons now get by without being detected.  This is because light is not continuous but made up of particles called photons. Turning down the intensity only lowers the number of photons given out by the light source. The particles that flash in one slit or the other behave like the bullets, while those that go undetected behave like waves.

      Well, we are not about to be outsmarted by an electron, so instead of lowering the intensity of the light, why don't we lower the frequency. The lower the frequency the less the electron will be disturbed, so we can finally see what is actually going on. Lower the frequency slightly and try the experiment again. We see the bullet curve . After lowering it for a while, we finally see a curve that looks somewhat like that of the waves! There is one problem, though. Lowering the frequency of light is the same as increasing it's wavelength , and by the time the frequency of the light is low enough to detect the wave pattern the wavelength is longer then the distance between the slits so we can no longer see which slit the electron went through .

      So have the electrons outsmarted us? Perhaps, but they have also taught us one of the most fundamental lessons in quantum physics - an observation is only valid in the context of the experiment in which it was performed . If you want to say that something behaves a certain way or even exists, you must give the context of this behavior or existence since in another context it may behave differently or not exist at all. We can't just say that an electron is a particle, since we have already seen proof that this is not always the case. We can only say that when we observe the electron in the two slit experiment it behaves like a particle. To see how it would behave under different conditions, we must perform a different experiment.

      So sometimes a particle acts like a particle and other times it acts like a wave. So which is it? According to Niels Bohr, who worked in Copenhagen when he presented what is now known as the Copenhagen interpretation of quantum theory, the particle is what you measure it to be. When it looks like a particle, it is a particle. When it looks like a wave, it is a wave. Furthermore, it is meaningless to ascribe any properties or even existence to anything that has not been measured. Bohr is basically saying that nothing is real unless it is observed.

      While there are many other interpretations of quantum physics, all based on the Copenhagen interpretation, the Copenhagen interpretation is by far the most widely used because it provides a "generic" interpretation that does not try to say any more then can be proven. Even so, the Copenhagen interpretation does have a flaw that we will discuss later. Still, since after 70 years no one has been able to come up with an interpretation that works better then the Copenhagen interpretation, that is the one we will use. We will discuss one of the alternatives later.

      In 1926, just weeks after several other physicists had published equations describing quantum physics in terms of matrices, Erwin Schrödinger created quantum equations based on wave mathematics , a mathematical system that corresponds to the world we know much more then the matrices. After the initial shock, first Schrödinger himself then others proved that the equations were mathematically equivalent . Bohr then invited Schrödinger to Copenhagen where they found that Schrödinger's waves were in fact nothing like real waves. For one thing, each particle that was being described as a wave required three dimensions . Even worse, from Schrödinger's point of view, particles still jumped from one quantum state to another; even expressed in terms of waves space was still not continuous. Upon discovering this, Schrödinger remarked to Bohr that "Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business." 

      Unfortunately, even today people try to imagine the atomic world as being a bunch of classical waves. As Schrödinger found out, this could not be further from the truth. The atomic world is nothing like our world, no matter how much we try to pretend it is. In many ways, the success of Schrödinger's equations has prevented people from thinking more deeply about the true nature of the atomic world .

      The Collapse of the Wave Function

      So why bring up the wave function at all if it hampers full appreciation of the atomic world? For one thing, the equations are much more familiar to physicists, so Schrödinger's equations are used much more often then the others. Also, it turns out that Bohr liked the idea and used it in his Copenhagen interpretation. Remember our experiment with electrons? Each possible route that the electron could take, called a ghost, could be described by a wave function . As we shall see later, the "damned quantum jumping" insures that there are only a finite, though large, number of possible routes. When no one is watching, the electron take every possible route and therefore interferes with itself. However, when the electron is observed, it is forced to choose one path. Bohr called this the "collapse of the wave function". The probability that a certain path will be chosen when the wave function collapses is, essentially, the square of the path's wave function .

      Bohr reasoned that nature likes to keep it possibilities open, and therefore follows every possible path. Only when observed is nature forced to choose only one path, so only then is just one path taken .

      The Uncertainty Principle

      Wait a minute… probability??? If we are going to destroy the wave pattern by observing the experiment, then we should at least be able to determine exactly where the electron goes. Newton figured that much out back in the early eighteenth century; just observe the position and momentum of the electron as it leaves the electron gun and we can determine exactly where it goes.

      Well, fine. But how exactly are we to determine the position and the momentum of the electron? If we disturb the electrons just in seeing if they are there or not, how are we possibly going to determine both their position and momentum? Still, a clever enough person, say Albert Einstein, should be able to come up with something, right?

      Unfortunately not. Einstein did actually spend a good deal of his life trying to do just that and failed. Furthermore, it turns out that if it were possible to determine both the position and the momentum at the same time, Quantum Physics would collapse . Because of the latter, Werner Heisenberg proposed in 1925 that it is in fact physically impossible to do so. As he stated it in what now is called the Heisenberg Uncertainty Principle, if you determine an object's position with uncertainty x, there must be an uncertainty in momentum, p, such that xp > h/4pi, where h is Planck's constant  (which we will discuss shortly). In other words, you can determine either the position or the momentum of an object as accurately as you like, but the act of doing so makes your measurement of the other property that much less. Human beings may someday build a device capable of transporting objects across the galaxy, but no one will ever be able to measure both the momentum and the position of an object at the same time. This applies not only to electrons but also to objects such as tennis balls and toasters, though for these objects the amount of uncertainty is so small compared to there size that it can safely be ignored under most circumstances.


      The EPR Experiment

      "God does not play dice" was Albert Einstein's reply to the Uncertainty Principle. Thus being his belief, he spent a good deal of his life after 1925 trying to determine both the position and the momentum of a particle. In 1935, Einstein and two other physicists, Podolski and Rosen, presented what is now known as the EPR paper in which they suggested a way to do just that. The idea is this: set up an interaction such that two particles are go off in opposite directions and do not interact with anything else. Wait until they are far apart, then measure the momentum of one and the position of the other. Because of conservation of momentum, you can determine the momentum of the particle not measured, so when you measure it's position you know both it's momentum and position. The only way quantum physics could be true is if the particles could communicate faster then the speed of light, which Einstein reasoned would be impossible because of his Theory of Relativity.

      In 1982, Alain Aspect, a French physicist, carried out the EPR experiment. He found that even if information needed to be communicated faster then light to prevent it, it was not possible to determine both the position and the momentum of a particle at the same time. This does not mean that it is possible to send a message faster then light, since viewing either one of the two particles gives no information about the other. It is only when both are seen that we find that quantum physics has agreed with the experiment. So does this mean relativity is wrong? No, it just means that the particles do not communicate by any means we know about. All we know is that every particle knows what every other particle it has ever interacted with is doing.

      So what is that h that was so important in the Uncertainty Principle? Well, technically speaking, it's 6.63 X 10-34 joule-seconds. It's call Planck's constant after Max Planck who, in 1900, introduced it in the equation E=hv where E is the energy of each quantum of radiation and v is it's frequency. What this says is that energy is not continuous as everyone had assumed but only comes in certain finite sizes based on Planck's constant.

      At first physicists thought that this was just a neat mathematical trick Planck used to explain experimental results that did not agree with classical physics. Then, in 1904, Einstein used this idea to explain certain properties of light--he said that light was in fact a particle with energy E=hv. After that the idea that energy isn't continuous was taken as a fact of nature - and with amazing results. There was now a reason why electrons were only found in certain energy levels around the nucleus of an atom. Ironically, Einstein gave quantum theory the push it needed to become the valid theory it is today, though he would spend the rest of his lift trying to prove that it was not a true description of nature.

      Also, by combining Planck's constant, the constant of gravity, and the speed of light, it is possible to create a quantum of length (about 10-35 meter) and a quantum of time (about 10-43 sec), called, respectively, Planck's length and Planck's time. While saying that energy is not continuous might not be too startling to the average person, since what we commonly think of as energy is not all that well defined anyway, it is startling to say that there are quantities of space and time that cannot be broken up into smaller pieces. Yet it is exactly this that gives nature a finite number of routes to take when an electron interferes with itself.

      Although it may seem like the idea that energy is quantized is a minor part of quantum physics when compared with ghost electrons and the uncertainty principle, it really is a fundamental statement about nature that caused everything else we've talked about to be discovered. And it is always true. In the strange world of the atom, anything that can be taken for granted is a major step towards an "atomic world view".

      Remember a while ago I said there was a problem with the Copenhagen interpretation? Well, you now know enough of what quantum physics is to be able to discuss what it isn't, and by far the biggest thing it isn't is complete. Sure, the math seems to be complete, but the theory includes absolutely nothing that would tie the math to any physical reality we could imagine. Furthermore, quantum physics leaves us with a rather large open question: what is reality? The Copenhagen interpretation attempts to solve this problem by saying that reality is what is measured. However, the measuring device itself is then not real until it is measured. The problem, which is known as the measurement problem, is when does the cycle stop?

      Remember that when we last left Schrödinger he was muttering about the "damned quantum jumping." He never did get used to quantum physics, but, unlike Einstein, he was able to come up with a very real demonstration of just how incomplete the physical view of our world given by quantum physics really is. Imagine a box in which there is a radioactive source, a Geiger counter (or anything that records the presence of radioactive particles), a bottle of cyanide, and a cat. The detector is turned on for just long enough that there is a fifty-fifty chance that the radioactive material will decay. If the material does decay, the Geiger counter detects the particle and crushes the bottle of cyanide, killing the cat. If the material does not decay, the cat lives. To us outside the box, the time of detection is when the box is open. At that point, the wave function collapses and the cat either dies or lives. However, until the box is opened, the cat is both dead and alive.

      On one hand, the cat itself could be considered the detector; it's presence is enough to collapse the wave function. But in that case, would the presence of a rat be enough? Or an ameba? Where is the line drawn? On the other hand, what if you replace the cat with a human (named "Wigner's friend" after Eugene Wigner, the physicist who developed many derivations of the Schrödinger's cat experiment). The human is certainly able to collapse the wave function, yet to us outside the box the measurement is not taken until the box is opened. If we try to develop some sort of "quantum relativity" where each individual has his own view of the world, then what is to prevent the world from getting "out of sync" between observers?

      While there are many different interpretations that solve the problem of Schrödinger’s Cat, one of which we will discuss shortly, none of them are satisfactory enough to have convinced a majority of physicists that the consequences of these interpretation s are better then the half dead cat. Furthermore, while these interpretations do prevent a half dead cat, they do not solve the underlying measurement problem. Until a better intrepretation surfaces, we are left with the Copenhagen interpretation and it's half dead cat. We can certainly understand how Schrödinger feels when he says, "I don't like it, and I'm sorry I ever had anything to do with it."Yet the problem doesn't go away; it is just left for the great thinkers of tomorrow._y0QVCBgUBA


      The Infinity Problem

      There is one last problem that we will discuss before moving on to the alternative interpretation. Unlike the others, this problem lies primarily in the mathematics of a certain part of quantum physics called quantum electrodynamics, or QED. This branch of quantum physics explains the electromagnetic interaction in quantum terms. The problem is, when you add the interaction particles and try to solve Schrödinger's wave equation, you get an electron with infinite mass, infinite energy, and infinite charge. There is no way to get rid of the infinities using valid mathematics, so, the theorists simply divide infinity by infinity and get whatever result the guys in the lab say the mass, energy, and charge should be. Even fudging the math, the other results of QED are so powerful that most physicists ignore the infinities and use the theory anyway. As Paul Dirac, who was one of the physicists who published quantum equations before Schrödinger, said, "Sensible mathematics involves neglecting a quantity when it turns out to be small - not neglecting it just because it is infinitely great and you do not want it!".

      Many Worlds

      One other interpretation, presented first by Hugh Everett III in 1957, is the many worlds or branching universe interpretation. In this theory, whenever a measurement takes place, the entire universe divides as many times as there are possible outcomes of the measurement. All universes are identical except for the outcome of that measurement. Unlike the science fiction view of "parallel universes", it is not possible for any of these worlds to interact with each other.

      While this creates an unthinkable number of different worlds, it does solve the problem of Schrödinger's cat. Instead of one cat, we now have two; one is dead, the other alive. However, it has still not solved the measurement problem! If the universe split every time there was more then one possibility, then we would not see the interference pattern in the electron experiment. So when does it split? No alternative interpretation has yet answered this question in a satisfactory way. And so the search continues…

        Introduction to Quantum Mechanics



      Quantum mechanics is a body of scientific principles describing the behavior of matter and its interactions on the atomic and subatomic scales.

      Just before 1900, it became clear that classical physics was unable to explain certain phenomena. Coming to terms with these limitations of classical physics led to the development of quantum mechanics in the early decades of the 20th century, a major revolution in physical theory. Much of the universe on the largest scale does not neatly conform to classical physics, because of general relativity. Similarly, quantum mechanics means that the universe in the small also does not neatly conform to classical mechanics. The principles of quantum mechanics are difficult for the human mind to understand, because humans are accustomed to reasoning about the world on a scale where classical physics is an excellent approximation. Quantum mechanics is counterintuitive; in the words of Richard Feynman, it deals with "Nature as She is—absurd."

      Many fundamental parts of the universe, such as photons (discrete units of light), behave in some ways like particles and in other ways like waves. Radiators of photons, such as neon lights, have emission spectra, but these spectra are discontinuous in that only certain frequencies are present. The laws of quantum mechanics predict the energies, the colours, and the spectral intensities of electromagnetic radiation. But the same laws ordain that the more closely one pins down one measure (such as the position of a particle), the less predictable another measure pertaining to the same particle (such as its momentum) must become. Put another way, measuring position first and then measuring momentum does not have the same outcome as measuring momentum first and then measuring position. Even more disconcerting, pairs of particles can be created as entangled twins -- which means that an action that pins down one characteristic of one particle will instantaneously pin down the same or other characteristic of its entangled twin, regardless of the distance separating the entangled twins.

      The first quantum theory: Max Planck and black body radiation

      Hot metalwork from a blacksmith. The yellow-orange glow is the visible part of the thermal radiation emitted due to the high temperature. Everything else in the picture is glowing with thermal radiation as well, but less brightly and at longer wavelengths that the human eye cannot see. A far-infrared camera will show this radiation.

      Thermal radiation is electromagnetic radiation emitted from the surface of an object which is due to the object's temperature. If an object is heated up sufficiently, it will start to emit light at the red end of the spectrum—it is "red hot". Heating it up further will cause the colour to change, as light at shorter wavelengths (higher frequencies) begins to be emitted. It turns out that a perfect emitter is also a perfect absorber. When it is cold, such an object will look perfectly black, as it will emit practically no visible light, but it will absorb all the light that falls on it. Consequently, an ideal thermal emitter is known as a black body, and the radiation it emits is also called black body radiation.

      In the late 19th century, thermal radiation had been fairly well characterized experimentally. The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. So, as temperature increases, the glow colour changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases monotonically with temperature. Physicists were searching for a theoretical explanation for these experimental results.

      The peak wavelength and total power radiated by a black body vary with temperature. Classical electromagnetism drastically overestimates these intensities, particularly at short wavelengths.

      The "answer" found using classical physics is the Rayleigh–Jeans law. This law agrees with experimental results at long wavelengths. At short wavelengths, however, classical physics predicts that energy will be emitted by a hot body at an infinite rate. This result, which is clearly wrong, is known as the ultraviolet catastrophe.

      The first model which was able to explain the full spectrum of thermal radiation was put forward by Max Planck in 1900. He modelled the thermal radiation as being in equilibrium with a set of harmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator had to produce an integral number of units of energy at its one characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy of each oscillator was "quantized". The quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator; the constant of proportionality is now known as the Planck constant. The Planck constant, usually written as h, has the value 6.63×10−34 J s, and so the energy, E, of an oscillator of frequency f is given by

      E = nhf,\, where n = 1,2,3,\ldots

      Planck's law was the first quantum theory in physics, and Planck won the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". At the time, however, Planck's view was that quantization was purely a mathematical trick, rather than (as we now know) a fundamental change in our understanding of the world.

      Photons: the quantisation of light

      Einstein's portrait by Harm Kamerlingh Onnes at the University of Leiden in 1920

      In 1905, ALBERT EINSTEIN took an extra step. He suggested that quantisation wasn't just a mathematical trick: the energy in a beam of light occurs in individual packets, which are now called photons. The energy of a single photon is given by its frequency multiplied by Planck's constant:

      E = hf.\,

      For centuries, scientists had debated between two possible theories of light: was it a wave or did it instead consist of a stream of tiny particles? By the 19th century, the debate was generally considered to have been settled in favour of the wave theory, as it was able to explain observed effects such as refraction, diffraction and polarization. James Clerk Maxwell had shown that electricity, magnetism and light are all manifestations of the same phenomenon: the electromagnetic field. Maxwell's equations, which are the complete set of laws of classical electromagnetism, describe light as wave: a combination of oscillating electric and magnetic fields. Because of the preponderance of evidence in favour of the wave theory, Einstein's ideas were met initially by great scepticism. Eventually, however, the photon model became favoured. One of the most significant pieces of evidence in favour of the photon model was its ability to explain several puzzling properties of the photoelectric effect, described in the following section. Nevertheless, the wave analogy remained indispensable for helping to understand other light phenomena, such as diffraction.

      The photoelectric effect

      Light (red arrows, left) is shone upon a metal. If the light is of sufficient frequency, electrons are ejected (blue arrows, right).

      In 1887, Heinrich Hertz observed that light can eject electrons from metal.In 1902, Philipp Lenard discovered that the maximum possible energy of an ejected electron is related to the frequency of the light, not to its intensity. Moreover, if the frequency of the light is too low, no electrons are ejected. The lowest frequency of light which causes electrons to be emitted, called the threshold frequency, is different for every metal. This appeared to be at odds with classical electromagnetism, which was thought to predict that the electron energy would be proportional to the intensity of the radiation.

      Einstein explained the effect by postulating that a beam of light is a stream of particles (photons), and that if the beam is of frequency f, each photon has an energy equal to hf. An electron is likely to be struck only by a single photon; this photon imparts at most an energy hf to the electron. Therefore, the intensity of the beam has no effect; only its frequency determines the maximum energy that can be imparted to the electrons.

      To explain the threshold frequency, Einstein argued that it takes a certain amount of energy, called the work function, denoted by φ, to remove an electron from the metal. This amount of energy is different for each metal. If the energy of the photon is less than the work function then it does not carry sufficient energy to remove the electron from the metal. The threshold frequency, f0, is the frequency of a photon whose energy is equal to the work function:

      \varphi = h f_0.\,

      If f is greater than f0, the energy hf is enough to remove an electron. The ejected electron has a kinetic energy EK which is at most equal to the photon's energy minus the energy needed to remove the electron from the metal:

      E_K = hf - \varphi = h(f - f_0).

      The relationship between frequency of radiation and the energy of each individual photon is why ultraviolet light can cause sunburn, but visible or infrared light cannot. A photon of ultraviolet light, which has a high frequency (short wavelength), will deliver a high amount of energy—enough to contribute to cellular damage such as a sunburn. A photon of infrared light, having a lower frequency (longer wavelength), will deliver a lower amount of energy—only enough to warm one's skin. So an infrared lamp can warm a large surface, perhaps large enough to keep people comfortable in a cold room, but it cannot give anyone a sunburn.

      Einstein's description of light as being composed of photons extended Planck's notion of quantised energy: a single photon of a given frequency f delivers an invariant amount of energy hf. In other words, individual photons can deliver more or less energy, but only depending on their frequencies. Though the photon is a particle, it was still said to have the wave-like property of frequency. Once again, the particle account of light had been "compromised."

      The quantisation of matter: the Bohr model of the atom

      By the early 20th century, it was known that atoms consisted of a diffuse cloud of negatively-charged electrons surrounding a small, dense, positively-charged nucleus. This suggested a model in which the electrons circled around the nucleus like planets orbiting the sun.

      Unfortunately, it was also known that the atom in this model would be unstable: the orbiting electrons should give off electromagnetic radiation, causing them to lose energy and spiral towards the nucleus, colliding with it in a fraction of a second.

      A second, related, puzzle was the emission spectrum of atoms. When a gas is heated, it gives off light at certain discrete frequencies. For example, the visible light given off by hydrogen consists of four different colours, as shown in the picture below. By contrast, white light contains light at the whole range of visible frequencies.

      Emission spectrum of hydrogen. When excited, hydrogen gas gives off light in four distinct colours (spectral lines) in the visible spectrum, as well as a number of lines in the infra-red and ultra-violet.

      In 1913, Niels Bohr proposed a new model of the atom that included quantized electron orbits. This solution became known as the Bohr model of the atom. In Bohr's model, electrons could inhabit only certain orbits around the atomic nucleus. When an atom emitted or absorbed energy, the electron did not move in a continuous trajectory from one orbit around the nucleus to another, as might be expected in classical theory. Instead, the electron would jump instantaneously from one orbit to another, giving off light in the form of a photon. The possible energies of photons given off by each element were determined by the differences in energy between the orbits, and so the emission spectrum for each element would contain a number of lines. The Bohr model was able to explain the emission spectrum of hydrogen, but wasn't able to make accurate predictions for multi-electron atoms, or to explain why some spectral lines are brighter than others.

      Atomic emission spectra

      By the end of the nineteenth century it was known that atomic hydrogen would glow when excited, for example in an electric discharge. This light was found to be made up of only four wavelengths: the visible portion of hydrogen's emission spectrum.

      In 1885 the Swiss mathematician Johann Balmer discovered that each wavelength λ in the visible spectrum of hydrogen is related to some integer n by the equation

      \lambda = B\left(\frac{n^2}{n^2-4}\right) \qquad\qquad n = 3,4,5,6

      where B is a constant which Balmer determined to be equal to 364.56 nm.

      In 1888, Johannes Rydberg generalized and greatly increased the explanatory utility of Balmer's formula. He supposed that hydrogen will emit light of wavelength λ if λ is related to two integers n and m according to what is now known as the Rydberg formula:

       \frac{1}{\lambda} = R \left(\frac{1}{m^2} - \frac{1}{n^2}\right),

      where R is the Rydberg constant, equal to 0.0110 nm−1, and n must be greater than m.

      Rydberg's formula accounts for the four visible wavelengths by setting m = 2 and n = 3,4,5,6. It also predicts additional wavelengths in the emission spectrum: for m = 1 and for n > 1, the emission spectrum should contain certain ultraviolet wavelengths, and for m = 3 and n > 3, it should also contain certain infrared wavelengths. Experimental observation of these wavelengths came several decades later: in 1908 Louis Paschen found some of the predicted infrared wavelengths, and in 1914 Theodore Lyman found some of the predicted ultraviolet wavelengths.Refinements were added by other researchers shortly after Bohr's work appeared. Arnold Sommerfeld showed that not all orbits could be perfectly circular, so a new "atomic number" was added for the shape of the orbit, k. Sommerfeld also showed that the orientation of orbit could be influenced by magnetic fields imposed on the radiating gas, which added a third quantum number, m.

      Bohr's theory represented electrons as orbiting the nucleus of an atom, much as planets orbit around the sun. However, we now envision electrons circulating around the nuclei of atoms in a way that is strikingly different from Bohr's atom, and what we see in the world of our everyday experience. Instead of orbits, electrons are said to inhabit "orbitals." An orbital is the "cloud" of possible locations in which an electron might be found, a distribution of probabilities rather than a precise location.

      Bohr's model of the atom was essentially two-dimensional: an electron orbiting in a plane around its nuclear "sun." Modern theory describes a three-dimensional arrangement of electronic shells and orbitals around atomic nuclei. The orbitals are spherical (s-type) or lobular (p, d and f-types) in shape. It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that determines the structure and strength of chemical bonds between atoms. Thus, the bizarre quantum nature of the atomic and sub-atomic world finds natural expression in the macroscopic world with which we are more familiar.

      Wave-particle duality

      In 1924, Louis de Broglie proposed the idea that just as light has both wave-like and particle-like properties, matter also has wave-like properties. The wavelength, λ, associated with a particle is related to its momentum, p:

       p = \frac{h}{\lambda}.

      The relationship, called the de Broglie hypothesis, holds for all types of matter. Thus all matter exhibits properties of both particles and waves.

      Three years later, the wave-like nature of electrons was demonstrated by showing that a beam of electrons could exhibit diffraction, just like a beam of light. At the University of Aberdeen, George Thomson passed a beam of electrons through a thin metal film and observed the predicted interference patterns. At Bell Labs, Davisson and Germer guided their beam through a crystalline grid. Similar wave-like phenomena were later shown for atoms and even small molecules. De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis; Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.

      This is the concept of wave-particle duality: neither the classical concepts of "particle" or "wave" can fully describe the behavior of quantum-scale objects, either photons or matter. Wave-particle duality is an example of the principle of complementarity in quantum physics. An elegant example of wave-particle duality, the double slit experiment, is discussed in the section below.

      De Broglie's treatment of quantum events served as a jumping off point for Schrödinger when he set about to construct a wave equation to describe quantum theoretical events.

      The double-slit experiment

      Light from one slit interferes with light from the other, producing an interference pattern (the 3 fringes shown at the right).

      In the double-slit experiment as originally performed by Thomas Young and Augustin Fresnel in 1827, a beam of light is directed through two narrow, closely spaced slits, producing an interference pattern of light and dark bands on a screen. If one of the slits is covered up, one might naively expect that the intensity of the fringes due to interference would be halved everywhere. In fact, a much simpler pattern is seen, a simple diffraction pattern. Closing one slit results in a much simpler pattern diametrically opposite the open slit. Exactly the same behaviour can be demonstrated in water waves, and so the double-slit experiment was seen as a demonstration of the wave nature of light.

      The diffraction pattern produced when light is shone through one slit (top) and the interference pattern produced by two slits (bottom). The interference pattern from two slits is much more complex, demonstrating the wave-like propagation of light.

      The double-slit experiment has also been performed using electrons, atoms, and even molecules, and the same type of interference pattern is seen. Thus all matter possesses both particle and wave characteristics.

      Even if the source intensity is turned down so that only one particle (e.g. photon or electron) is passing through the apparatus at a time, the same interference pattern develops over time. The quantum particle acts as a wave when passing through the double slits, but as a particle when it is detected. This is a typical feature of quantum complementarity: a quantum particle will act as a wave when we do an experiment to measure its wave-like properties, and like a particle when we do an experiment to measure its particle-like properties. Where on the detector screen any individual particle shows up will be the result of an entirely random process.

      Application to the Bohr model

      De Broglie expanded the Bohr model of the atom by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an electron will be observed only in situations that permit a standing wave around a nucleus. An example of a standing wave is a string fixed at both ends and made to vibrate (as in a string instrument). Hence a standing wave must have zero amplitude at each fixed end. The waves created by a stringed instrument also appear to oscillate in place, moving from crest to trough in an up-and-down motion. The wavelength of a standing wave is related to the length of the vibrating object and the boundary conditions. (For example, the fixed/fixed string can carry standing waves of wavelengths 2l/n where l is the length and n is a positive integer.) In a vibrating medium that traces out a simple closed curve, the wave must be a continuous formation of crests and troughs all around the curve. Since electron orbitals are simple closed curves, each electron must be its own standing wave, occupying a unique orbital.

      Development of modern quantum mechanics

      Full quantum mechanical theory

      An electron falling from energy state 3 to energy state 2 emits one red photon.

      For a somewhat more sophisticated look at how Heisenberg got from the old quantum mechanics and classical physics to the new quantum mechanics, see Heisenberg's entryway to matrix mechanics.

      To make a long and rather complicated story short, Werner Heisenberg used the idea that since classical physics is correct when it applies to phenomena in the world of things larger than atoms and molecules, it must stand as a special case of a more inclusive quantum theoretical model. So he hoped that he could modify quantum physics in such a way that when the parameters were on the scale of everyday objects it would look just like classical physics, but when the parameters were pulled down to the atomic scale the discontinuities seen in things like the widely spaced frequencies of the visible hydrogen bright line spectrum would come back into sight.

      By means of an intense series of mathematical analogies that some physicists have termed "magical," Heisenberg wrote out an equation that is the quantum mechanical analogue for the classical computation of intensities. Remember that the one thing that people at that time most wanted to understand about hydrogen radiation was how to predict or account for the intensities of the lines in its spectrum. Although Heisenberg did not know it at the time, the general format he worked out to express his new way of working with quantum theoretical calculations can serve as a recipe for two matrices and how to multiply them.

      Heisenberg's groundbreaking paper of 1925 neither uses nor even mentions matrices. Heisenberg's great advance was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)"of hydrogen radiation.

      After Heisenberg wrote his ground breaking paper, he turned it over to one of his senior colleagues for any needed corrections and went on a well-deserved vacation. Max Born puzzled over the equations and the non-commuting equations that Heisenberg had found troublesome and disturbing. After several days he realized that these equations amounted to directions for writing out matrices. Matrices were a bit off the beaten track, even for mathematicians of that time, but how to do math with them was already clearly established. He and a few others had the job of working everything out in matrix form before Heisenberg returned from his time off, and within a few months the new quantum mechanics in matrix form formed the basis for another paper.

      When quantities such as position and momentum are mentioned in the context of Heisenberg's matrix mechanics, it is essential to keep in mind that a statement such as pqqp does not refer to a single value of p and a single value q but to a matrix (grid of values arranged in a defined way) of values of position and a matrix of values of momentum. So multiplying p times q or q times p is really talking about the matrix multiplication of the two matrices. When two matrices are multiplied, the answer is a third matrix.

      Max Born saw that when the matrices that represent pq and qp were calculated they would not be equal. Heisenberg had already seen the same thing in terms of his original way of formulating things, and Heisenberg may have guessed what was almost immediately obvious to Born — that the difference between the answer matrices for pq and for qp would always involve two factors that came out of Heisenberg's original math: Planck's constant h and i, which is the square root of negative one. So the very idea of what Heisenberg preferred to call the "indeterminacy principle" (usually known as the uncertainty principle) was lurking in Heisenberg's original equations.

      Paul Dirac decided that the essence of Heisenberg's work lay in the very feature that Heisenberg had originally found problematical — the fact of non-commutativity such as that between multiplication of a momentum matrix by a displacement matrix and multiplication of a displacement matrix by a momentum matrix. That insight led Dirac in new and productive directions.

      Schrödinger wave equation

      In 1925, building on De Broglie's theoretical model of particles as waves, Erwin Schrödinger analyzed how an electron would behave if it were assumed to be a wave surrounding a nucleus. Rather than explaining the atom by an analogy to satellites orbiting a planet, he treated electrons as waves with each electron having a unique wavefunction. The mathematical wavefunction is called the "Schrödinger equation" after its creator. Schrödinger's equation describes a wavefunction by three properties (Wolfgang Pauli later added a fourth: spin):

      1. An "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy;
      2. The "shape" of the orbital, spherical or otherwise;
      3. The "inclination" of the orbital, determining the magnetic moment of the orbital around the z-axis.

      The collective name for these three properties is the "wavefunction of the electron," describing the quantum state of the electron. The quantum state of an electron refers to its collective properties, which describe what can be said about the electron at a point in time. The quantum state of the electron is described by its wavefunction, denoted by ψ.

      The three properties of Schrödinger's equation describing the wavefunction of the electron (and thus its quantum state) are each called quantum numbers. The first property describing the orbital is the principal quantum number, numbered according to Bohr's model, in which n denotes the energy of each orbital.

      The next quantum number, the azimuthal quantum number, denoted l (lower case L), describes the shape of the orbital. The shape is a consequence of the angular momentum of the orbital. The angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number l represents the orbital angular momentum of an electron around its nucleus. However, the shape of each orbital has its own letter as well. The first shape is denoted by the letter s (for "spherical"). The next shape is denoted by the letter p and has the form of a dumbbell. The other orbitals have more complicated shapes (see Atomic Orbitals), and are denoted by the letters d, f, and g.

      The third quantum number in Schrödinger's equation describes the magnetic moment of the electron. This number is denoted by either m or ml, because the magnetic moment depends on the second quantum number l.

      In May 1926, Schrödinger proved that Heisenberg's matrix mechanics and his own wave mechanics made the same predictions about the properties and behaviour of the electron; mathematically, the two theories were identical. Yet both men disagreed on the physical interpretations of their respective theories. Heisenberg saw no problem in the existence of discontinuous quantum jumps, while Schrödinger hoped that a theory based on continuous wave-like properties could avoid what he called (in the words of Wilhelm Wien.), "this nonsense about quantum jumps."

      Uncertainty principle

      One of Heisenberg's seniors, Max Born explained how he took his strange "recipe" given above and discovered something ground breaking:

      By consideration of ...examples...[Heisenberg] found this rule.... This was in the summer of 1925. Heisenberg...took leave of absence...and handed over his paper to me for publication....

      Heisenberg's rule of multiplication left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory....Such quadratic arrays are quite familiar to mathematicians and are called matrices, in association with a definite rule of multiplication. I applied this rule to Heisenberg's quantum condition and found that it agreed for the diagonal elements. It was easy to guess what the remaining elements must be, namely, null; and immediately there stood before me the strange formula

      {QP - PQ = \frac{ih}{2\pi}}
      [The symbol Q is the matrix for displacement, P is the matrix for momentum, i stands for the square root of negative one, and h is Planck's constant.]

      This is the Heisenberg uncertainty principle, derived from the mathematics. Quantum mechanics strongly limits the precision with which the properties of moving subatomic particles can be measured. An observer can precisely measure either position or momentum, but not both. In the limit, measuring either variable with complete precision would entail a complete absence of precision in the measurement of the other.

      Wavefunction collapse is a forced term for whatever happened when it becomes appropriate to replace the description of an uncertain state of a system by a description of the system in a definite state. Explanations for the nature of the process of becoming certain are controversial. At any time before an electron "shows up" on a detection screen it can only be described by a set of probabilities for where it might show up. When it does show up, for instance in the CCD of an electronic camera, the time and the space where it interacted with the device are known within very tight limits. However, the photon has disappeared, and the wave function has disappeared with it. In its place some physical change in the detection screen has appeared, e.g., an exposed spot in a sheet of photographic film.

      Eigenstates and eigenvalues

      Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Therefore it is necessary to formulate clearly the difference between the state of something that is indeterminate, such as an electron in a probability cloud, and the state of something having a definite value. When an object can definitely be "pinned-down" in some respect, it is said to possess an eigenstate.

      The Pauli exclusion principle

      Wolfgang Pauli proposed the following concise statement of his principle: "There cannot exist an atom in such a quantum state that two electrons within have the same set of quantum numbers."

      He developed the exclusion principle from what he called a "two-valued quantum degree of freedom" to account for the observation of a doublet, meaning a pair of lines differing by a small amount (e.g., on the order of 0.15Å), in the spectrum of atomic hydrogen. The existence of these closely spaced lines in the bright-line spectrum meant that there was more energy in the electron orbital from magnetic moments than had previously been described.

      In early 1925, Uhlenbeck and Goudsmit proposed that electrons rotate about an axis in the same way that the earth rotates on its axis. They proposed to call this property spin. Spin would account for the missing magnetic moment, and allow two electrons in the same orbital to occupy distinct quantum states if they "spun" in opposite directions, thus satisfying the Exclusion Principle. A new quantum number was then needed, one to represent the momentum embodied in the rotation of each electron.

      By this time an electron within an atomic orbital was recognized to have four kinds of fundamental characteristics that came to be identified by the four quantum numbers. The exclusion principle demands that no two electrons within an atom may have the same values of all four numbers.

      The chemist Linus Pauling wrote, by way of example:

      In the case of a helium atom with two electrons in the 1 s orbital, the Pauli Exclusion Principle requires that the two electrons differ in the value of one quantum number. Their values of n, l, and ml are the same; moreover, they have the same spin, s = 12. Accordingly they must differ in the value of ms, which can have the value of +12 for one electron and −12 for the other."

      Dirac wave equation

      In 1928, Paul Dirac extended the Pauli equation, which described spinning electrons, to account for special relativity. The result was a theory that dealt properly with events, such as the speed at which an electron orbits the nucleus, occurring at a substantial fraction of the speed of light. By using the simplest electromagnetic interaction, Dirac was able to predict the value of the magnetic moment associated with the electron's spin, and found the experimentally observed value, which was too large to be that of a spinning charged sphere governed by classical physics. He was able to solve for the spectral lines of the hydrogen atom, and to reproduce from physical first principles Sommerfeld's successful formula for the fine structure of the hydrogen spectrum.

      Dirac's equations sometimes yielded a negative value for energy, for which he proposed a novel solution: he posited the existence of an antielectron and of a dynamical vacuum. This led to many-particle quantum field theory.

      Quantum entanglement

      Superposition of two quantum characteristics, and two resolution possibilities.

      The Pauli exclusion principle says that two electrons in one system cannot be in the same state. Nature leaves open the possibility, however, that two electrons can have both states "superimposed" over them. Recall that the wave functions that emerge simultaneously from the double slits arrive at the detection screen in a state of superposition. Nothing is certain until the superimposed waveforms "collapse," At that instant an electron shows up somewhere in accordance with the probabilities that are the squares of the amplitudes of the two superimposed waveforms. The situation there is already very abstract. A concrete way of thinking about entangled photons, photons in which two contrary states are superimposed on each of them in the same event, is as follows:

      Imagine that the superposition of a state that can be mentally labeled as blue and another state that can be mentally labeled as red will then appear (in imagination, of course) as a purple state. Two photons are produced as the result of the same atomic event. Perhaps they are produced by the excitation of a crystal that characteristically absorbs a photon of a certain frequency and emits two photons of half the original frequency. So the two photons come out "purple." If the experimenter now performs some experiment that will determine whether one of the photons is either blue or red, then that experiment changes the photon involved from one having a superposition of "blue" and "red" characteristics to a photon that has only one of those characteristics. The problem that Einstein had with such an imagined situation was that if one of these photons had been kept bouncing between mirrors in a laboratory on earth, and the other one had traveled halfway to the nearest star, when its twin was made to reveal itself as either blue or red, that meant that the distant photon now had to lose its "purple" status too. So whenever it might be investigated, it would necessarily show up, instantaneously, in the opposite state to whatever its twin had revealed.

      Suppose that some species of animal life carries both male and female characteristics in its genetic potential. It will become either male or female depending on some environmental change. Perhaps it will remain indeterminate until the weather either turns very hot or very cold. Then it will show one set of sexual characteristics and will be locked into that sexual status by epigenetic changes, the presence in its system of high levels of androgen or estrogen, etc. There are actually situations in nature that are similar to this scenario, but now imagine that if twins are born, then they are forbidden by nature to both manifest the same sex. So if one twin goes to Antarctica and changes to become a female, then the other twin will turn into a male despite the fact that local weather has done nothing special to it. Such a world would be very hard to explain. How can something that happens to one animal in Antarctica affect its twin in Redwood, California? Is it mental telepathy? What? How can the change be instantaneous? Even a radio message from Antarctica would take a certain amount of time.

      In trying to show that quantum mechanics was not a complete theory, Einstein started with the theory's prediction that two or more particles that have interacted in the past can appear strongly correlated when their various properties are later measured. He sought to explain this seeming interaction in a classical way, through their common past, and preferably not by some "spooky action at a distance." The argument is worked out in a famous paper, Einstein, Podolsky, and Rosen (1935; abbreviated EPR), setting out what is now called the EPR paradox. Assuming what is now usually called local realism, EPR attempted to show from quantum theory that a particle has both position and momentum simultaneously, while according to the Copenhagen interpretation, only one of those two properties actually exists and only at the moment that it is being measured. EPR concluded that quantum theory is incomplete in that it refuses to consider physical properties which objectively exist in nature. (Einstein, Podolsky, & Rosen 1935 is currently Einstein's most cited publication in physics journals.) In the same year, Erwin Schrödinger used the word "entanglement" and declared: "I would not call that one but rather the characteristic trait of quantum mechanics." The question of whether entanglement is a real condition is still in dispute. The Bell inequalities are the most powerful challenge to Einstein's claims.

      Quantum electrodynamics

      Quantum electrodynamics (QED) is the name of the quantum theory of the electromagnetic force. Understanding QED begins with understanding electromagnetism. Electromagnetism can be called "electrodynamics" because it is a dynamic interaction between electrical and magnetic forces. Electromagnetism begins with the electric charge.

      Electric charges are the sources of, and create, electric fields. An electric field is a field which exerts a force on any particles that carry electric charges, at any point in space. This includes the electron, proton, and even quarks, among others. As a force is exerted, electric charges move, a current flows and a magnetic field is produced. The magnetic field, in turn causes electric current (moving electrons). The interacting electric and magnetic field is called an electromagnetic field.

      The physical description of interacting charged particles, electrical currents, electrical fields, and magnetic fields is called electromagnetism.

      In 1928 Paul Dirac produced a relativistic quantum theory of electromagnetism. This was the progenitor to modern quantum electrodynamics, in that it had essential ingredients of the modern theory. However, the problem of unsolvable infinities developed in this relativistic quantum theory. Years later, renormalization solved this problem. Initially viewed as a suspect, provisional procedure by some of its originators, renormalization eventually was embraced as an important and self-consistent tool in QED and other fields of physics. Also, in the late 1940s Feynman's diagrams showed all possible interactions of a given event. The diagrams showed that the electromagnetic force is the interactions of photons between interacting particles.

      An example of a prediction of quantum electrodynamics which has been verified experimentally is the Lamb shift. This refers to an effect whereby the quantum nature of the electromagnetic field causes the energy levels in an atom or ion to deviate slightly from what they would otherwise be. As a result, spectral lines may shift or split.

      In the 1960s physicists realized that QED broke down at extremely high energies. From this inconsistency the Standard Model of particle physics was discovered, which remedied the higher energy breakdown in theory. The Standard Model unifies the electromagnetic and weak interactions into one theory. This is called the electroweak theory.


      The physical measurements, equations, and predictions pertinent to quantum mechanics are all consistent and hold a very high level of confirmation. However, the question of what these abstract models say about the underlying nature of the real world has received competing answers.

      In classical mechanics, the energy of an oscillation or vibration can take on any value at any frequency. However, in quantum mechanics, energy of a photon is related to its frequency by a conversion factor called Planck's constant h. Atomic electrons can exist in states with discrete "energy levels" or in a superposition of such states. Just as the velocity with which an object approaches a sun will determine the distance from the sun at which it can establish a stable orbit, so too the energy carried by an electron will automatically assign it to a given orbital around the nucleus of an atom. Moving from one energy state to another either requires that more energy be supplied to the electron (moving it to a higher energy state) or the electron must lose a certain amount of energy as a photon (moving it to a lower energy state). Four such transitions from a higher to a lower energy state give visible lines in the bright line spectrum of hydrogen. Other transitions give lines outside the visible spectrum.

      The four visible lines in the spectrum of hydrogen were known for some time before scientists knew anything more than their empirically determined wavelengths. Then, Balmer figured out a mathematical rule by which he could make quantum theoretical predictions of the observed wavelengths. The same basic rule was improved in two stages, first by writing it in terms of the inverse values of all of the numbers used by Balmer, and second by generalizing the rule and replacing

        {\left( \frac{1}{4} - \frac{1}{n^2} \right)} with   {\left( \frac{1}{m^2} - \frac{1}{n^2} \right)}

      This additional level of generality permitted the entire hydrogen bright-line spectrum, from infrared through the visible colours to ultraviolet and higher frequencies, to be predicted: m and n could take any integer value as long as n was larger than m. Using Planck's constant, one could assign energies to individual frequencies (or wavelengths) of electromagnetic radiation. To predict the intensities of these bright lines, physicists needed to use matrix mathematics, Schrödinger's equation, or some other computational scheme involving higher mathematics. There were not only the basic six energy levels of hydrogen, but also other factors that created additional energy levels. The very first calculation that Heisenberg made in his new theory involved an infinite series, and the more factors involved (the more "quantum numbers" were involved) the more complex the mathematics. But the basic insight into the structure of the hydrogen atom was encoded in the simple formula that Balmer guessed from a list of wavelengths.

      The photoelectric effect was discovered soon after Balmer made his rule, and in 1905 Einstein first depicted light as being made of photons to account for that effect.

      Bohr explained the Rydberg formula in terms of atomic structure. Two years later, in 1925, Heisenberg removed the last traces of classical physics from the new quantum theory by making the breakthrough that led to the matrix formulation of quantum mechanics, and Pauli enunciated his exclusion principle. Further advances came by closing in on some of the elusive details: (1) adding the ml (spin) quantum numbers (discovered by Pauli), (2) adding the Ms quantum numbers (discovered by Goudsmit and Uhlenbeck), (3) broadening the quantum picture to account for relativistic effects (Dirac's work), (4) showing that particles such as electrons, and even larger entities, have a wave nature (the matter waves of de Broglie).

      Dirac introduced a new theoretical formulation "which if interrogated in a particle-like way gave particle behavior and if interrogated in a wave-like way gave wave behavior."

      Several improvements in mathematical formulation have also furthered quantum mechanics:

      De Broglie's quantum theoretical description based on waves was followed upon by Schrödinger. Schrödinger's method of representing the state of each atomic entity is a generally more practical scheme to use than Heisenberg's. It makes it possible to conceptualize a "wave function" that passes through both sides of a double-slit experiment and then arrives at the detection screen as two parts of itself that are superimposed but a little shifted (a little out of phase). It also makes it possible to understand how two photons or other things of that order of magnitude might be created in the same event or otherwise closely linked in history and so carry identical copies of superimposed wave functions. That mental picture can then be used to explain how when one of them is coerced into revealing itself, it must manifest one or the other superimposed wave nature, and its twin (regardless of its distance away in space or time) must manifest the complementary wave nature.

      Prominent among later scientists who increased the elegance and accuracy of quantum-theoretical formulations was Richard Feynman who followed up on Dirac's work. The basic picture given in the original Balmer formula remained true, but it has been qualified by revelation of many details, such as angular momentum and spin, and extended to descriptions that go beyond a mere explanation of the electron and its behavior while bound to an atomic nucleus. Active research still continues to resolve some remaining issues.