# Introduction to quantum mechanics: Map

### Map showing all locations mentioned on Wikipedia article: Everyday experience creates preconceptions that fail drastically when that experience is extended to the very massive and the very fast, or when extended to the very small and the very cold. The large scale requires relativity theory, and the small scale requires quantum mechanics. Quantum physics deals with "Nature as She is—absurd."

Quantum physics deals with situations where the usual picture of reality breaks down. Photons (discrete units of light) and other very small things have some behaviors that resemble classical particles like billiard balls and other behaviors that resemble classical waves like water waves. Radiators of photons such as neon lights have spectra, but the spectra are chopped up instead of being continuous. The energies carried by photons form a discontinuous and color coded series. The energies, the colors, and the spectral intensities of electromagnetic radiation produced are all interconnected by law. But the same laws ordain that the more closely one pins down one measure (such as the position of a particle) the more wildly another measure relating to the same thing must fluctuate (such as momentum). Put another way, measuring position first and then measuring momentum is not the same as measuring momentum first and then measuring position. Even more disconcerting, particles can be created as twins and therefore as entangled entities -- which means that doing something that pins down one characteristic of one particle will reveal something about its entangled twin even if it is millions and millions of miles away.

## An elegant example The most elegant character on the quantum stage is the double-slit experiment. It demonstrates particle-wave duality, and it highlights several features of quantum mechanics. Photons emitted from some source such as a laser will behave differently depending on whether one or two slits lie in their path. When only one slit is present, the light observed on the screen will appear as a rather narrow diffraction pattern (see the photo below).

However, things get really strange when two slits are introduced into the experiment. With two slits present, what arrives at a remote detection screen will be a superposition of two waves. As the illustration shows, a wave from the top slit and another from the bottom slit will fall on top of each other on the detection screen, and so they are superimposed. The same basic experiment can be done by shooting an electron at a double slit. The wave nature of light causes the light waves passing through both slits to interfere with themselves, creating an interference pattern of bright and dark bands on the screen. However, at the screen, the light is always found to be absorbed as discrete particles, called photons.

What is even stranger is what happens when the light source is "turned down" to where only one photon will be emitted at a time. Everyday experience leads one to expect that the photon will travel through one or the other slit as a particle, and strike the screen as a particle. However, any single photon travels through both slits as a wave, and creates two wave patterns that interfere with each other. Even more strangely, that photon is then detected as a particle on the screen.

Where a photon or electron shows up on the detection screen will depend on the probabilities as calculated by adding the amplitudes of the two waves at each point, and squaring that sum. However, where any individual photon or electron strikes the screen will be the result of an entirely random process. The end result will be in accord with the probabilities that can be calculated.

Photons function as though they are waves as they go through the slits. When two slits are present, the "wave function" pertaining to each photon goes through each slit. The wave functions are superimposed all across the detection screen, yet at the detection screen only one particle, a photon, shows up and its position is in accord with strict probability laws. So what humans interpret as the wave nature of photons and as the particle nature of photons must both appear in the final results. All of these points will be revisited below.

## How the unexpected came to light A black-body heated to 5000 kelvins emits different wavelengths of radiation with different intensities (blue).
Classical electromagnetism drastically overestimates these intensities (black).

The 1827 experiments of Thomas Young and Augustin Fresnel demonstrated that light is subject to interference patterns. By the end of the nineteenth century, classical electromagnetism treated light as a wave phenomenon: a combination of oscillating electric and magnetic fields.

Around the turn of the twentieth century, it became clear that classical electromagnetism fails to explain several physical phenomena. The first is black-body radiation, for which classical electromagnetism predicts that a heated object should produce enormous intensities of high-energy ultraviolet light. The way out of this "ultraviolet catastrophe" was prepared by Max Planck, who determined that the energy content of black-body radiation of frequency f comes as multiples of a quantum of energy hf. He was not aware of the ultraviolet catastrophe, however, and only was trying to fix the problem he knew about—that the existing models did not yield the right energy distribution for black-body radiation.

The second unexplainable phenomenon was the photoelectric effect, wherein light could sometimes eject electrons from a metal. Contrary to the predictions of classical mechanics, however, it was not the intensity but the frequency of the light that determined if it could eject electrons. In 1905, Albert Einstein successfully explained the effect by positing that light is composed of particles called photons.

Around the same time, scientists were trying to understand the physics of emission spectra. It was known that a gaseous atomic hydrogen would glow when exposed to electric discharge, and that this glow was composed of a small number of wavelengths of light. Johann Balmer and Johannes Rydberg had empirically determined a formula to fit the observed wavelengths, but in 1913 Niels Bohr developed a new theory of the atom to explain the observed emissions. In the Bohr model, the electron orbits the nucleus only at certain allowed distances, and with certain allowed energies. A photon is emitted when the electron jumps from a higher-energy orbit to a lower-energy orbit.

### Planck and the constant h

Max Planck's work established an empirical equation to represent the observed energy distribution in the frequencies present in black-body radiation. According to Planck's model, the radiating body is composed of an enormous number of elementary oscillators, each having its own unique frequency. According to this model, the energy E emitted by the black-body must be some integer multiple of hf, where h is the Planck constant, a proportionality constant equal to , and f is the frequency of the light.

In classical physics, and in everyday experience of things like guitar strings, a vibrating object has an amplitude that is proportional to the force applied to it, e.g., to how hard the guitar string is plucked. In his model, Planck's oscillators each had its fundamental frequency, to that fundamental frequency was associated a lowest energy given by hf, and the oscillator was then said to be able to radiate any integer multiple of that fundamental energy, yielding the equation

In terms of photons radiated by something like a neon light, this law could be interpreted to explain that the change of energy state by any electron yields a photon of a frequency and an energy related to the change of energy of the electron, and that the energy of the neon light at that frequency is the sum of the n number of electrons making that change. In other words, there is no single oscillator emitting energy at multiples of some lowest energy, but multiples of single oscillators each having a single energy and frequency. Rather than itself being some lowest possible amount of energy, h is a proportionality constant that properly associates frequencies stated in some unit with energies stated in some unit. In some unit systems such as Planck units, h is numerically equal to one, and the number representing the frequency is the same as the number representing the energy. The quantum of energy is then written f, and the energy of the radiation is .

### The photoelectric effect In 1887, Heinrich Hertz observed that light can eject electrons from metal. In 1902, Philipp Lenard discovered that the maximum possible energy of each ejected electron is related to the frequency of the light, not its intensity. Moreover, if the frequency of the light is too low, no electrons are ejected. The lowest frequency which still ejects electrons from a metal is called the threshold frequency, and it is unique to the metal.

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 Actually there can be intensity-dependent effects , but at light intensities achievable with non-laser sources these effects are unobservable. ; only its frequency determines the maximum energy that can be imparted to the electrons.

To explain the threshold frequency f0, Einstein argued that it takes a certain amount of energy φ ("phi") simply to remove the electron from the metal. Called the work function, φ is equal to hf0 and is specific to the metal. If the frequency f of the photon is less than f0 its energy hf is less than the energy φ, and so the photon does not carry sufficient energy to remove the electron from the metal. If f is greater than f0, the energy hf is enough to remove an electron; an ejected electron has a kinetic energy K which is at most equal to the difference of hf and φ:
K = hf - \varphi = hf - hf_0.

For example, a photon of ultraviolet light, having a 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 long wavelength, will deliver a low amount of energy—only enough to warm one's skin. So a very large infrared light can warm a large surface, perhaps large enough to keep people comfortable in a cold room or even make people too hot, but it cannot give anyone a sunburn. What is needed for an electron to be freed from its original atom and jump a spark gap or do something else requiring a certain voltage to be present is for the frequency of the incident light to be high enough.

Einstein's description of light as being composed of photons extended Planck's notion of quantized 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."

Despite this, though, Einstein's photoelectric effect equation can be derived and explained without resorting to "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, and the results are quantitavely 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

### Atomic emission spectra When excited by an electric discharge, atomic hydrogen produces several wavelengths of light.
By the end of the nineteenth century it was known that atomic hydrogen would glow in an electric discharge. When light from this glow was passed through a diffraction grating, it was found to be made up of only four wavelengths. These constitute the visible portion of hydrogen's emission spectrum. The four visible wavelengths of light in the emission spectrum of atomic hydrogen.
In 1885 the Swedish mathematician Johann Balmer discovered that each wavelength λ in the visible spectrum of hydrogen is related to some integer n by the equation
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 and . It also predicts additional wavelengths in the emission spectrum: for and for , the emission spectrum should contain certain ultraviolet wavelengths, and for and , 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.

In 1908, Walter Ritz discovered what has come to be known as the Ritz combination principle that demonstrates how new intervals among frequencies in a bright line spectrum can be discovered because there are several differences of frequencies between the energy states (or orbits) of electrons that keep repeating themselves. This principle is implicit in Heisenberg's breakthrough formulation of the new quantum mechanics in 1925.

### The Bohr model of the atom The Bohr model of the atom, showing electron quantum jumping to ground state .

In 1897, a research team headed by J J Thompson discovered and named the electron, the carrier of negative charge. By means of the gold foil experiment, physicists discovered that matter is mostly empty space. Once that was clear, it was hypothesized that negatively charged electrons orbit a positively charged nucleus, so that all atoms resemble a miniature solar system. But that simple analogy predicted that electrons would take only about one hundredth of a microsecond to crash into the nucleus. Hence the great question of early 20th century physics was: "How do electrons normally remain in stable orbits around the nucleus?"

In 1913, Niels Bohr solved this substantial problem by applying the notion of discrete (non-continuous) quanta to electron orbits.Bohr theorized that an electron can only inhabit certain orbits around the nucleus, and that they may have only certain energies. He arrived at this conclusion by assuming that the angular momentum L of an electron must be quantized:
L = n\frac{h}{2\pi},
where n is a positive integer and h is the Planck constant. Together with Coulomb's law and the equations of circular motion, this assumption implies that an electron orbits a proton at a distance r given by
r = n^2 a_{\mathrm{B}},\!
where n is a positive integer and aB is the Bohr radius, equal to 0.0529 nm. Additionally, the energy of the electron is given by
E = -\frac{k_{\mathrm{e}}e^2}{2a_{\mathrm{B}}} \frac{1}{n^2},
where ke is the Coulomb constant and e is the elementary charge.

A consequence of these constraints is that the electron will not crash into the nucleus: it cannot continuously emit energy, and it cannot come closer to the nucleus than aB. An electron changes its energy by suddenly disappearing from its original orbit and reappearing in another orbit. An electron that absorbs a photon gains a quantum of energy, so it jumps to an orbit that is farther from the nucleus, while an electron that emits a photon loses a quantum of energy and so jumps to an orbital closer to the nucleus. An electron cannot gain or lose a fractional quantum of energy, and hence it cannot be found at some fraction of the distance between allowed orbits.

With these assumptions, the Bohr model explains why the Rydberg formula is true: each photon from glowing atomic hydrogen is due to an electron moving from a higher orbit rn to a lower orbit rm. The energy Eγ of this photon is the difference in the energies En and Em of the electron:
E_{\gamma} = E_n - E_m = \frac{k_{\mathrm{e}}e^2}{2a_{\mathrm{B}}}\left(\frac{1}{m^2}-\frac{1}{n^2}\right)
Then, since the photon's energy is related to its wavelength by ,
\frac{1}{\lambda} = \frac{k_{\mathrm{e}}e^2}{2a_{\mathrm{B}}hc}\left(\frac{1}{m^2}-\frac{1}{n^2}\right).
The quantity kee2/2aBhc is equal to the Rydberg constant. Therefore, the Bohr model of the atom explains the emission spectrum of hydrogen in terms of fundamental constants and the behavior of the electron. The model can be easily modified to account of the emission spectrum of any hydrogen-like atom (that is, ions such as He+ or O7+ which contain only one electron).

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.

### Consequences of contemporary research Red: (high amperage) low voltage. Violet: high voltage.

Other experiments showed that when light shone upon a metal surface it would drive electrons away. The result can be seen in the light meters designed for photographic use. A beam of light creates an electrical potential (a voltage), and that voltage will cause a current with a certain amperage to flow through an external part of the circuit. In a light meter, the voltage induced by a beam of light induces an electric current of a certain voltage and amperage that powers a little electromagnet that moves the needle in the light meter. Contrary to classical electromagnetism, the voltage produced by a beam of light did not change when the intensity of the beam of light was changed. Only the current in the circuit changed. But if one substituted beams of a single frequency, red beams and violet beams of equal intensity would produce different voltages. These experiments showed that longer wavelength light produces lower voltage, i.e., it puts less of a "kick" on individual electrons, and shorter wavelength light produces a greater force on individual electrons. Infrared light warms; ultraviolet penetrates.

### The double-slit experiment Scientists were forced to draw a seemingly very self-contradictory conclusion. Light behaves like a wave in some situations, and yet it performs like particles in other situations. The quantum physicists enunciated the principle of complementarity, i.e., the idea that light cannot be adequately characterized by the wave interpretation, but it also cannot be adequately characterized by the particle interpretation. One cannot stand without the other, at least not when talking about things on an atomic scale. In the quantum world, a photon may be emitted as the result of an interaction within a single atom, and end up by being absorbed by a single atom in the detection screen. But where on that detection screen it appears depends very strongly on whether there is a single path between the point of origin and the screen, or there are two or more paths. If there are two or more paths then something wavelike passes through both slits and then interferes with itself. The fact of self interference determines the probabilities of the photon's potential points of appearance.

The double-slit experiment is a very compelling example of quantum effects and complementarity. Originally performed by Young and Fresnel in 1827, a beam of light is directed through two narrow, closely spaced slits. The two resulting beams interfere with each other and produce light and dark bands on a detector—a consequence of the wave nature of light. Later repetitions of the experiment involved an apparatus which could shoot only one photon at a time at the slits. One would naively expect that each photon must pass through either one slit or the other; thus, one should observe only two bands on the detector. In reality, the same interference pattern of light and dark bands emerges. Therefore, the photon cannot adequately be explained as being simply a particle: it is also some kind of wave phenomenon which interferes with itself as it passes through the slits.

This same behavior is exhibited by not just photons, but also electrons, atoms, and even some molecules. Thus all matter possesses both particle and wave characteristics.

### Wave-particle duality

Both the idea of a wave and the idea of a particle are models derived from our everyday experience. We cannot see individual photons, and can only investigate their properties indirectly. Take, for example, the rainbow of colours we see reflected from a puddle of water when a thin film of oil rests on its surface. We can explain that phenomenon by modelling light as waves. Other phenomena, such as the working of the photoelectric meters in our cameras, may be explained by thinking in terms of particles of light colliding with the detection screen inside the meter. In both cases, we take concepts from our everyday experience and apply them to a world we will never see or otherwise experience directly.

Neither wave nor particle is an entirely satisfactory explanation. In general, any model can only approximate that which it models. A model is useful only within the range of conditions where it makes accurate predictions. Newtonian physics remains a good predictor of most everyday (macroscopic) phenomena. To remind us that both "wave" and "particle" are concepts imported from our macro world to explain atomic-scale phenomena, some physicists have used the term "wavicle" to refer to whatever it is that is "really there." Astrophysicist A.S. Eddington proposed in 1927 that "We can scarcely describe such an entity as a wave or as a particle; perhaps as a compromise we had better call it a 'wavicle' ". In the following discussion, "wave" and "particle" may both be used depending on which aspect of quantum mechanical phenomena is under discussion.

Niels Bohr showed that neither the wave analogy nor the particle analogy, taken individually, fully describe the empirical properties of light. All forms of electromagnetic radiation were found to behave in certain experiments as though they were particles, and in other experiments as though they were waves. With these facts in mind, Bohr enunciated the principle of complementarity, which pairs concepts such as wave and particle, or position and momentum.

In 1923, Louis de Broglie explored the mathematical consequences of Bohr's findings and discovered the theory of wave-particle duality, which states that subatomic particles also have simultaneous wave and particle properties. 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. A standing wave requires that the wavelength be an integer fraction of the length of the vibrating object. (In other words, a harmonic frequency must be an integer multiple of the fundamental frequency of the vibrating object.) 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.

In 1924, de Broglie extended his treatment of the wave-particle duality and derived a relationship between the momentum p and wavelength λ of a particle:
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.

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.

## Development of modern quantum mechanics

### Full quantum mechanical theory An electron falling from energy state 3 to energy state 2 emits one red photon.

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 analog 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 groundbreaking 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: Plank'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 the Greek letter \psi ("psi," pronounced "sigh").

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 m with a subscript l, 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.]

That is the Heisenberg uncertainty principle, and it came out of the math! 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

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.

### Eigenstates and eigenvalues

For a more detailed introduction to this subject, see: Introduction to eigenstates

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 was recognized to have four kinds of fundamental characteristics that came to be identified by the four quantum numbers:

The chemist Linus Pauling wrote, by way of example:

### 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 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.

## Interpretations

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.

## Summary

In classical mechanics, the energy of an oscillation or vibration can take on any value. However, in quantum mechanics, energy of an oscillator is gained and lost in "chunks" whose size is given by Planck's constant h times the frequency at which the oscillator would oscillate according to classical mechanics. 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 originally observed, but for a time scientists did not know anything more than their 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 involved, 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 colors to ultraviolet and higher frequencies, to be predicted because m could be a whole range of integers as long as any m was always larger than the corresponding n. 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.

Persons important for discovering and elaborating quantum theory:

The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus.

• Richard Feynman, 1985. QED: The Strange Theory of Light and Matter, Princeton University Press. ISBN 0-691-08388-6
• Ghirardi, GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra-ket notation can be passed over on a first reading.
• N. David Mermin, 1990, “Spooky actions at a distance: mysteries of the QT” in his Boojums all the way through. Cambridge Univ. Press: 110-176. The author is a rare physicist who tries to communicate to philosophers and humanists.
• Roland Omnes (1999) Understanding Quantum Mechanics. Princeton Univ. Press.
• Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 5-8.
• Veltman, M. J. G., 2003. Facts and Mysteries in Elementary Particle Physics. World Scientific Publishing Company.

## Notes

1. Richard P. Feynman, QED, p. 10
2. Aezel, Amir D., Entanglement, p. 35. (Penguin, 2003) ISBN 0-452-28457
3. Dicke and Wittke, Introduction to Quantum Mechanics, p. 12
4. Robert H. Dicke and James P. Wittke, 1960. Introduction to Quantum Mechanics. Addison-Wesley: 9f.
5. For the length of time involved, see George Gamow's One, Two, Three...Infinity, p. 140.
6. In this case, the energy of the electron is the sum of its kinetic and potential energies. The electron has kinetic energy by virtue of its motion around the nucleus, and potential energy because of its electromagnetic interaction with the nucleus.
7. This follows from Planck's equation , since the wavelength of light is related to its frequency by .
8. See Linus Pauling, The Nature of the Chemical Bond,
9. A very clear explanation of interference in thin films may be found in Sears, op. cit., p. 203ff.
10. A.S. Eddington, The Nature of the Physical World, the course of Gifford Lectures that Eddington delivered in the University of Edinburgh in January to March 1927, Kessinger Publishing, 2005, p. 201.
11. Aezel, Amir D., Entanglrment, p. 51f. (Penguin, 2003) ISBN 0-452-28457
12. Heisenberg's paper of 1925 is translated in B. L. Van der Waerden's Sources of Quantum Mechanics, where it appears as chapter 12.
13. Aitchison, et al., "Understanding Heisenberg's 'magical' paper of July 1925: a new look at the calculational details," p. 2
14. Thomas F. Jordan, Quantum Mechanics in Simmple Matrix Form, p. 149
15. W. Moore, Schrödinger: Life and Thought, Cambridge University Press (1989), p. 222.
16. Born's Nobel lecture quoted in Thomas F. Jordan's Quantum Mechanics in Simple Matrix Form, p. 6
17. See Introduction to quantum mechanics. by Henrik Smith, p. 58 for a readable introduction. See Ian J. R. Aitchison, et al., "Understanding Heisenberg's 'magical' paper of July 1925," Appendix A, for a mathematical derivation of this relationship.
18. Linus Pauling, The Nature of the Chemical Bond, p. 47
19. E. Schrödinger, Proceedings of the Cambridge Philosophical Society, 31 (1935), p. 555says: "When two systems, of which we know the states by their respective representation, enter into a temporary physical interaction due to known forces between them and when after a time of mutual influence the systems separate again, then they can no longer be described as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics."

## References

• Bernstein, Jeremy, 2005, "Max Born and the quantum theory," Am. J. Phys. 73(11).
• Beller, Mara, 2001. Quantum Dialogue: The Making of a Revolution. University of Chicago Press.
• Louis de Broglie, 1953. The Revolution in Physics. Noonday Press.
• Albert Einstein, 1934. Essays in Science. Philosophical Library.
• Herbert Feigl and May Brodbeck, 1953. Readings in the Philosophy of Science, Appleton-Century-Crofts.
• Fowler, Michael, 1999. The Bohr Atom. Lecture series, University of Virginia.
• Werner Heisenberg, 1958. Physics and Philosophy. Harper and Brothers.
• Lakshmibala, S., 2004, "Heisenberg, Matrix Mechanics and the Uncertainty Principle," Resonance, Journal of Science Education 9(8).
• Richard L. Liboff, 1992. Introductory Quantum Mechanics, 2nd ed.
• Lindsay, Robert Bruce and Henry Margenau, 1936. Foundations of Physics. Dover.
• McEvoy, J.P., and Zarate, Oscar. Introducing Quantum Theory, ISBN 1874166374
• Nave, Carl Rod, 2005. Hyperphysics-Quantum Physics, Department of Physics and Astronomy, Georgia State University, CD.
• Peat, F. David, 2002. From Certainty to Uncertainty: The Story of Science and Ideas in the Twenty-First Century. Joseph Henry Press.
• Hans Reichenbach, 1944. Philosophic Foundations of Quantum Mechanics. University of California Press.
• Paul Arthur Schilpp, 1949. Albert Einstein: Philosopher-Scientist. Tudor Publishing Company.
• Sears, Francis Weston, 1949. Optics. Addison-Wesley.
• ; cited in:
• Van Vleck, J. H.,1928, "The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics," Proc. Nat. Acad. Sci. 14: 179.
• Wieman, Carl, and Perkins, Katherine, 2005, "Transforming Physics Education," Physics Today.
• Westmoreland, M. D., and Schumacher, B., 1998, " Quantum Entanglement and the Nonexistence of Superluminal Signals."
• Bronner, P., Strunz, A. et al. (2009). " Demonstrating quantum random with single photons." European Journal of Physics 30(5): 1189-1200

1. Richard P. Feynman, QED, p. 10
2. Aezel, Amir D., Entanglement, p. 35. (Penguin, 2003) ISBN 0-452-28457
3. Dicke and Wittke, Introduction to Quantum Mechanics, p. 12
4. Robert H. Dicke and James P. Wittke, 1960. Introduction to Quantum Mechanics. Addison-Wesley: 9f.
5. For the length of time involved, see George Gamow's One, Two, Three...Infinity, p. 140.
6. In this case, the energy of the electron is the sum of its kinetic and potential energies. The electron has kinetic energy by virtue of its motion around the nucleus, and potential energy because of its electromagnetic interaction with the nucleus.
7. This follows from Planck's equation , since the wavelength of light is related to its frequency by .
8. See Linus Pauling, The Nature of the Chemical Bond,
9. A very clear explanation of interference in thin films may be found in Sears, op. cit., p. 203ff.
10. A.S. Eddington, The Nature of the Physical World, the course of Gifford Lectures that Eddington delivered in the University of Edinburgh in January to March 1927, Kessinger Publishing, 2005, p. 201.
11. Aezel, Amir D., Entanglrment, p. 51f. (Penguin, 2003) ISBN 0-452-28457
12. Heisenberg's paper of 1925 is translated in B. L. Van der Waerden's Sources of Quantum Mechanics, where it appears as chapter 12.
13. Aitchison, et al., "Understanding Heisenberg's 'magical' paper of July 1925: a new look at the calculational details," p. 2
14. Thomas F. Jordan, Quantum Mechanics in Simmple Matrix Form, p. 149
15. W. Moore, Schrödinger: Life and Thought, Cambridge University Press (1989), p. 222.
16. Born's Nobel lecture quoted in Thomas F. Jordan's Quantum Mechanics in Simple Matrix Form, p. 6
17. See Introduction to quantum mechanics. by Henrik Smith, p. 58 for a readable introduction. See Ian J. R. Aitchison, et al., "Understanding Heisenberg's 'magical' paper of July 1925," Appendix A, for a mathematical derivation of this relationship.
18. Linus Pauling, The Nature of the Chemical Bond, p. 47
19. E. Schrödinger, Proceedings of the Cambridge Philosophical Society, 31 (1935), p. 555says: "When two systems, of which we know the states by their respective representation, enter into a temporary physical interaction due to known forces between them and when after a time of mutual influence the systems separate again, then they can no longer be described as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics."