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Quantum mechanics
\Delta x\, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction · Mathematical formulation

Quantum mechanics is the set of scientific principles describing the behavior of energy and matter on the atomic and subatomatic scale. Much like the universe on the large and very vast scale (i.e., general relativity), so the universe on the small scale (i.e., quantum mechanics) does not neatly conform to the rules of classical physics. As such, it presents a set of rules that is counterintuitive and difficult to understand for the human mind, as humans are accustomed to the world on a scale dominated by classical physics. In other words, quantum mechanics deals with "Nature as She is—absurd."[1]

Many elementary parts of the universe, such as photons (discrete units of light) have some behaviours which resemble a particle but other behaviours that resemble a wave. 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 colour coded series. The energies, the colours, and the spectral intensities of electromagnetic radiation produced are all interconnected by laws. 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 (such as momentum) must fluctuate. 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 determine something about its entangled twin even if it is millions and millions of miles away.

Around the turn of the twentieth century, it became clear that classical physics was unable to explain several phenomena. Understanding these limitations of classical physics led to a revolution in physics: the development of quantum mechanics in the early decades of the last century.


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.[2] 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.[3] 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".[note 1] The quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator; the constant of proportionality is 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[4]

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".[5] 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.[6]

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.[7] 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.[8] 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.[8] An electron is likely to be struck only by a single photon; this photon imparts at most an energy hf to the electron.[8] Therefore, the intensity of the beam has no effect;[note 2] only its frequency determines the maximum energy that can be imparted to the electrons.[8]

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.[8] 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 less 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."[10][note 3]

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.[note 4] 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.[12] 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.[13] 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:[14]

 \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.[14]

The Bohr model of the atom

The Bohr model of the atom, showing electron quantum jumping to ground state n = 1.

In 1913, Niels Bohr applied the notion of quantisation to electron orbits, particularly in the case of the hydrogen atom.[15] Bohr theorised that the angular momentum, L, of an electron is quantised:

L = n\frac{h}{2\pi},

where n is a positive integer and h is the Planck constant. Starting from this assumption, Coulomb's law and the equations of circular motion show that an electron with n units of angular momentum will orbit a proton at a distance r given by

r = \frac{n^2 h^2}{4 \pi^2 k_e m e^2},

where ke is the Coulomb constant, m is the mass of an electron, and e is the charge on an electron. For simplicity this is written as

r = n^2 a_0,\!

where a0, called the Bohr radius, is equal to 0.0529 nm. The Bohr radius is the radius of the smallest allowed orbit.

The energy of the electron[note 5] can also be calculated, and is given by

E = -\frac{k_{\mathrm{e}}e^2}{2a_0} \frac{1}{n^2}.

Thus Bohr's assumption that angular momentum is quantised means that an electron can only inhabit certain orbits around the nucleus, and that it may have only certain energies. 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 a0.

An electron can lose energy by jumping instantaneously from its original orbit to a lower orbit; the extra energy is emitted in the form of a photon. Conversely, an electron that absorbs a photon gains energy, so it jumps to an orbit that is farther from the nucleus.

Each photon from glowing atomic hydrogen is due to an electron moving from a higher orbit, with radius 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_0}\left(\frac{1}{m^2}-\frac{1}{n^2}\right)

Since Planck's equation shows that the photon's energy is related to its wavelength by Eγ = hc/λ, the wavelengths of light which can be emitted are given by

\frac{1}{\lambda} = \frac{k_{\mathrm{e}}e^2}{2 a_0 h c}\left(\frac{1}{m^2}-\frac{1}{n^2}\right).

This equation has the same form as the Rydberg formula, and predicts that the constant R should be given by

R = \frac{k_{\mathrm{e}}e^2}{2 a_0 h c} .

Therefore the Bohr model of the atom can predict the emission spectrum of hydrogen in terms of fundamental constants.[note 6]

Later refinements

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.[16] 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.[17]

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[note 7] 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.[18] The wavelength, λ, associated with a particle is related to its momentum, p:[19][20]

 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 interferencee 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.[21]

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)"[22] 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: 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.[23]

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 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[24]), "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:[25]

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.[26]]

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.

Wave function 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."[27]

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:

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 quantum number, 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."[27]

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." [28] The question of whether entanglement is a real condition is still in dispute.[29] 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).[30] 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."[31]

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.

See also

Persons important for discovering and elaborating quantum theory:


  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,[9] 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[11]
  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.
  5. ^ 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.
  6. ^ The model can be easily modified to account of the emission spectrum of any system consisting of a nucleus and a single electron (that is, ions such as He+ or O7+ which contain only one electron).
  7. ^ See Linus Pauling, The Nature of the Chemical Bond.


  1. ^ Richard P. Feynman, QED, p. 10
  2. ^ Landau, L. D.; E. M. Lifshitz (1996). Statistical Physics (3rd Edition Part 1 ed.). Oxford: Butterworth-Heinemann. 
  3. ^ This was published (in German) as Planck, Max (1901), "Ueber das Gesetz der Energieverteilung im Normalspectrum", Ann. Phys. 309 (3): 553–63, doi:10.1002/andp.19013090310, . English translation: "On the Law of Distribution of Energy in the Normal Spectrum".
  4. ^ Francis Weston Sears (1958). Mechanics, Wave Motion, and Heat. Addison-Wesley. pp. 33–43. ISBN 537f. 
  5. ^ "The Nobel Prize in Physics 1918". The Nobel Foundation. Retrieved 2009-08-01. 
  6. ^ Kragh, Helge (1 December 2000), Max Planck: the reluctant revolutionary,, 
  7. ^ Einstein, Albert (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt". Annalen der Physik 17: 132–148. , translated into English as On a Heuristic Viewpoint Concerning the Production and Transformation of Light. The term "photon" was introduced in 1926.
  8. ^ a b c d e Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. pp. 127–9. 
  9. ^
  10. ^ Dicke and Wittke, Introduction to Quantum Mechanics, p. 12
  11. ^
  12. ^ World Book Encyclopedia, page 6, 2007.
  13. ^ Dicke and Wittke, Introduction to Quantum Mechanics, p. 10f.
  14. ^ a b Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. pp. 147–8. 
  15. ^ McEvoy, J. P.; Zarate, O. (2004). Introducing Quantum Theory. Totem Books. pp. 70–89, especially p. 89. ISBN 1840465778. 
  16. ^ J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 90f. ISBN 1-84046-577-8. 
  17. ^ J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 92f. ISBN 1-84046-577-8. 
  18. ^ J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 110f. ISBN 1-84046-577-8. 
  19. ^ Aezel, Amir D., Entanglrment, p. 51f. (Penguin, 2003) ISBN 0-452-28457
  20. ^ J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 114. ISBN 1-84046-577-8. 
  21. ^ Heisenberg's paper of 1925 is translated in B. L. Van der Waerden's Sources of Quantum Mechanics, where it appears as chapter 12.
  22. ^ Aitchison, et al., "Understanding Heisenberg's 'magical' paper of July 1925: a new look at the calculational details," p. 2
  23. ^ Thomas F. Jordan, Quantum Mechanics in Simple Matrix Form, p. 149
  24. ^ W. Moore, Schrödinger: Life and Thought, Cambridge University Press (1989), p. 222.
  25. ^ Born's Nobel lecture quoted in Thomas F. Jordan's Quantum Mechanics in Simple Matrix Form, p. 6
  26. ^ 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.
  27. ^ a b Linus Pauling, The Nature of the Chemical Bond, p. 47
  28. ^ 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."
  29. ^ "Quantum Nonlocality and the Possibility of Superluminal Effects", John G. Cramer,
  30. ^ J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. pp. 152110–115. ISBN 1-84046-577-8. 
  31. ^ J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 152f. ISBN 1-84046-577-8. 

Further reading

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.

External links

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