Introduction to quantum mechanics

This article is a non-technical introduction to the subject.For the main encyclopedia article, see Quantum mechan-ics.Quantum mechanics is the science of the very small:

Refereed version

the body of scientific principles that explains the be-haviour of matter and its interactions with energy on thescale of atoms and subatomic particles.Classical physics explains matter and energy on a scalefamiliar to human experience, including the behaviour ofastronomical bodies. It remains the key to measurementfor much of modern science and technology. However,towards the end of the 19th century, scientists discoveredphenomena in both the large (macro) and the small (mi-cro) worlds that classical physics could not explain.[1] AsThomas Kuhn explains in his analysis of the philosophyof science, The Structure of Scientific Revolutions, comingto terms with these limitations led to two major revolu-tions in physics which created a shift in the original sci-entific paradigm: the theory of relativity and the develop-ment of quantummechanics.[2] This article describes howphysicists discovered the limitations of classical physicsand developed the main concepts of the quantum the-ory that replaced it in the early decades of the 20th cen-tury. These concepts are described in roughly the orderin which they were first discovered. For a more completehistory of the subject, see History of quantum mechanics.In this sense, the word quantum means the minimumamount of any physical entity involved in an interaction.Certain characteristics of matter can take only discretevalues.Light behaves in some respects like particles and in other

respects like waves. Matter—particles such as electronsand atoms—exhibits wavelike behaviour too. Some lightsources, including neon lights, give off only certain dis-crete frequencies of light. Quantum mechanics showsthat light, along with all other forms of electromagneticradiation, comes in discrete units, called photons, andpredicts its energies, colours, and spectral intensities.Some aspects of quantum mechanics can seem counter-intuitive or even paradoxical, because they describe be-haviour quite different from that seen at larger lengthscales. In the words of Richard Feynman, quantum me-chanics deals with “nature as She is – absurd”.[3] For ex-ample, the uncertainty principle of quantum mechanicsmeans that the more closely one pins down one measure-ment (such as the position of a particle), the less pre-cise another measurement pertaining to the same particle(such as its momentum) must become.

1 The first quantum theory: MaxPlanck and black-body radiation

Hot metalwork. The yellow-orange glow is the visible part of thethermal radiation emitted due to the high temperature. Every-thing else in the picture is glowing with thermal radiation as well,but less brightly and at longer wavelengths than the human eyecan detect. A far-infrared camera can observe this radiation.

Thermal radiation is electromagnetic radiation emittedfrom the surface of an object due to the object’s inter-nal energy. If an object is heated sufficiently, it starts toemit light at the red end of the spectrum, as it becomesred hot.Heating it further causes the colour to change from redto yellow, white, and blue, as light at shorter wavelengths

1

2 2 PHOTONS: THE QUANTISATION OF LIGHT

(higher frequencies) begins to be emitted. A perfect emit-ter is also a perfect absorber: when it is cold, such an ob-ject looks perfectly black, because it absorbs all the lightthat falls on it and emits none. Consequently, an idealthermal emitter is known as a black body, and the radia-tion it emits is called black-body radiation.In the late 19th century, thermal radiation had been fairlywell characterized experimentally.[note 1] However, classi-cal physics was unable to explain the relationship betweentemperatures and predominant frequencies of radiation.Physicists searched for a single theory that explained allthe experimental results.

1e-29

1e-28

1e-27

1e-26

1e-25

1e-24

1e-23

1e+8 1e+9

Rad

ianc

e / J

m -2

sr -1

Frequency / Hz

Wien

Planck

Rayleigh-Jeans

Predictions of the amount of thermal radiation of different fre-quencies emitted by a body. Correct values predicted by Planck’slaw (green) contrasted against the classical values of Rayleigh-Jeans law (red) and Wien approximation (blue).

The first model that was able to explain the full spec-trum of thermal radiation was put forward byMax Planckin 1900.[4] He proposed a mathematical model in whichthe thermal radiation was in equilibrium with a set ofharmonic oscillators. To reproduce the experimental re-sults, he had to assume that each oscillator produced aninteger number of units of energy at its single character-istic frequency, rather than being able to emit any arbi-trary amount of energy. In other words, the energy ofeach oscillator was quantized.[note 2] The quantum of en-ergy for each oscillator, according to Planck, was pro-portional to the frequency of the oscillator; the constantof proportionality is now known as the Planck constant.The Planck constant, usually written as h, has the valueof 6.63×10−34 J s. So, the energy E of an oscillator offrequency f is given by

E = nhf, where n = 1, 2, 3, . . . [5]

To change the colour of such a radiating body, it is nec-essary to change its temperature. Planck’s law explainswhy: increasing the temperature of a body allows it toemit more energy overall, and means that a larger propor-tion of the energy is towards the violet end of the spec-trum.

Planck’s law was the first quantum theory in physics, andPlanck won the Nobel Prize in 1918 “in recognition ofthe services he rendered to the advancement of Physics byhis discovery of energy quanta”.[6] At the time, however,Planck’s view was that quantization was purely a mathe-matical construct, rather than (as is now believed) a fun-damental change in our understanding of the world.[7]

2 Photons: the quantisation oflight

Albert Einstein in around 1905.

In 1905, Albert Einstein took an extra step. He suggestedthat quantisation was not just a mathematical construct,but that the energy in a beam of light actually occurs inindividual packets, which are now called photons.[8] Theenergy of a single photon is given by its frequency multi-plied by Planck’s constant:

E = hf

For centuries, scientists had debated between two pos-sible theories of light: was it a wave or did it insteadcomprise a stream of tiny particles? By the 19th cen-tury, the debate was generally considered to have beensettled in favour of the wave theory, as it was able toexplain observed effects such as refraction, diffraction,interference and polarization. James Clerk Maxwell hadshown that electricity, magnetism and light are all mani-festations of the same phenomenon: the electromagnetic

2.2 Consequences of the light being quantised 3

field. Maxwell’s equations, which are the complete setof laws of classical electromagnetism, describe light aswaves: a combination of oscillating electric and mag-netic fields. Because of the preponderance of evidencein favour of the wave theory, Einstein’s ideas were metinitially with great skepticism. Eventually, however, thephoton model became favoured. One of the most sig-nificant pieces of evidence in its favour was its ability toexplain several puzzling properties of the photoelectriceffect, described in the following section. Nonethe-less, the wave analogy remained indispensable for helpingto understand other characteristics of light: diffraction,refraction and interference.

2.1 The photoelectric effect

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

Main article: Photoelectric effect

In 1887, Heinrich Hertz observed that when light withsufficient frequency hits a metallic surface, it emitselectrons.[9] In 1902, Philipp Lenard discovered that themaximum possible energy of an ejected electron is re-lated to the frequency of the light, not to its intensity: ifthe frequency is too low, no electrons are ejected regard-less of the intensity. Strong beams of light toward thered end of the spectrum might produce no electrical po-tential at all, while weak beams of light toward the vio-let end of the spectrum would produce higher and highervoltages. The lowest frequency of light that can causeelectrons to be emitted, called the threshold frequency, isdifferent for different metals. This observation is at oddswith classical electromagnetism, which predicts that theelectron’s energy should be proportional to the intensityof the radiation.[10]:24 So when physicists first discovereddevices exhibiting the photoelectric effect, they initiallyexpected that a higher intensity of light would produce ahigher voltage from the photoelectric device.Einstein explained the effect by postulating that a beamof light is a stream of particles ("photons") and that, if the

beam is of frequency f, then each photon has an energyequal to hf.[9] An electron is likely to be struck only by asingle photon, which imparts at most an energy hf to theelectron.[9] Therefore, the intensity of the beam has noeffect[note 3] and only its frequency determines the maxi-mum energy that can be imparted to the electron.[9]

To explain the threshold effect, Einstein argued that ittakes a certain amount of energy, called the work func-tion and denoted by φ, to remove an electron from themetal.[9] This amount of energy is different for eachmetal. If the energy of the photon is less than the workfunction, then it does not carry sufficient energy to re-move the electron from the metal. The threshold fre-quency, f0, is the frequency of a photon whose energyis equal to the work function:

φ = hf0.

If f is greater than f0, the energy hf is enough to removean electron. The ejected electron has a kinetic energy,EK, which is, at most, equal to the photon’s energy minusthe energy needed to dislodge the electron from themetal:

EK = hf − φ = h(f − f0).

Einstein’s description of light as being composed of parti-cles extended Planck’s notion of quantised energy, whichis that a single photon of a given frequency, f, deliversan invariant amount of energy, hf. In other words, indi-vidual photons can deliver more or less energy, but onlydepending on their frequencies. In nature, single photonsare rarely encountered. The Sun and emission sourcesavailable in the 19th century emit vast numbers of pho-tons every second, and so the importance of the energycarried by each individual photon was not obvious. Ein-stein’s idea that the energy contained in individual unitsof light depends on their frequency made it possible to ex-plain experimental results that had hitherto seemed quitecounterintuitive. However, although the photon is a par-ticle, it was still being described as having the wave-likeproperty of frequency. Once again, the particle accountof light was being compromised[11][note 4].

2.2 Consequences of the light being quan-tised

The relationship between the frequency of electromag-netic radiation and the energy of each individual photonis why ultraviolet light can cause sunburn, but visible orinfrared light cannot. A photon of ultraviolet light willdeliver a high amount of energy – enough to contributeto cellular damage such as occurs in a sunburn. A photonof infrared light will deliver a lower amount of energy –only enough to warm one’s skin. So, an infrared lamp can

4 3 THE QUANTISATION OF MATTER: THE BOHR MODEL OF THE ATOM

warm a large surface, perhaps large enough to keep peo-ple comfortable in a cold room, but it cannot give anyonea sunburn.All photons of the same frequency have identical energy,and all photons of different frequencies have proportion-ally (order 1, E ₒ ₒ = hf ) different energies. However,although the energy imparted by photons is invariant atany given frequency, the initial energy state of the elec-trons in a photoelectric device prior to absorption of lightis not necessarily uniform. Anomalous results may oc-cur in the case of individual electrons. For instance, anelectron that was already excited above the equilibriumlevel of the photoelectric device might be ejected whenit absorbed uncharacteristically low frequency illumina-tion. Statistically, however, the characteristic behaviourof a photoelectric device will reflect the behaviour of thevast majority of its electrons, which will be at their equi-librium level. This point is helpful in comprehending thedistinction between the study of individual particles inquantum dynamics and the study of massed particles inclassical physics.

3 The quantisation of matter: theBohr model of the atom

By the dawn of the 20th century, evidence required amodel of the atom with a diffuse cloud of negativelycharged electrons surrounding a small, dense, positivelycharged nucleus. These properties suggested a model inwhich the electrons circle around the nucleus like planetsorbiting a sun.[note 5] However, it was also known that theatom in this model would be unstable: according to classi-cal theory, orbiting electrons are undergoing centripetalacceleration, and should therefore give off electromag-netic radiation, the loss of energy also causing them tospiral toward the nucleus, colliding with it in a fraction ofa second.A second, related, puzzle was the emission spectrum ofatoms. When a gas is heated, it gives off light only at dis-crete frequencies. For example, the visible light given offby hydrogen consists of four different colours, as shownin the picture below. The intensity of the light at differ-ent frequencies is also different. By contrast, white lightconsists of a continuous emission across the whole rangeof visible frequencies. By the end of the nineteenth cen-tury, a simple rule known as Balmer’s formula had beenfound which showed how the frequencies of the differ-ent lines were related to each other, though without ex-plaining why this was, or making any prediction aboutthe intensities. The formula also predicted some addi-tional spectral lines in ultraviolet and infrared light whichhad not been observed at the time. These lines were laterobserved experimentally, raising confidence in the valueof the formula.The mathematical formula describing hydrogen’s emis-

Emission spectrum of hydrogen. When excited, hydrogen gasgives off light in four distinct colours (spectral lines) in the vis-ible spectrum, as well as a number of lines in the infrared andultraviolet.

sion spectrum.

In 1885 the Swiss mathematician Johann Balmer discov-ered that each wavelength λ (lambda) in the visible spec-trum of hydrogen is related to some integer n by the equa-tion

λ = B

(n2

n2 − 4

)n = 3, 4, 5, 6

where B is a constant which Balmer determined to beequal to 364.56 nm.In 1888 Johannes Rydberg generalized and greatly in-creased the explanatory utility of Balmer’s formula. Hepredicted that λ is related to two integers n andm accord-ing to what is now known as the Rydberg formula:[13]

1

λ= R

(1

m2− 1

n2

),

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 wave-lengths of hydrogen by setting m = 2 and n = 3, 4, 5,6. It also predicts additional wavelengths in the emissionspectrum: for m = 1 and for n > 1, the emission spectrumshould contain certain ultraviolet wavelengths, and for m= 3 and n > 3, it should also contain certain infrared wave-lengths. Experimental observation of these wavelengthscame two decades later: in 1908 Louis Paschen foundsome of the predicted infrared wavelengths, and in 1914Theodore Lyman found some of the predicted ultravioletwavelengths.[13]

Note that both Balmer and Rydberg’s formulas involveintegers: in modern terms, they imply that some prop-erty of the atom is quantised. Understanding exactly whatthis property was, and why it was quantised, was a majorpart in the development of quantum mechanics, as willbe shown in the rest of this article.In 1913 Niels Bohr proposed a new model of the atomthat included quantized electron orbits: electrons still or-bit the nucleus much as planets orbit around the sun, butthey are only permitted to inhabit certain orbits, not toorbit at any distance.[14] When an atom emitted (or ab-sorbed) energy, the electron did not move in a continuoustrajectory from one orbit around the nucleus to another,

5

n = 1

n = 2

n = 3Increasing energy

of orbits

A photon is emittedwith energy E = hf

The Bohr model of the atom, showing an electron transitioningfrom one orbit to another by emitting a photon.

as might be expected classically. Instead, the electronwould jump instantaneously from one orbit to another,giving off the emitted light in the form of a photon.[15]The possible energies of photons given off by each ele-ment were determined by the differences in energy be-tween the orbits, and so the emission spectrum for eachelement would contain a number of lines.[16]

Niels Bohr as a young man

Starting from only one simple assumption about the rulethat the orbits must obey, the Bohr model was able to re-late the observed spectral lines in the emission spectrumof hydrogen to previously known constants. In Bohr’smodel the electron simply wasn't allowed to emit energy

continuously and crash into the nucleus: once it was inthe closest permitted orbit, it was stable forever. Bohr’smodel didn't explain why the orbits should be quantisedin that way, nor was it able to make accurate predictionsfor atoms with more than one electron, or to explain whysome spectral lines are brighter than others.Although some of the fundamental assumptions of theBohr model were soon found to be wrong, the key re-sult that the discrete lines in emission spectra are due tosome property of the electrons in atoms being quantisedis correct. The way that the electrons actually behave isstrikingly different from Bohr’s atom, and from what wesee in the world of our everyday experience; this mod-ern quantum mechanical model of the atom is discussedbelow.A more detailed explanation of the Bohr model.

Bohr theorised that the angular momentum, L, of an elec-tron is quantised:

L = nh

2π= nℏ

where n is an integer and h is the Planck constant. Startingfrom this assumption, Coulomb’s law and the equations ofcircular motion show that an electron with n units of an-gular momentum will orbit a proton at a distance r givenby

r =n2h2

4π2keme2

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

r = n2a0,

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 6] can also be calculated,and is given by

E = −kee2

2a0

1

n2

Thus Bohr’s assumption that angular momentum is quan-tised means that an electron can only inhabit certain or-bits around the nucleus, and that it can have only certainenergies. A consequence of these constraints is that theelectron will not crash into the nucleus: it cannot con-tinuously emit energy, and it cannot come closer to thenucleus than a0 (the Bohr radius).An electron loses energy by jumping instantaneouslyfrom its original orbit to a lower orbit; the extra energy

6 4 WAVE-PARTICLE DUALITY

is emitted in the form of a photon. Conversely, an elec-tron that absorbs a photon gains energy, hence it jumpsto an orbit that is farther from the nucleus.Each photon from glowing atomic hydrogen is due to anelectron moving from a higher orbit, with radius rn, toa lower orbit, rm. The energy Eᵧ of this photon is thedifference in the energies En and Em of the electron:

Eγ = En − Em =kee2

2a0

(1

m2− 1

n2

)Since Planck’s equation shows that the photon’s energy isrelated to its wavelength by Eᵧ = hc/λ, the wavelengths oflight that can be emitted are given by

1

λ=

kee2

2a0hc

(1

m2− 1

n2

).

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

R =kee2

2a0hc.

Therefore, the Bohr model of the atom can predict theemission spectrum of hydrogen in terms of fundamentalconstants.[note 7] However, it was not able to make accu-rate predictions for multi-electron atoms, or to explainwhy some spectral lines are brighter than others.

4 Wave-particle duality

Main article: Wave-particle dualityJust as light has both wave-like and particle-like proper-ties, matter also has wave-like properties.[17]

Matter behaving as a wave was first demonstrated exper-imentally for electrons: a beam of electrons can exhibitdiffraction, just like a beam of light or a water wave.[note 8]Similar wave-like phenomena were later shown for atomsand even small molecules.The wavelength, λ, associated with any object is related toits momentum, p, through the Planck constant, h:[18][19]

p =h

λ.

The relationship, called the de Broglie hypothesis, holdsfor all types of matter: all matter exhibits properties ofboth particles and waves.The concept of wave–particle duality says that neitherthe classical concept of “particle” nor of “wave” canfully describe the behaviour of quantum-scale objects,

Louis de Broglie in 1929. De Broglie won the Nobel Prize inPhysics for his prediction that matter acts as a wave, made in his1924 PhD thesis.

either photons or matter. Wave–particle duality is an ex-ample of the principle of complementarity in quantumphysics.[20][21][22][23][24] An elegant example of wave–particle duality, the double slit experiment, is discussedin the section below.

4.1 The double-slit experiment

Main article: Double-slit experimentIn the double-slit experiment, as originally performed by

The diffraction pattern produced when light is shone throughone slit (top) and the interference pattern produced by two slits(bottom). The much more complex pattern from two slits, withits small-scale interference fringes, demonstrates the wave-likepropagation of light.

7

Thomas Young and Augustin Fresnel in 1827, a beam oflight is directed through two narrow, closely spaced slits,producing an interference pattern of light and dark bandson a screen. If one of the slits is covered up, one mightnaively expect that the intensity of the fringes due to in-terference would be halved everywhere. In fact, a muchsimpler pattern is seen, a simple diffraction pattern. Clos-ing one slit results in a much simpler pattern diametricallyopposite the open slit. Exactly the same behaviour can bedemonstrated in water waves, and so the double-slit ex-periment was seen as a demonstration of the wave natureof light.

The double slit experiment for a classical particle, a wave, and aquantum particle demonstrating wave-particle duality

The double-slit experiment has also been performed us-ing electrons, atoms, and even molecules, and the sametype of interference pattern is seen. Thus it has beendemonstrated that all matter possesses both particle andwave characteristics.Even if the source intensity is turned down, so that onlyone particle (e.g. photon or electron) is passing throughthe apparatus at a time, the same interference pattern de-velops over time. The quantum particle acts as a wavewhen passing through the double slits, but as a particlewhen it is detected. This is a typical feature of quantumcomplementarity: a quantum particle will act as a wavein an experiment to measure its wave-like properties, andlike a particle in an experiment to measure its particle-like properties. The point on the detector screen whereany individual particle shows up will be the result of a ran-dom process. However, the distribution pattern of manyindividual particles will mimic the diffraction pattern pro-duced by waves.

4.2 Application to the Bohr model

De Broglie expanded the Bohr model of the atom byshowing that an electron in orbit around a nucleus couldbe thought of as having wave-like properties. In partic-ular, an electron will be observed only in situations thatpermit a standing wave around a nucleus. An example ofa standing wave is a violin string, which is fixed at bothends and can be made to vibrate. The waves created bya stringed instrument appear to oscillate in place, mov-ing from crest to trough in an up-and-down motion. The

wavelength of a standing wave is related to the length ofthe vibrating object and the boundary conditions. Forexample, because the violin string is fixed at both ends,it can carry standing waves of wavelengths 2l/n, wherel is the length and n is a positive integer. De Brogliesuggested that the allowed electron orbits were those forwhich the circumference of the orbit would be an integernumber of wavelengths. The electron’s wavelength there-fore determines that only Bohr orbits of certain distancesfrom the nucleus are possible. In turn, at any distancefrom the nucleus smaller than a certain value it would beimpossible to establish an orbit. The minimum possibledistance from the nucleus is called the Bohr radius.[25]

De Broglie’s treatment of quantum events served as astarting point for Schrödinger when he set out to constructa wave equation to describe quantum theoretical events.

5 Spin

Quantum spin versus classical magnet in the Stern-Gerlach ex-periment.

Main article: Spin (physics)See also: Stern–Gerlach experiment

In 1922, Otto Stern andWalther Gerlach shot silver atomsthrough an (inhomogeneous) magnetic field. In classi-cal mechanics, a magnet thrown through a magnetic fieldmay be, depending on its orientation (if it is pointingwith its northern pole upwards or down, or somewherein between), deflected a small or large distance upwardsor downwards. The atoms that Stern and Gerlach shotthrough the magnetic field acted in a similar way. How-ever, while the magnets could be deflected variable dis-tances, the atoms would always be deflected a constantdistance either up or down. This implied that the prop-erty of the atom which corresponds to the magnet’s orien-tation must be quantised, taking one of two values (eitherup or down), as opposed to being chosen freely from anyangle.Ralph Kronig originated the idea that particles such asatoms or electrons behave as if they rotate, or “spin”,about an axis. Spin would account for the missingmagnetic moment, and allow two electrons in the same

8 7 COPENHAGEN INTERPRETATION

orbital to occupy distinct quantum states if they “spun” inopposite directions, thus satisfying the exclusion princi-ple. The quantum number represented the sense (positiveor negative) of spin.The choice of orientation of the magnetic field used in theStern-Gerlach experiment is arbitrary. In the animationshown here, the field is vertical and so the atoms are de-flected either up or down. If the magnet is rotated a quar-ter turn, the atoms will be deflected either left or right.Using a vertical field shows that the spin along the verti-cal axis is quantised, and using a horizontal field showsthat the spin along the horizontal axis is quantised.If, instead of hitting a detector screen, one of the beamsof atoms coming out of the Stern-Gerlach apparatus ispassed into another (inhomogeneous) magnetic field ori-ented in the same direction, all of the atoms will be de-flected the same way in this second field. However, if thesecond field is oriented at 90° to the first, then half ofthe atoms will be deflected one way and half the other, sothat the atom’s spin about the horizontal and vertical axesare independent of each other. However, if one of thesebeams (e.g. the atoms that were deflected up then left) ispassed into a third magnetic field, oriented the same wayas the first, half of the atoms will go one way and halfthe other. The action of measuring the atoms’ spin withrespect to a horizontal field has changed their spin withrespect to a vertical field.The Stern-Gerlach experiment demonstrates a number ofimportant features of quantum mechanics:

• a feature of the natural world has been demonstratedto be quantised, and only able to take certain discretevalues

• particles possess an intrinsic angular momentumthat is closely analogous to the angular momentumof a classically spinning object

• measurement changes the system being measured inquantum mechanics. Only the spin of an object inone direction can be known, and observing the spinin another direction will destroy the original infor-mation about the spin.

• quantum mechanics is probabilistic: whether thespin of any individual atom sent into the apparatusis positive or negative is random.

6 Development of modern quan-tum mechanics

In 1925, Werner Heisenberg attempted to solve one ofthe problems that the Bohr model left unanswered, ex-plaining the intensities of the different lines in the hy-drogen emission spectrum. Through a series of mathe-matical analogies, he wrote out the quantum mechanical

analogue for the classical computation of intensities.[26]Shortly afterwards, Heisenberg’s colleague Max Born re-alised that Heisenberg’s method of calculating the proba-bilities for transitions between the different energy levelscould best be expressed by using the mathematical con-cept of matrices.[note 9]

In the same year, building on de Broglie’s hypothesis,Erwin Schrödinger developed the equation that describesthe behaviour of a quantum mechanical wave.[27] Themathematical model, called the Schrödinger equationafter its creator, is central to quantum mechanics, de-fines the permitted stationary states of a quantum sys-tem, and describes how the quantum state of a phys-ical system changes in time.[28] The wave itself is de-scribed by a mathematical function known as a "wavefunction". Schrödinger said that the wave function pro-vides the “means for predicting probability of measure-ment results”.[29]

Schrödinger was able to calculate the energy levels of hy-drogen by treating a hydrogen atom’s electron as a classi-cal wave, moving in a well of electrical potential createdby the proton. This calculation accurately reproduced theenergy levels of the Bohr model.In May 1926, Schrödinger proved that Heisenberg’smatrix mechanics and his own wave mechanics made thesame predictions about the properties and behaviour ofthe electron; mathematically, the two theories had anunderlying common form. Yet the two men disagreedon the interpretation of their mutual theory. For in-stance, Heisenberg accepted the theoretical prediction ofjumps of electrons between orbitals in an atom,[30] butSchrödinger hoped that a theory based on continuouswave-like properties could avoid what he called (as para-phrased byWilhelmWien) “this nonsense about quantumjumps.”[31]

7 Copenhagen interpretation

Main article: Copenhagen interpretationBohr, Heisenberg and others tried to explain whatthese experimental results and mathematical models re-ally mean. Their description, known as the Copenhageninterpretation of quantum mechanics, aimed to describethe nature of reality that was being probed by the mea-surements and described by the mathematical formula-tions of quantum mechanics.The main principles of the Copenhagen interpretationare:

1. A system is completely described by a wave func-tion, usually represented by the Greek letter ψ(“psi”). (Heisenberg)

2. How ψ changes over time is given by theSchrödinger equation.

7.1 Uncertainty principle 9

The Niels Bohr Institute in Copenhagen, which was a focal pointfor researchers in quantum mechanics and related subjects in the1920s and 1930s. Most of the world’s best known theoreticalphysicists spent time there.

3. The description of nature is essentially probabilistic.The probability of an event – for example, where onthe screen a particle will show up in the two slit ex-periment – is related to the square of the absolutevalue of the amplitude of its wave function. (Bornrule, due to Max Born, which gives a physical mean-ing to the wave function in the Copenhagen interpre-tation: the probability amplitude)

4. It is not possible to know the values of all ofthe properties of the system at the same time;those properties that are not known with precisionmust be described by probabilities. (Heisenberg’suncertainty principle)

5. Matter, like energy, exhibits a wave–particle dual-ity. An experiment can demonstrate the particle-likeproperties of matter, or its wave-like properties; butnot both at the same time. (Complementarity prin-ciple due to Bohr)

6. Measuring devices are essentially classical devices,andmeasure classical properties such as position andmomentum.

7. The quantum mechanical description of large sys-tems should closely approximate the classical de-scription. (Correspondence principle of Bohr andHeisenberg)

Various consequences of these principles are discussed inmore detail in the following subsections.

7.1 Uncertainty principle

Main article: Uncertainty principleSuppose it is desired to measure the position and speed ofan object – for example a car going through a radar speedtrap. It can be assumed that the car has a definite positionand speed at a particular moment in time. How accuratelythese values can be measured depends on the quality of

Werner Heisenberg at the age of 26. Heisenberg won the NobelPrize in Physics in 1932 for the work that he did at around thistime.[32]

the measuring equipment – if the precision of the mea-suring equipment is improved, it will provide a result thatis closer to the true value. It might be assumed that thespeed of the car and its position could be operationally de-fined and measured simultaneously, as precisely as mightbe desired.In 1927, Heisenberg proved that this last assumption isnot correct.[33] Quantum mechanics shows that certainpairs of physical properties, such as for example posi-tion and speed, cannot be simultaneously measured, nordefined in operational terms, to arbitrary precision: themore precisely one property is measured, or defined inoperational terms, the less precisely can the other. Thisstatement is known as the uncertainty principle. The un-certainty principle isn't only a statement about the accu-racy of our measuring equipment, but, more deeply, isabout the conceptual nature of the measured quantities– the assumption that the car had simultaneously definedposition and speed does not work in quantum mechan-ics. On a scale of cars and people, these uncertaintiesare negligible, but when dealing with atoms and electronsthey become critical.[34]

Heisenberg gave, as an illustration, the measurement ofthe position and momentum of an electron using a photonof light. In measuring the electron’s position, the higherthe frequency of the photon, themore accurate is themea-

10 7 COPENHAGEN INTERPRETATION

surement of the position of the impact, but the greater isthe disturbance of the electron, which absorbs a randomamount of energy, rendering the measurement obtainedof its momentum increasingly uncertain (momentum isvelocity multiplied by mass), for one is necessarily mea-suring its post-impact disturbed momentum from the col-lision products and not its original momentum. With aphoton of lower frequency, the disturbance (and henceuncertainty) in the momentum is less, but so is the accu-racy of the measurement of the position of the impact.[35]

The uncertainty principle shows mathematically that theproduct of the uncertainty in the position and momentumof a particle (momentum is velocity multiplied by mass)could never be less than a certain value, and that this valueis related to Planck’s constant.

7.2 Wave function collapse

Main article: Wave function collapse

Wave function collapse is a forced expression for what-ever just happened when it becomes appropriate to re-place the description of an uncertain state of a system bya description of the system in a definite state. Explana-tions for the nature of the process of becoming certainare controversial. At any time before a photon “showsup” on a detection screen it can only be described by aset of probabilities for where it might show up. When itdoes show up, for instance in the CCD of an electroniccamera, the time and the space where it interacted withthe device are known within very tight limits. However,the photon has disappeared, and the wave function hasdisappeared with it. In its place some physical change inthe detection screen has appeared, e.g., an exposed spotin a sheet of photographic film, or a change in electricpotential in some cell of a CCD.

7.3 Eigenstates and eigenvaluesFor a more detailed introduction to this subject,see: Introduction to eigenstates

Because of the uncertainty principle, statements aboutboth the position and momentum of particles can onlyassign a probability that the position or momentum willhave some numerical value. Therefore, it is necessaryto formulate clearly the difference between the state ofsomething that is indeterminate, such as an electron ina probability cloud, and the state of something having adefinite value. When an object can definitely be “pinned-down” in some respect, it is said to possess an eigenstate.In the Stern-Gerlach experiment discussed above, thespin of the atom about the vertical axis has two eigen-states: up and down. Before measuring it, we can onlysay that any individual atom has equal probability of be-ing found to have spin up or spin down. Themeasurement

process causes the wavefunction to collapse into one ofthe two states.The eigenstates of spin about the vertical axis are not si-multaneously eigenstates of spin about the horizontal axis,so this atom has equal probability of being found to haveeither value of spin about the horizontal axis. As de-scribed in the section above, measuring the spin about thehorizontal axis can allow an atom which was spin up tobecome spin down: measuring its spin about the horizon-tal axis collapses its wave function into one of the eigen-states of this measurement, which means it is no longer inan eigenstate of spin about the vertical axis, so can takeeither value.

7.4 The Pauli exclusion principle

Wolfgang Pauli

Main article: Pauli exclusion principle

In 1924, Wolfgang Pauli proposed a new quantum de-gree of freedom (or quantum number), with two possi-ble values, to resolve inconsistencies between observedmolecular spectra and the predictions of quantum me-chanics. In particular, the spectrum of atomic hydro-gen had a doublet, or pair of lines differing by a smallamount, where only one line was expected. Pauli formu-lated his exclusion principle, stating that “There cannotexist an atom in such a quantum state that two electronswithin [it] have the same set of quantum numbers.”[36]

A year later, Uhlenbeck and Goudsmit identified Pauli’snew degree of freedom with the property called spinwhose effects were observed in the Stern–Gerlach exper-iment.

7.5 Application to the hydrogen atom 11

7.5 Application to the hydrogen atom

Main article: Atomic orbital model

Bohr’s model of the atom was essentially a planetaryone, with the electrons orbiting around the nuclear “sun.”However, the uncertainty principle states that an electroncannot simultaneously have an exact location and velocityin the way that a planet does. Instead of classical orbits,electrons are said to inhabit atomic orbitals. An orbitalis the “cloud” of possible locations in which an electronmight be found, a distribution of probabilities rather thana precise location.[36] Each orbital is three dimensional,rather than the two dimensional orbit, and is often de-picted as a three-dimensional region within which thereis a 95 percent probability of finding the electron.[37]

Schrödinger was able to calculate the energy levels of hy-drogen by treating a hydrogen atom’s electron as a wave,represented by the "wave function" Ψ, in an electric po-tential well, V, created by the proton. The solutions toSchrödinger’s equation are distributions of probabilitiesfor electron positions and locations. Orbitals have a rangeof different shapes in three dimensions. The energies ofthe different orbitals can be calculated, and they accu-rately match the energy levels of the Bohr model.Within Schrödinger’s picture, each electron has fourproperties:

1. An “orbital” designation, indicating whether the par-ticle wave is one that is closer to the nucleus with lessenergy or one that is farther from the nucleus withmore energy;

2. The “shape” of the orbital, spherical or otherwise;

3. The “inclination” of the orbital, determining themagnetic moment of the orbital around the z-axis.

4. The “spin” of the electron.

The collective name for these properties is the quantumstate of the electron. The quantum state can be describedby giving a number to each of these properties; these areknown as the electron’s quantum numbers. The quantumstate of the electron is described by its wave function. ThePauli exclusion principle demands that no two electronswithin an atommay have the same values of all four num-bers.

The shapes of the first five atomic orbitals: 1s, 2s, 2px, 2py, and2pz. The colours show the phase of the wave function.

The first property describing the orbital is the principalquantum number, n, which is the same as in Bohr’s model.n denotes the energy level of each orbital. The possiblevalues for n are integers:

n = 1, 2, 3 . . .

The next quantum number, the azimuthal quantum num-ber, denoted l, describes the shape of the orbital. Theshape is a consequence of the angular momentum of theorbital. The angular momentum represents the resistanceof a spinning object to speeding up or slowing down underthe influence of external force. The azimuthal quantumnumber represents the orbital angular momentum of anelectron around its nucleus. The possible values for l areintegers from 0 to n − 1:

l = 0, 1, . . . , n− 1.

The shape of each orbital has its own letter as well. Thefirst shape is denoted by the letter s (a mnemonic being"sphere”). The next shape is denoted by the letter p andhas the form of a dumbbell. The other orbitals have morecomplicated shapes (see atomic orbital), and are denotedby the letters d, f, and g.The third quantum number, the magnetic quantum num-ber, describes the magnetic moment of the electron, andis denoted by ml (or simply m). The possible values forml are integers from −l to l:

ml = −l,−(l − 1), . . . , 0, 1, . . . , l.

The magnetic quantum number measures the compo-nent of the angular momentum in a particular direction.The choice of direction is arbitrary, conventionally thez-direction is chosen.The fourth quantum number, the spin quantum number(pertaining to the “orientation” of the electron’s spin) isdenoted ms, with values +1⁄2 or −1⁄2.The chemist Linus Pauling wrote, by way of example:

In the case of a helium atom with two elec-trons in the 1s orbital, the Pauli Exclusion Prin-ciple requires that the two electrons differ inthe value of one quantum number. Their val-ues of n, l, and ml are the same. Accordinglythey must differ in the value of ms, which canhave the value of +1⁄2 for one electron and −1⁄2for the other.”[36]

It is the underlying structure and symmetry of atomic or-bitals, and the way that electrons fill them, that leads tothe organisation of the periodic table. The way the atomicorbitals on different atoms combine to formmolecular or-bitals determines the structure and strength of chemicalbonds between atoms.

12 9 QUANTUM ENTANGLEMENT

8 Dirac wave equation

Main article: Dirac equationIn 1928, Paul Dirac extended the Pauli equation, which

Paul Dirac (1902–1984)

described spinning electrons, to account for special rel-ativity. The result was a theory that dealt properly withevents, such as the speed at which an electron orbits thenucleus, occurring at a substantial fraction of the speedof light. By using the simplest electromagnetic interac-tion, Dirac was able to predict the value of the magneticmoment associated with the electron’s spin, and found theexperimentally observed value, which was too large to bethat of a spinning charged sphere governed by classicalphysics. He was able to solve for the spectral lines ofthe hydrogen atom, and to reproduce from physical firstprinciples Sommerfeld's successful formula for the finestructure of the hydrogen spectrum.Dirac’s equations sometimes yielded a negative valuefor energy, for which he proposed a novel solution: heposited the existence of an antielectron and of a dynami-cal vacuum. This led to the many-particle quantum fieldtheory.

9 Quantum entanglement

Main article: Quantum entanglementThe Pauli exclusion principle says that two electrons inone system cannot be in the same state. Nature leavesopen the possibility, however, that two electrons can haveboth states “superimposed” over each of them. Recall

&

&

Sol Alpha Centauri

Alpha CentauriSol

or

Characteristic 1

Characteristic 2

Superposition of two quantum characteristics, and two resolutionpossibilities.

that the wave functions that emerge simultaneously fromthe double slits arrive at the detection screen in a stateof superposition. Nothing is certain until the superim-posed waveforms “collapse”. At that instant an electronshows up somewhere in accordance with the probabil-ity that is the square of the absolute value of the sum ofthe complex-valued amplitudes of the two superimposedwaveforms. The situation there is already very abstract. Aconcrete way of thinking about entangled photons, pho-tons in which two contrary states are superimposed oneach of them in the same event, is as follows:Imagine that the superposition of a state that can be men-tally labeled as blue and another state that can be men-tally labeled as red will then appear (in imagination, ofcourse) as a purple state. Two photons are produced asthe result of the same atomic event. Perhaps they areproduced by the excitation of a crystal that characteris-tically absorbs a photon of a certain frequency and emitstwo photons of half the original frequency. So the twophotons come out “purple.” If the experimenter now per-forms some experiment that will determine whether oneof the photons is either blue or red, then that experimentchanges the photon involved from one having a superpo-sition of “blue” and “red” characteristics to a photon thathas only one of those characteristics. The problem thatEinstein had with such an imagined situation was that ifone of these photons had been kept bouncing betweenmirrors in a laboratory on earth, and the other one hadtraveled halfway to the nearest star, when its twin wasmade to reveal itself as either blue or red, that meant thatthe distant photon now had to lose its “purple” status too.So whenever it might be investigated after its twin hadbeen measured, it would necessarily show up in the op-posite state to whatever its twin had revealed.In trying to show that quantummechanics was not a com-plete theory, Einstein started with the theory’s predic-tion that two or more particles that have interacted inthe past can appear strongly correlated when their vari-ous properties are later measured. He sought to explainthis seeming interaction in a classical way, through theircommon past, and preferably not by some “spooky ac-tion at a distance.” The argument is worked out in a fa-mous paper, Einstein, Podolsky, and Rosen (1935; ab-breviated EPR), setting out what is now called the EPRparadox. Assuming what is now usually called local re-alism, EPR attempted to show from quantum theory thata particle has both position and momentum simultane-

13

ously, while according to the Copenhagen interpretation,only one of those two properties actually exists and onlyat the moment that it is being measured. EPR concludedthat quantum theory is incomplete in that it refuses to con-sider physical properties which objectively exist in nature.(Einstein, Podolsky, & Rosen 1935 is currently Einstein’smost cited publication in physics journals.) In the sameyear, Erwin Schrödinger used the word “entanglement”and declared: “I would not call that one but rather thecharacteristic trait of quantum mechanics.”[38] The ques-tion of whether entanglement is a real condition is stillin dispute.[39] The Bell inequalities are the most powerfulchallenge to Einstein’s claims.

10 Quantum field theory

Main article: Quantum field theory

The idea of quantum field theory began in the late 1920swith British physicist Paul Dirac, when he attempted toquantise the electromagnetic field – a procedure for con-structing a quantum theory starting from a classical the-ory.A field in physics is “a region or space in which agiven effect (such as magnetism) exists.”[40] Other ef-fects that manifest themselves as fields are gravitationand static electricity.[41] In 2008, physicist Richard Ham-mond wrote that

Sometimes we distinguish between quan-tum mechanics (QM) and quantum field the-ory (QFT). QM refers to a system in which thenumber of particles is fixed, and the fields (suchas the electromechanical field) are continuousclassical entities. QFT ... goes a step furtherand allows for the creation and annihilation ofparticles . . . .

He added, however, that quantummechanics is often usedto refer to “the entire notion of quantum view.”[42]:108

In 1931, Dirac proposed the existence of particles thatlater became known as anti-matter.[43] Dirac shared theNobel Prize in Physics for 1933 with Schrödinger, “forthe discovery of new productive forms of atomic the-ory.”[44]

On its face, quantum field theory allows infinite numbersof particles, and leaves it up to the theory itself to pre-dict how many and with which probabilities or numbersthey should exist. When developed further, the theoryoften contradicts observation, so that its creation and an-nihilation operators can be empirically tied down. Fur-thermore, empirical conservation laws like that of mass-energy suggest certain constraints on the mathematicalform of the theory, which are mathematically speakingfinicky. The latter fact both serves to make quantum field

theories difficult to handle, but has also lead to furtherrestrictions on admissible forms of the theory; the com-plications are mentioned below under the rubrik of renor-malization.

11 Quantum electrodynamics

Main article: Quantum electrodynamics

Quantum electrodynamics (QED) is the name ofthe quantum theory of the electromagnetic force.Understanding QED begins with understandingelectromagnetism. Electromagnetism can be called“electrodynamics” because it is a dynamic interactionbetween electrical and magnetic forces. Electromag-netism begins with the electric charge.Electric charges are the sources of, and create, electricfields. An electric field is a field which exerts a forceon any particles that carry electric charges, at any pointin space. This includes the electron, proton, and evenquarks, among others. As a force is exerted, electriccharges move, a current flows and a magnetic field isproduced. The changing magnetic field, in turn causeselectric current (often moving electrons). The physi-cal description of interacting charged particles, electri-cal currents, electrical fields, and magnetic fields is calledelectromagnetism.In 1928 Paul Dirac produced a relativistic quantum the-ory of electromagnetism. This was the progenitor tomodern quantum electrodynamics, in that it had essentialingredients of the modern theory. However, the prob-lem of unsolvable infinities developed in this relativisticquantum theory. Years later, renormalization largelysolved this problem. Initially viewed as a suspect, pro-visional procedure by some of its originators, renormal-ization 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 depicted all possi-ble interactions pertaining to a given event. The diagramsshowed that the electromagnetic force is the interaction ofphotons between interacting particles.An example of a prediction of quantum electrodynamicswhich has been verified experimentally is the Lamb shift.This refers to an effect whereby the quantum nature of theelectromagnetic field causes the energy levels in an atomor ion to deviate slightly from what they would otherwisebe. As a result, spectral lines may shift or split.Similarly, within a freely propagating electromagneticwave, the current can also be just an abstract displacementcurrent, instead of involving charge carriers. In QED, itsfull description makes essential use of short lived virtualparticles. There, QED again validates an earlier, rathermysterious concept.

14 16 NOTES

12 Standard Model

Main article: Standard model

In the 1960s physicists realized that QED broke downat extremely high energies. From this inconsistency theStandardModel of particle physics was discovered, whichremedied the higher energy breakdown in theory. It isanother, extended quantum field theory which unifies theelectromagnetic and weak interactions into one theory.This is called the electroweak theory.Additionally the Standard Model contains a high energyunification of the electroweak theory with the strongforce, described by quantum chromodynamics. It alsopostulates a connection with gravity as yet another gaugetheory, but the connection is as of 2015 still poorly un-derstood. The theory’s prediction of the Higgs particle toexplain inertial mass has stood recent empirical tests atthe Large hadron collider, and thus the Standard modelis now considered the basic and more or less completedescription of particle physics as we know it.

13 Interpretations

Main article: Interpretations of quantum mechanics

The physical measurements, equations, and predictionspertinent to quantum mechanics are all consistent andhold a very high level of confirmation. However, thequestion of what these abstract models say about the un-derlying nature of the real world has received competinganswers.

14 Applications

Main article: Quantum mechanics: applications

Applications of quantummechanics include the laser, thetransistor, the electron microscope, and magnetic reso-nance imaging. A special class of quantum mechanicalapplications is related to macroscopic quantum phenom-ena such as superfluid helium and superconductors. Thestudy of semiconductors led to the invention of the diodeand the transistor, which are indispensable for modernelectronics.In even the simple light switch, quantum tunnelling is ab-solutely vital, as otherwise the electrons in the electriccurrent could not penetrate the potential barrier made upof a layer of oxide. Flash memory chips found in USBdrives also use quantum tunnelling, to erase their mem-ory cells.[45]

15 See also

16 Notes[1] A number of formulae had been created which were able

to describe some of the experimental measurements ofthermal radiation: how the wavelength at which the ra-diation is strongest changes with temperature is given byWien’s displacement law, the overall power emitted perunit area is given by the Stefan-Boltzmann law. The besttheoretical explanation of the experimental results was theRayleigh-Jeans law, which agrees with experimental re-sults well at large wavelengths (or, equivalently, low fre-quencies), but strongly disagrees at short wavelengths (orhigh frequencies). In fact, at short wavelengths, classicalphysics predicted that energy will be emitted by a hot bodyat an infinite rate. This result, which is clearly wrong, isknown as the ultraviolet catastrophe.

[2] The word quantum comes from the Latin word for “howmuch” (as does quantity). Something which is quan-tized, like the energy of Planck’s harmonic oscillators, canonly take specific values. For example, in most coun-tries money is effectively quantized, with the quantum ofmoney being the lowest-value coin in circulation. Me-chanics is the branch of science that deals with the actionof forces on objects. So, quantum mechanics is the partof mechanics that deals with objects for which particularproperties are quantized.

[3] Actually, there can be intensity-dependent effects, but atintensities achievable with non-laser sources, these effectsare unobservable.

[4] Einstein’s photoelectric effect equation can be derivedand explained without requiring the concept of “photons”.That is, the electromagnetic radiation can be treated as aclassical electromagnetic wave, as long as the electrons inthe material are treated by the laws of quantum mechan-ics. The results are quantitatively correct for thermal lightsources (the sun, incandescent lamps, etc) both for the rateof electron emission as well as their angular distribution.For more on this point, see [12]

[5] The classical model of the atom is called the planetarymodel, or sometimes the Rutherford model after ErnestRutherford who proposed it in 1911, based on the Geiger–Marsden gold foil experiment which first demonstrated theexistence of the nucleus.

[6] In this case, the energy of the electron is the sum of itskinetic and potential energies. The electron has kineticenergy by virtue of its actual motion around the nucleus,and potential energy because of its electromagnetic inter-action with the nucleus.

[7] The model can be easily modified to account for the emis-sion spectrum of any system consisting of a nucleus anda single electron (that is, ions such as He+ or O7+ whichcontain only one electron) but cannot be extended to anatom with two electrons like neutral helium.

[8] Electron diffraction was first demonstrated three years af-ter de Broglie published his hypothesis. At the University

15

of Aberdeen, George Thomson passed a beam of elec-trons through a thin metal film and observed diffractionpatterns, as would be predicted by the de Broglie hypothe-sis. At Bell Labs, Davisson andGermer guided an electronbeam through a crystalline grid. De Broglie was awardedthe Nobel Prize in Physics in 1929 for his hypothesis;Thomson and Davisson shared the Nobel Prize for Physicsin 1937 for their experimental work.

[9] For a somewhat more sophisticated look at how Heisen-berg transitioned from the old quantum theory andclassical physics to the new quantum mechanics, seeHeisenberg’s entryway to matrix mechanics.

17 References[1] Quantum Mechanics from National Public Radio

[2] Kuhn, Thomas S. The Structure of Scientific Revolutions.Fourth ed. Chicago; London: The University of ChicagoPress, 2012. Print.

[3] Feynman, Richard P. (1988). QED : the strange theory oflight and matter (1st Princeton pbk., seventh printing withcorrections. ed.). Princeton, N.J.: Princeton UniversityPress. p. 10. ISBN 978-0691024172.

[4] This result was published (in German) as Planck,Max (1901). “Ueber das Gesetz der Energiev-erteilung im Normalspectrum” (PDF). Ann. Phys.309 (3): 553–63. Bibcode:1901AnP...309..553P.doi:10.1002/andp.19013090310.. English translation:"On the Law of Distribution of Energy in the NormalSpectrum".

[5] Francis Weston Sears (1958). Mechanics, Wave Motion,and Heat. Addison-Wesley. p. 537.

[6] “The Nobel Prize in Physics 1918”. Nobel Foundation.Retrieved 2009-08-01.

[7] Kragh, Helge (1 December 2000). “Max Planck: the re-luctant revolutionary”. PhysicsWorld.com.

[8] Einstein, Albert (1905). "Über einen die Erzeu-gung und Verwandlung des Lichtes betreffenden heuris-tischen Gesichtspunkt” (PDF). Annalen der Physik17 (6): 132–148. Bibcode:1905AnP...322..132E.doi:10.1002/andp.19053220607., translated into Englishas On a Heuristic Viewpoint Concerning the Productionand Transformation of Light. The term “photon” was in-troduced in 1926.

[9] Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004).Modern Physics for Scientists and Engineers. PrenticeHall. pp. 127–9. ISBN 0-13-589789-0.

[10] Stephen Hawking, The Universe in a Nutshell, Bantam,2001.

[11] Dicke and Wittke, Introduction to Quantum Mechanics, p.12

[12] NTRS.NASA.gov

[13] Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004).Modern Physics for Scientists and Engineers. PrenticeHall. pp. 147–8. ISBN 0-13-589789-0.

[14] McEvoy, J. P.; Zarate, O. (2004). Introducing QuantumTheory. Totem \Books. pp. 70–89, especially p. 89.ISBN 1-84046-577-8.

[15] World Book Encyclopedia, page 6, 2007.

[16] Dicke and Wittke, Introduction to Quantum Mechanics, p.10f.

[17] J. P.McEvoy andOscar Zarate (2004). Introducing Quan-tum Theory. Totem Books. p. 110f. ISBN 1-84046-577-8.

[18] Aczel, Amir D., Entanglement, p. 51f. (Penguin, 2003)ISBN 978-1-5519-2647-6

[19] J. P.McEvoy andOscar Zarate (2004). Introducing Quan-tum Theory. Totem Books. p. 114. ISBN 1-84046-577-8.

[20] Zettili, Nouredine (2009). Quantum Mechanics: Conceptsand Applications. JohnWiley and Sons. pp. 26–27. ISBN0470026782.

[21] Selleri, Franco (2012). Wave-Particle Duality. SpringerScience and Business Media. p. 41. ISBN 1461533325.

[22] Podgorsak, Ervin B. (2013). Compendium to RadiationPhysics for Medical Physicists. Springer Science and Busi-ness Media. p. 88. ISBN 3642201865.

[23] Halliday, David; Resnick, Robert (2013). Fundamentalsof Physics, 10th Ed. JohnWiley and Sons. p. 1272. ISBN1118230612.

[24] Myers, Rusty L. (2006). The Basics of Physics. Green-wood Publishing Group. p. 172. ISBN 0313328579.

[25] Introducing Quantum Theory, p. 87

[26] Van der Waerden, B. L. (1967). Sources of QuantumMechanics (in German translated to English). Mineola,New York: Dover Publications. pp. 261–276. ReceivedJuly 29, 1925 SeeWerner Heisenberg’s paper, “Quantum-Theoretical Re-interpretation of Kinematic and Mechan-ical Relations” pp. 261-276

[27] Nobel Prize Organization. “Erwin Schrödinger - Bio-graphical”. Retrieved 28 March 2014. His great discov-ery, Schrödinger’s wave equation, was made at the end ofthis epoch-during the first half of 1926.

[28] “Schrodinger Equation (Physics),” Encyclopædia Britan-nica

[29] Erwin Schrödinger, “The Present Situation in QuantumMechanics”, p. 9. “This translation was originally pub-lished in Proceedings of the American Philosophical So-ciety, 124, 323-38, and then appeared as Section I.11of Part I of Quantum Theory and Measurement (J.A.Wheeler and W.H. Zurek, eds., Princeton universityPress, New Jersey 1983). This paper can be downloadedfrom http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html.”

16 18 BIBLIOGRAPHY

[30] Heisenberg, W. (1955). The development of the interpre-tation of the quantum theory, pp. 12–29 inNiels Bohr andthe Development of Physics: Essays dedicated to Niels Bohron the occasion of his seventieth birthday, edited by Pauli,W. with the assistance of Rosenfeld, L. and Weisskopf,V., Pergamon, London, p. 13: “the single quantum jump... is “factual” in nature”.

[31] W. Moore, Schrödinger: Life and Thought, CambridgeUniversity Press (1989), p. 222. See p. 227 forSchrödinger’s own words.

[32] Heisenberg’s Nobel Prize citation

[33] Heisenberg first published his work on the uncer-tainty principle in the leading German physics jour-nal Zeitschrift für Physik: Heisenberg, W. (1927)."Über den anschaulichen Inhalt der quantentheo-retischen Kinematik und Mechanik”. Z. Phys. 43(3–4): 172–198. Bibcode:1927ZPhy...43..172H.doi:10.1007/BF01397280.

[34] Nobel Prize in Physics presentation speech, 1932

[35] “Uncertainty principle,” Encyclopædia Britannica

[36] Linus Pauling, The Nature of the Chemical Bond, p. 47

[37] “Orbital (chemistry and physics),” Encyclopædia Britan-nica

[38] E. Schrödinger, Proceedings of the Cambridge Philosoph-ical Society, 31 (1935), p. 555, says: “When two systems,of which we know the states by their respective represen-tation, enter into a temporary physical interaction due toknown forces between them and when after a time of mu-tual influence the systems separate again, then they can nolonger be described as before, viz., by endowing each ofthem with a representative of its own. I would not call thatone but rather the characteristic trait of quantum mechan-ics.”

[39] “Quantum Nonlocality and the Possibility of Superlumi-nal Effects”, John G. Cramer, npl.washington.edu

[40] “Mechanics,” Merriam-Webster Online Dictionary

[41] “Field”, Encyclopædia Britannica

[42] Richard Hammond, The Unknown Universe, New PageBooks, 2008. ISBN 978-1-60163-003-2

[43] The Physical World website

[44] “The Nobel Prize in Physics 1933”. Nobel Foundation.Retrieved 2007-11-24.

[45] Durrani, Z. A. K.; Ahmed, H. (2008). Vijay Kumar, ed.Nanosilicon. Elsevier. p. 345. ISBN 978-0-08-044528-1.

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• Wieman, Carl; Perkins, Katherine (2005). “Trans-forming Physics Education”. Physics Today58 (11): 36. Bibcode:2005PhT....58k..36W.doi:10.1063/1.2155756.

19 Further reading

The following titles, all by working physicists, attempt tocommunicate quantum theory to lay people, using a min-imum of technical apparatus.

• Jim Al-Khalili (2003) Quantum: A Guide for thePerplexed. Weidenfield & Nicholson. ISBN 978-1780225340

• Chester, Marvin (1987) Primer of Quantum Me-chanics. John Wiley. ISBN 0-486-42878-8

• Brian Cox and Jeff Forshaw (2011) The QuantumUniverse. Allen Lane. ISBN 978-1-84614-432-5

• Richard Feynman (1985) QED: The Strange The-ory of Light and Matter. Princeton University Press.ISBN 0-691-08388-6

• Ford, Kenneth (2005) The Quantum World. Har-vard Univ. Press. Includes elementary particlephysics.

• Ghirardi, GianCarlo (2004) Sneaking a Look atGod’s Cards, Gerald Malsbary, trans. PrincetonUniv. Press. The most technical of the works citedhere. Passages using algebra, trigonometry, andbra–ket notation can be passed over on a first read-ing.

• Tony Hey and Walters, Patrick (2003) The NewQuantum Universe. Cambridge Univ. Press. In-cludes much about the technologies quantum theoryhas made possible. ISBN 978-0521564571

• Vladimir G. Ivancevic, Tijana T. Ivancevic (2008)Quantum leap: from Dirac and Feynman, across theuniverse, to human body and mind. World Scien-tific Publishing Company. Provides an intuitive in-troduction in non-mathematical terms and an intro-duction in comparatively basic mathematical terms.ISBN 978-9812819277

• N. David Mermin (1990) “Spooky actions at a dis-tance: mysteries of the QT” in his Boojums all theway through. Cambridge Univ. Press: 110–176.The author is a rare physicist who tries to commu-nicate to philosophers and humanists. ISBN 978-0521388801

• Roland Omnès (1999) Understanding Quantum Me-chanics. Princeton Univ. Press. ISBN 978-0691004358

• Victor Stenger (2000) Timeless Reality: Symme-try, Simplicity, and Multiple Universes. BuffaloNY: Prometheus Books. Chpts. 5–8. ISBN 978-1573928595

• Martinus Veltman (2003) Facts and Mysteries in El-ementary Particle Physics. World Scientific Publish-ing Company. ISBN 978-9812381491

• J. P. McEvoy and Oscar Zarate (2004). IntroducingQuantum Theory. Totem Books. ISBN 1-84046-577-8

18 20 EXTERNAL LINKS

20 External links• "Microscopic World – Introduction to QuantumMechanics." by Takada, Kenjiro, Emeritus profes-sor at Kyushu University

• Quantum Theory. at encyclopedia.com

• The spooky quantum

• The Quantum Exchange (tutorials and open sourcelearning software).

• Atoms and the Periodic Table

• Single and double slit interference

• Time-Evolution of a Wavepacket in a Square WellAn animated demonstration of a wave packet dis-persion over time.

• Experiments with single photons An introductioninto quantum physics with interactive experiments

• Carroll, Sean M. “Quantum Mechanics (an embar-rassment)". Sixty Symbols. Brady Haran for the Uni-versity of Nottingham.

• Comprehensive animations

19

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