Euler's identity

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Euler%27s_identity

In mathematics, Euler's identity (also known as Euler's equation) is the equality

$${\displaystyle e^{i\pi }+1=0}$$

where

e is Euler's number, the base of natural logarithms,

i is the imaginary unit, which by definition satisfies i² = −1, and

π is pi, the ratio of the circumference of a circle to its diameter.

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula ${\displaystyle e^{ix}=\cos x+i\sin x}$ when evaluated for x = π. Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that π is transcendental, which implies the impossibility of squaring the circle.

Mathematical beauty

Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

• The number 0, the additive identity.
• The number 1, the multiplicative identity.
• The number π (π = 3.1415...), the fundamental circle constant.
• The number e (e = 2.718...), also known as Euler's number, which occurs widely in mathematical analysis.
• The number i, the imaginary unit of the complex numbers.

Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Stanford University mathematics professor Keith Devlin has said, "like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence". And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being "of exquisite beauty".

Mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics". And Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity "is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth".

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler's identity as the "most beautiful theorem in mathematics". In another poll of readers that was conducted by Physics World in 2004, Euler's identity tied with Maxwell's equations (of electromagnetism) as the "greatest equation ever".

At least three books in popular mathematics have been published about Euler's identity:

• Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, by Paul Nahin (2011)
• A Most Elegant Equation: Euler's formula and the beauty of mathematics, by David Stipp (2017)
• Euler's Pioneering Equation: The most beautiful theorem in mathematics, by Robin Wilson (2018).

Explanations

Imaginary exponents

Main article: Euler's formula

See also: Complex exponents with a positive real base

In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)^N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ/N)^N. It can be seen that as N gets larger (1 + iπ/N)^N approaches a limit of −1.

Fundamentally, Euler's identity asserts that ${\displaystyle e^{i\pi }}$ is equal to −1. The expression ${\displaystyle e^{i\pi }}$ is a special case of the expression ${\displaystyle e^z}$, where z is any complex number. In general, ${\displaystyle e^z}$ is defined for complex z by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition is:

${\displaystyle e^z=\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{z}{n}\right)}^n.}$

Euler's identity therefore states that the limit, as n approaches infinity, of ${\displaystyle (1+i\pi /n{)}^n}$ is equal to −1. This limit is illustrated in the animation to the right.

Euler's formula for a general angle

Euler's identity is a special case of Euler's formula, which states that for any real number x,

${\displaystyle e^{ix}=\cos x+i\sin x}$

where the inputs of the trigonometric functions sine and cosine are given in radians.

In particular, when x = π,

${\displaystyle e^{i\pi }=\cos \pi +i\sin \pi .}$

Since

${\displaystyle \cos \pi =-1}$

and

${\displaystyle \sin \pi =0,}$

it follows that

${\displaystyle e^{i\pi }=-1+0i,}$

which yields Euler's identity:

${\displaystyle e^{i\pi }+1=0.}$

Geometric interpretation

Any complex number ${\displaystyle z=x+iy}$ can be represented by the point ${\displaystyle (x,y)}$ on the complex plane. This point can also be represented in polar coordinates as ${\displaystyle (r,\theta )}$, where r is the absolute value of z (distance from the origin), and ${\displaystyle \theta }$ is the argument of z (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of ${\displaystyle (r\cos \theta ,r\sin \theta )}$, implying that ${\displaystyle z=r(\cos \theta +i\sin \theta )}$. According to Euler's formula, this is equivalent to saying ${\displaystyle z=r{e}^{i\theta }}$.

Euler's identity says that ${\displaystyle -1=e^{i\pi }}$. Since ${\displaystyle e^{i\pi }}$ is ${\displaystyle r{e}^{i\theta }}$ for r = 1 and ${\displaystyle \theta =\pi }$, this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive x-axis is ${\displaystyle \pi }$ radians.

Additionally, when any complex number z is multiplied by ${\displaystyle e^{i\theta }}$, it has the effect of rotating z counterclockwise by an angle of ${\displaystyle \theta }$ on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point ${\displaystyle \pi }$ radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting ${\displaystyle \theta }$ equal to ${\displaystyle 2\pi }$ yields the related equation ${\displaystyle e^{2\pi i}=1,}$ which can be interpreted as saying that rotating any point by one turn around the origin returns it to its original position.

Generalizations

Euler's identity is also a special case of the more general identity that the nth roots of unity, for n > 1, add up to 0:

${\displaystyle \sum _{k=0}^{n-1}{e}^{2\pi i\frac{k}{n}}=0.}$

Euler's identity is the case where n = 2.

In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions. Let {i, j, k} be the basis elements; then,

${\displaystyle e^{\frac{1}{\sqrt{3}}(i\pm j\pm k)\pi }+1=0.}$

In general, given real a₁, a₂, and a₃ such that a₁² + a₂² + a₃² = 1, then,

${\displaystyle e^{\left(a_{1}i+a_{2}j+a_{3}k\right)\pi }+1=0.}$

For octonions, with real a_n such that a₁² + a₂² + ... + a₇² = 1, and with the octonion basis elements {i₁, i₂, ..., i₇},

${\displaystyle e^{\left(a_1{i}_1+a_2{i}_2+\cdots +a_7{i}_7\right)\pi }+1=0.}$

History

While Euler's identity is a direct result of Euler's formula, published in his monumental work of mathematical analysis in 1748, Introductio in analysin infinitorum, it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.

Robin Wilson states the following.

We've seen how it [Euler's identity] can easily be deduced from results of Johann Bernoulli and Roger Cotes, but that neither of them seem to have done so. Even Euler does not seem to have written it down explicitly – and certainly it doesn't appear in any of his publications – though he must surely have realized that it follows immediately from his identity [i.e. Euler's formula], e^ix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly….

Diamagnetism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Diamagnetism 

Pyrolytic carbon has one of the largest diamagnetic constants of any room temperature material. Here a pyrolytic carbon sheet is levitated by its repulsion from the strong magnetic field of neodymium magnets

Diamagnetism is the property of materials that are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attracted by a magnetic field. Diamagnetism is a quantum mechanical effect that occurs in all materials; when it is the only contribution to the magnetism, the material is called diamagnetic. In paramagnetic and ferromagnetic substances, the weak diamagnetic force is overcome by the attractive force of magnetic dipoles in the material. The magnetic permeability of diamagnetic materials is less than the permeability of vacuum, μ₀. In most materials, diamagnetism is a weak effect which can be detected only by sensitive laboratory instruments, but a superconductor acts as a strong diamagnet because it entirely expels any magnetic field from its interior (the Meissner effect).

Diamagnetism was first discovered when Anton Brugmans observed in 1778 that bismuth was repelled by magnetic fields. In 1845, Michael Faraday demonstrated that it was a property of matter and concluded that every material responded (in either a diamagnetic or paramagnetic way) to an applied magnetic field. On a suggestion by William Whewell, Faraday first referred to the phenomenon as diamagnetic (the prefix dia- meaning through or across), then later changed it to diamagnetism.

A simple rule of thumb is used in chemistry to determine whether a particle (atom, ion, or molecule) is paramagnetic or diamagnetic: If all electrons in the particle are paired, then the substance made of this particle is diamagnetic; If it has unpaired electrons, then the substance is paramagnetic.

Materials

Diamagnetic material interaction in magnetic field. On keeping diamagnetic materials in a magnetic field, the electron orbital motion changes in such a way that magnetic dipole moments are induced on the atoms / molecules in the direction opposite to the external magnetic field

Diamagnetism is a property of all materials, and always makes a weak contribution to the material's response to a magnetic field. However, other forms of magnetism (such as ferromagnetism or paramagnetism) are so much stronger such that, when different forms of magnetism are present in a material, the diamagnetic contribution is usually negligible. Substances where the diamagnetic behaviour is the strongest effect are termed diamagnetic materials, or diamagnets. Diamagnetic materials are those that some people generally think of as non-magnetic, and include water, wood, most organic compounds such as petroleum and some plastics, and many metals including copper, particularly the heavy ones with many core electrons, such as mercury, gold and bismuth. The magnetic susceptibility values of various molecular fragments are called Pascal's constants (named after Paul Pascal [fr]).

Diamagnetic materials, like water, or water-based materials, have a relative magnetic permeability that is less than or equal to 1, and therefore a magnetic susceptibility less than or equal to 0, since susceptibility is defined as χ_v = μ_v − 1. This means that diamagnetic materials are repelled by magnetic fields. However, since diamagnetism is such a weak property, its effects are not observable in everyday life. For example, the magnetic susceptibility of diamagnets such as water is χ_v = −9.05×10⁻⁶. The most strongly diamagnetic material is bismuth, χ_v = −1.66×10⁻⁴, although pyrolytic carbon may have a susceptibility of χ_v = −4.00×10⁻⁴ in one plane. Nevertheless, these values are orders of magnitude smaller than the magnetism exhibited by paramagnets and ferromagnets. Because χ_v is derived from the ratio of the internal magnetic field to the applied field, it is a dimensionless value.

In rare cases, the diamagnetic contribution can be stronger than paramagnetic contribution. This is the case for gold, which has a magnetic susceptibility less than 0 (and is thus by definition a diamagnetic material), but when measured carefully with X-ray magnetic circular dichroism, has an extremely weak paramagnetic contribution that is overcome by a stronger diamagnetic contribution.

Notable diamagnetic materials

Material	χv [× 10−5 (SI units)]
Superconductor	−105
Pyrolytic carbon	−40.9
Bismuth	−16.6
Neon	−6.74
Mercury	−2.9
Silver	−2.6
Carbon (diamond)	−2.1
Lead	−1.8
Carbon (graphite)	−1.6
Copper	−1.0
Water	−0.91

Superconductors

Transition from ordinary conductivity (left) to superconductivity (right). At the transition, the superconductor expels the magnetic field and then acts as a perfect diamagnet.

Superconductors may be considered perfect diamagnets (χ_v = −1), because they expel all magnetic fields (except in a thin surface layer) due to the Meissner effect.

Demonstrations

Curving water surfaces

If a powerful magnet (such as a supermagnet) is covered with a layer of water (that is thin compared to the diameter of the magnet) then the field of the magnet significantly repels the water. This causes a slight dimple in the water's surface that may be seen by a reflection in its surface.

Levitation

Main article: Magnetic levitation § Diamagnetic levitation

A live frog levitates inside a 32 mm (1.26 in) diameter vertical bore of a Bitter solenoid in a magnetic field of about 16 teslas at the Nijmegen High Field Magnet Laboratory.

Diamagnets may be levitated in stable equilibrium in a magnetic field, with no power consumption. Earnshaw's theorem seems to preclude the possibility of static magnetic levitation. However, Earnshaw's theorem applies only to objects with positive susceptibilities, such as ferromagnets (which have a permanent positive moment) and paramagnets (which induce a positive moment). These are attracted to field maxima, which do not exist in free space. Diamagnets (which induce a negative moment) are attracted to field minima, and there can be a field minimum in free space.

A thin slice of pyrolytic graphite, which is an unusually strongly diamagnetic material, can be stably floated in a magnetic field, such as that from rare earth permanent magnets. This can be done with all components at room temperature, making a visually effective and relatively convenient demonstration of diamagnetism.

The Radboud University Nijmegen, the Netherlands, has conducted experiments where water and other substances were successfully levitated. Most spectacularly, a live frog (see figure) was levitated.

In September 2009, NASA's Jet Propulsion Laboratory (JPL) in Pasadena, California announced it had successfully levitated mice using a superconducting magnet, an important step forward since mice are closer biologically to humans than frogs. JPL said it hopes to perform experiments regarding the effects of microgravity on bone and muscle mass.

Recent experiments studying the growth of protein crystals have led to a technique using powerful magnets to allow growth in ways that counteract Earth's gravity.

A simple homemade device for demonstration can be constructed out of bismuth plates and a few permanent magnets that levitate a permanent magnet.

Theory

The electrons in a material generally settle in orbitals, with effectively zero resistance and act like current loops. Thus it might be imagined that diamagnetism effects in general would be common, since any applied magnetic field would generate currents in these loops that would oppose the change, in a similar way to superconductors, which are essentially perfect diamagnets. However, since the electrons are rigidly held in orbitals by the charge of the protons and are further constrained by the Pauli exclusion principle, many materials exhibit diamagnetism, but typically respond very little to the applied field.

The Bohr–Van Leeuwen theorem proves that there cannot be any diamagnetism or paramagnetism in a purely classical system. However, the classical theory of Langevin for diamagnetism gives the same prediction as the quantum theory. The classical theory is given below.

Langevin diamagnetism

Paul Langevin's theory of diamagnetism (1905) applies to materials containing atoms with closed shells (see dielectrics). A field with intensity B, applied to an electron with charge e and mass m, gives rise to Larmor precession with frequency ω = eB / 2m. The number of revolutions per unit time is ω / 2π, so the current for an atom with Z electrons is (in SI units)

${\displaystyle I=-\frac{Z{e}^{2}B}{4\pi m}.}$

The magnetic moment of a current loop is equal to the current times the area of the loop. Suppose the field is aligned with the z axis. The average loop area can be given as ${\displaystyle \pi ⟨{\rho }^2⟩}$, where ${\displaystyle ⟨{\rho }^2⟩}$ is the mean square distance of the electrons perpendicular to the z axis. The magnetic moment is therefore

${\displaystyle \mu =-\frac{Z{e}^{2}B}{4m}⟨{\rho }^2⟩.}$

If the distribution of charge is spherically symmetric, we can suppose that the distribution of x,y,z coordinates are independent and identically distributed. Then ${\displaystyle ⟨x^2⟩=⟨y^2⟩=⟨z^2⟩=\frac{1}{3}⟨r^2⟩}$, where ${\displaystyle ⟨r^2⟩}$ is the mean square distance of the electrons from the nucleus. Therefore, ${\displaystyle ⟨{\rho }^2⟩=⟨x^2⟩+⟨y^2⟩=\frac{2}{3}⟨r^2⟩}$. If ${\displaystyle n}$ is the number of atoms per unit volume, the volume diamagnetic susceptibility in SI units is

${\displaystyle \chi =\frac{{\mu }_{0}n\mu }{B}=-\frac{{\mu }_0{e}^{2}Zn}{6m}⟨r^2⟩.}$

In atoms, Langevin susceptibility is of the same order of magnitude as Van Vleck paramagnetic susceptibility.

In metals

The Langevin theory is not the full picture for metals because there are also non-localized electrons. The theory that describes diamagnetism in a free electron gas is called Landau diamagnetism, named after Lev Landau, and instead considers the weak counteracting field that forms when the electrons' trajectories are curved due to the Lorentz force. Landau diamagnetism, however, should be contrasted with Pauli paramagnetism, an effect associated with the polarization of delocalized electrons' spins. For the bulk case of a 3D system and low magnetic fields, the (volume) diamagnetic susceptibility can be calculated using Landau quantization, which in SI units is

${\displaystyle \chi =-{\mu }_0\frac{e^2}{12{\pi }^{2}m\hslash }\sqrt{2m{E}_{\mathrm{F}}},}$

where ${\displaystyle E_{\mathrm{F}}}$ is the Fermi energy. This is equivalent to ${\displaystyle -{\mu }_0{\mu }_{\mathrm{B}}^{2}g(E_{\mathrm{F}})/3}$, exactly $-1/3$ times Pauli paramagnetic susceptibility, where ${\displaystyle {\mu }_{\mathrm{B}}=e\hslash /2m}$ is the Bohr magneton and ${\displaystyle g(E)}$ is the density of states (number of states per energy per volume). This formula takes into account the spin degeneracy of the carriers (spin ½ electrons).

In doped semiconductors the ratio between Landau and Pauli susceptibilities may change due to the effective mass of the charge carriers differing from the electron mass in vacuum, increasing the diamagnetic contribution. The formula presented here only applies for the bulk; in confined systems like quantum dots, the description is altered due to quantum confinement. Additionally, for strong magnetic fields, the susceptibility of delocalized electrons oscillates as a function of the field strength, a phenomenon known as the De Haas–Van Alphen effect, also first described theoretically by Landau.