Angular momentum
In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r and its momentum vector p = mv; this definition can be applied to each point in physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and the orbital angular momentum; the spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin; the total angular momentum of an object is the sum of orbital angular momenta.
The orbital angular momentum vector of a particle is always parallel and directly proportional to the orbital angular velocity vector ω of the particle, where the constant of proportionality depends on both the mass of the particle and its distance from origin. However, the spin angular momentum of the object is proportional but not always parallel to the spin angular velocity Ω, making the constant of proportionality a second-rank tensor rather than a scalar. Angular momentum is additive. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density over the entire body. Torque can be defined as the rate of change of angular momentum, analogous to force; the net external torque on any system is always equal to the total torque on the system. Therefore, for a closed system, the total torque on the system must be 0, which means that the total angular momentum of the system is constant; the conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, the precession of gyroscopes.
In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is. In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning that at any time, only one component can be measured with definite precision; because of this, it turns out that the notion of an elementary particle "spinning" about an axis does not exist. For technical reasons, elementary particles still possess a spin angular momentum, but this angular momentum does not correspond to spinning motion in the ordinary sense. Angular momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, treat it as a scalar. Angular momentum can be considered a rotational analog of linear momentum.
Thus, where linear momentum p is proportional to mass m and linear speed v, p = m v, angular momentum L is proportional to moment of inertia I and angular speed ω, L = I ω. Unlike mass, which depends only on amount of matter, moment of inertia is dependent on the position of the axis of rotation and the shape of the matter. Unlike linear speed, which does not depend upon the choice of origin, angular velocity is always measured with respect to a fixed origin; therefore speaking, L should be referred to as the angular momentum relative to that center. Because I = r 2 m for a single particle and ω = v r for circular motion, angular momentum can be expanded, L = r 2 m ⋅ v r, reduced to, L = r m v, the product of the radius of rotation r and the linear momentum of the particle p = m v, where v in this case is the equivalent linear speed at the radius; this simple analysis can apply to non-circular motion if only the component of the motion, perpendicular to the radius vector is considered. In that case, L
Quantum mechanics
Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large scale. Quantum mechanics differs from classical physics in that energy, angular momentum and other quantities of a bound system are restricted to discrete values. Quantum mechanics arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others; the modern theory is formulated in various specially developed mathematical formalisms.
In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position and other physical properties of a particle. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the laser, the transistor and semiconductors such as the microprocessor and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he described in a paper titled On the nature of light and colours.
This experiment played a major role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays; these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, the 1900 quantum hypothesis of Max Planck. Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics. Following Max Planck's solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect.
Around 1900–1910, the atomic theory and the corpuscular theory of light first came to be accepted as scientific fact. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, for which Niels Bohr developed his theory of the atomic structure, confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept introduced by Arnold Sommerfeld; this phase is known as old quantum theory. According to Planck, each energy element is proportional to its frequency: E = h ν, where h is Planck's constant. Planck cautiously insisted that this was an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.
In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material, he won the 1921 Nobel Prize in Physics for this work. Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle, with a discrete quantum of energy, dependent on its frequency; the foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wi
Periodic table
The periodic table known as the periodic table of elements, is a tabular display of the chemical elements, which are arranged by atomic number, electron configuration, recurring chemical properties. The structure of the table shows periodic trends; the seven rows of the table, called periods have metals on the left and non-metals on the right. The columns, called groups, contain elements with similar chemical behaviours. Six groups have accepted names as well as assigned numbers: for example, group 17 elements are the halogens. Displayed are four simple rectangular areas or blocks associated with the filling of different atomic orbitals; the organization of the periodic table can be used to derive relationships between the various element properties, to predict chemical properties and behaviours of undiscovered or newly synthesized elements. Russian chemist Dmitri Mendeleev published the first recognizable periodic table in 1869, developed to illustrate periodic trends of the then-known elements.
He predicted some properties of unidentified elements that were expected to fill gaps within the table. Most of his forecasts proved to be correct. Mendeleev's idea has been expanded and refined with the discovery or synthesis of further new elements and the development of new theoretical models to explain chemical behaviour; the modern periodic table now provides a useful framework for analyzing chemical reactions, continues to be used in chemistry, nuclear physics and other sciences. The elements from atomic numbers 1 through 118 have been discovered or synthesized, completing seven full rows of the periodic table; the first 94 elements all occur though some are found only in trace amounts and a few were discovered in nature only after having first been synthesized. Elements 95 to 118 have only been synthesized in nuclear reactors; the synthesis of elements having higher atomic numbers is being pursued: these elements would begin an eighth row, theoretical work has been done to suggest possible candidates for this extension.
Numerous synthetic radionuclides of occurring elements have been produced in laboratories. Each chemical element has a unique atomic number representing the number of protons in its nucleus. Most elements have differing numbers of neutrons among different atoms, with these variants being referred to as isotopes. For example, carbon has three occurring isotopes: all of its atoms have six protons and most have six neutrons as well, but about one per cent have seven neutrons, a small fraction have eight neutrons. Isotopes are never separated in the periodic table. Elements with no stable isotopes have the atomic masses of their most stable isotopes, where such masses are shown, listed in parentheses. In the standard periodic table, the elements are listed in order of increasing atomic number Z. A new row is started. Columns are determined by the electron configuration of the atom. Elements with similar chemical properties fall into the same group in the periodic table, although in the f-block, to some respect in the d-block, the elements in the same period tend to have similar properties, as well.
Thus, it is easy to predict the chemical properties of an element if one knows the properties of the elements around it. Since 2016, the periodic table has 118 confirmed elements, from element 1 to 118. Elements 113, 115, 117 and 118, the most recent discoveries, were confirmed by the International Union of Pure and Applied Chemistry in December 2015, their proposed names, moscovium and oganesson were announced by the IUPAC in June 2016 and made official in November 2016. The first 94 elements occur naturally. Of the 94 occurring elements, 83 are primordial and 11 occur only in decay chains of primordial elements. No element heavier than einsteinium has been observed in macroscopic quantities in its pure form, nor has astatine. A group or family is a vertical column in the periodic table. Groups have more significant periodic trends than periods and blocks, explained below. Modern quantum mechanical theories of atomic structure explain group trends by proposing that elements within the same group have the same electron configurations in their valence shell.
Elements in the same group tend to have a shared chemistry and exhibit a clear trend in properties with increasing atomic number. In some parts of the periodic table, such as the d-block and the f-block, horizontal similarities can be as important as, or more pronounced than, vertical similarities. Under an international naming convention, the groups are numbered numerically from 1 to 18 from the leftmost column to the rightmost column, they were known by roman numerals. In America, the roman numerals were followed by either an "A" if the group was in the s- or p-block, or a "B" if the group was in the d-block; the roman numerals used correspond to the last digit of today's naming convention (e.g. the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, its discovery was a significant landmark in the development of the subject; the equation is named after Erwin Schrödinger, who derived the equation in 1925, published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. In classical mechanics, Newton's second law is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force F on the system; those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation; the concept of a wave function is a fundamental postulate of quantum mechanics.
Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be unitary, must therefore be generated by the exponential of a self-adjoint operator, the quantum Hamiltonian. This derivation is explained below. In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular and subatomic systems, but macroscopic systems even the whole universe. Schrödinger's equation is central to all applications of quantum mechanics including quantum field theory which combines special relativity with quantum mechanics. Theories of quantum gravity, such as string theory do not modify Schrödinger's equation; the Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, the path integral formulation, developed chiefly by Richard Feynman.
Paul Dirac incorporated the Schrödinger equation into a single formulation. The form of the Schrödinger equation depends on the physical situation; the most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time: where i is the imaginary unit, ℏ = h 2 π is the reduced Planck constant, Ψ is the state vector of the quantum system, t is time, H ^ is the Hamiltonian operator. The position-space wave function of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector | r ⟩, it is a scalar function, expressed as Ψ = ⟨ r | Ψ ⟩. The momentum-space wave function can be defined as Ψ ~ = ⟨ p | Ψ ⟩, where | p ⟩ is the momentum eigenvector; the most famous example is the nonrelativistic Schrödinger equation for the wave function in position space Ψ of a single particle subject to a potential V, such as that due to an electric field. Where m is the particle's mass, ∇ 2 is the Laplacian.
This is a diffusion equation, but unlike the heat equation, this one is a wave equation given the imaginary unit present in the transient term. The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version; the general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is a classical approximation to reality and yields accurate results in many situations, but only to a certain extent. To apply the Schrödinger equation, write down the Hamiltonian for the system, accounting for the kinetic and potential energies of the particles constituting the system insert it into the Schrödinger equation; the resulting partial differential equation is solved for the wave function, which contains information about the system. The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states.
These states are important as their individual study simplifies the task of solving the time-dependent Schrödinger equation for any state. Stationary states can be described by a simpler form of the Schrödinger equation, the time-independe
Introduction to quantum mechanics
Quantum mechanics is the science of the small. It explains the behavior of matter and its interactions with energy on the scale of atoms and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the Moon. Classical physics is still used in much of modern technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large and the small worlds that classical physics could not explain; the desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the theory of relativity and the development of quantum mechanics. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century, it describes these concepts in the order in which they were first discovered.
For a more complete history of the subject, see History of quantum mechanics. Light behaves in other aspects like waves. Matter—the "stuff" of the universe consisting of particles such as electrons and atoms—exhibits wavelike behavior too; some light sources, such as neon lights, give off only certain frequencies of light. Quantum mechanics shows that light, along with all other forms of electromagnetic radiation, comes in discrete units, called photons, predicts its energies and spectral intensities. A single photon is a quantum, or smallest observable amount, of the electromagnetic field because a partial photon has never been observed. More broadly, quantum mechanics shows that many quantities, such as angular momentum, that appeared continuous in the zoomed-out view of classical mechanics, turn out to be quantized. Angular momentum is required to take on one of a set of discrete allowable values, since the gap between these values is so minute, the discontinuity is only apparent at the atomic level.
Many aspects of quantum mechanics are counterintuitive and can seem paradoxical, because they describe behavior quite different from that seen at larger scales. In the words of quantum physicist Richard Feynman, quantum mechanics deals with "nature as She is – absurd". For example, the uncertainty principle of quantum mechanics means that the more one pins down one measurement, the less accurate another measurement pertaining to the same particle must become. Thermal radiation is electromagnetic radiation emitted from the surface of an object due to the object's internal energy. If an object is heated sufficiently, it starts to emit light at the red end of the spectrum, as it becomes red hot. Heating it further causes the color to change from red to yellow and blue, as it emits light at shorter wavelengths. A perfect emitter is a perfect absorber: when it is cold, such an object looks black, because it absorbs all the light that falls on it and emits none. An ideal thermal emitter is known as a black body, the radiation it emits is called black-body radiation.
In the late 19th century, thermal radiation had been well characterized experimentally. However, classical physics led to the Rayleigh–Jeans law, which, as shown in the figure, agrees with experimental results well at low frequencies, but disagrees at high frequencies. Physicists searched for a single theory; the first model, able to explain the full spectrum of thermal radiation was put forward by Max Planck in 1900. He proposed a mathematical model in which the thermal radiation was in equilibrium with a set of harmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator emitted an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy emitted by an oscillator was quantized; the quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator. The Planck constant written as h, has the value of 6.63×10−34 J s.
So, the energy E of an oscillator of frequency f is given by E = n h f, where n = 1, 2, 3, … To change the color of such a radiating body, it is necessary to change its temperature. Planck's law explains why: increasing the temperature of a body allows it to emit more energy overall, means that a larger proportion of the energy is towards the violet end of the spectrum. Planck's law was the first quantum theory in physics, Planck won the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". At the time, Planck's view was that quantization was purely a heuristic mathematical construct, rather than a fundamental change in our understanding of the world. In 1905, Albert Einstein took an extra step, he suggested that quantization was not just a mathematical construct, but that the energy in a beam of light occurs in individual packets, which are now called photons. The energy of a single photon is given by its frequency multiplied by Planck's constant: E = h f For centuri
Planck constant
The Planck constant is a physical constant, the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant; the Planck constant is of fundamental importance in quantum mechanics, in metrology it is the basis for the definition of the kilogram. At the end of the 19th century, physicists were unable to explain why the observed spectrum of black body radiation, which by had been measured, diverged at higher frequencies from that predicted by existing theories. In 1900, Max Planck empirically derived a formula for the observed spectrum, he assumed that a hypothetical electrically charged oscillator in a cavity that contained black body radiation could only change its energy in a minimal increment, E, proportional to the frequency of its associated electromagnetic wave. He was able to calculate the proportionality constant, h, from the experimental measurements, that constant is named in his honor.
In 1905, the value E was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave, it was called a photon. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". Since energy and mass are equivalent, the Planck constant relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to replacing the metal cylinder, called the International Prototype of the Kilogram, that had defined the kilogram since 1889; the new definition was unanimously approved at the General Conference on Weights and Measures on 16 November 2018 as part of the 2019 redefinition of SI base units. For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.62607015×10−34 J⋅s exactly.
The kilogram was the last SI base unit to be re-defined by a fundamental physical property to replace a physical artefact. In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation first posed by Kirchhoff some 40 years earlier; every physical body continuously emits electromagnetic radiation. At low frequencies, Planck's law tends to the Rayleigh–Jeans law, while in the limit of high frequencies it tends to the Wien approximation but there was no overall expression or explanation for the shape of the observed emission spectrum. Approaching this problem, Planck hypothesized that the equations of motion for light describe a set of harmonic oscillators, one for each possible frequency, he examined how the entropy of the oscillators varied with the temperature of the body, trying to match Wien's law, was able to derive an approximate mathematical function for black-body spectrum. To create Planck's law, which predicts blackbody emissions by fitting the observed curves, he multiplied the classical expression by a complex factor that involves a constant, h, in both the numerator and the denominator, which subsequently became known as the Planck Constant.
The spectral radiance of a body, Bν, describes the amount of energy it emits at different radiation frequencies. It is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency. Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by B ν = 2 h ν 3 c 2 1 e h ν k B T − 1 where kB is the Boltzmann constant, h is the Planck constant, c is the speed of light in the medium, whether material or vacuum; the spectral radiance can be expressed per unit wavelength λ instead of per unit frequency. In this case, it is given by B λ = 2 h c 2 λ 5 1 e h c λ k B T − 1. Showing how radiated energy emitted at shorter wavelengths increases more with temperature than energy emitted at longer wavelengths; the law may be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation. The SI units of Bν are W·sr−1·m−2·Hz−1, while those of Bλ are W·sr−1·m−3.
Planck soon realized. There were several different solutions, each of which gave a different value for the entropy of the oscillators. To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics, which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics." One of his new boundary conditions was to interpret UN [the vibrational energy
Stern–Gerlach experiment
The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent through a spatially varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment are deflected, due to the magnetic field gradient, from a straight path; the screen reveals discrete points of accumulation, rather than a continuous distribution, owing to their quantized spin. This experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale systems; the experiment was first conducted by the German physicists Otto Stern and Walter Gerlach in 1922. The Stern–Gerlach experiment involves sending a beam of silver atoms through an inhomogeneous magnetic field and observing their deflection; the results show that particles possess an intrinsic angular momentum, analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values.
Another important result is that only one component of a particle's spin can be measured at one time, meaning that the measurement of the spin along the z-axis destroys information about a particle's spin along the x and y axis. The experiment is conducted using electrically neutral particles such as silver atoms; this avoids the large deflection in the path of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate. If the particle is treated as a classical spinning magnetic dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole. If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. However, if the magnetic field is inhomogeneous the force on one end of the dipole will be greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory.
If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by an amount proportional to its magnetic moment, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount; this was a measurement of the quantum observable now known as spin angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values or point spectrum. Although some discrete quantum phenomena, such as atomic spectra, were observed much earlier, the Stern–Gerlach experiment allowed scientists to observe separation between discrete quantum states for the first time in the history of science. By now, it is known that, quantum angular momentum of any kind has a discrete spectrum, sometimes expressed as "angular momentum is quantized".
If the experiment is conducted using charged particles like electrons, there will be a Lorentz force that tends to bend the trajectory in a circle. This force can be cancelled by an electric field of appropriate magnitude oriented transverse to the charged particle's path.i Electrons are spin-1⁄2 particles. These have only two possible spin angular momentum values measured along any axis, + ℏ 2 or − ℏ 2, a purely quantum mechanical phenomenon; because its value is always the same, it is regarded as an intrinsic property of electrons, is sometimes known as "intrinsic angular momentum". If one measures the spin along a vertical axis, electrons are described as "spin up" or "spin down", based on the magnetic moment pointing up or down, respectively. To mathematically describe the experiment with spin + 1 2 particles, it is easiest to use Dirac's bra–ket notation; as the particles pass through the Stern–Gerlach device, they are deflected either up or down, observed by the detector which resolves to either spin up or spin down.
These are described by the angular momentum quantum number j, which can take on one of the two possible allowed values, either + ℏ 2 or − ℏ 2. The act of observing the momentum along the z axis corresponds to the operator J z. In mathematical terms, the initial state of the particles is | ψ ⟩ = c 1 | ψ j = + ℏ 2 ⟩ + c 2 | ψ j = − ℏ 2 ⟩ where constants c 1 and c 2 are complex numbers; this initial state spin can point in any direction. The sq