The total angular momentum of the collection of particles is the sum of the angular momentum of each particle,. The first term is the angular momentum of the center of mass relative to the origin. Similar to Single particle , below, it is the angular momentum of one particle of mass M at the center of mass moving with velocity V. The second term is the angular momentum of the particles moving relative to the center of mass, similar to Fixed center of mass , below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass.

The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body. Rearranging equation 2 by vector identities, multiplying both terms by "one", and grouping appropriately,. In modern 20th century theoretical physics, angular momentum not including any intrinsic angular momentum — see below is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes , unless it happens to be asymptotically rotationally invariant.

In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element:. This has the advantage of a clearer geometric interpretation as a plane element, defined from the x and p vectors, and the expression is true in any number of dimensions two or higher. In Cartesian coordinates:.

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The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor: [26]. In relativistic mechanics , the relativistic angular momentum of a particle is expressed as an antisymmetric tensor of second order:. In each of the above cases, for a system of particles, the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system. Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics.

In relativistic quantum mechanics , it differs even more, in which the above relativistic definition becomes a tensorial operator. L is then an operator , specifically called the orbital angular momentum operator. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so 3. Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. However, in quantum physics, there is another type of angular momentum, called spin angular momentum , represented by the spin operator S. Almost all elementary particles have spin.

Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. Finally, there is total angular momentum J , which combines both the spin and orbital angular momentum of all particles and fields.

Conservation of angular momentum applies to J , but not to L or S ; for example, the spin—orbit interaction allows angular momentum to transfer back and forth between L and S , with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have fractional values.

In quantum mechanics , angular momentum is quantized — that is, it cannot vary continuously, but only in " quantum leaps " between certain allowed values. There are additional restrictions as well, see angular momentum operator for details. However, it is very important in the microscopic world.

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For example, the structure of electron shells and subshells in chemistry is significantly affected by the quantization of angular momentum. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision.

Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis. For the precise commutation relations , see angular momentum operator. The "exp" in the formula refers to operator exponential To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO 3 will act on it.

There is then an associated action of the Lie algebra so 3 of SO 3 ; the operators describing the action of so 3 on our Hilbert space are the total angular momentum operators. The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.

When describing the motion of a charged particle in an electromagnetic field , the canonical momentum P derived from the Lagrangian for this system is not gauge invariant. Instead, the momentum that is physical, the so-called kinetic momentum used throughout this article , is in SI units. The gauge-invariant angular momentum, that is kinetic angular momentum , is given by. The interplay with quantum mechanics is discussed further in the article on canonical commutation relations. In classical Maxwell electrodynamics the Poynting vector is a linear momentum density of electromagnetic field.

The above identities are valid locally , i. Newton , in the Principia , hinted at angular momentum in his examples of the First Law of Motion ,.

## Electron, Spin and Momentum Densities and Chemical Reactivity | Paul G. Mezey | Springer

However, his geometric proof of the law of areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force. As a planet orbits the Sun , the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's gravity was the cause of all of Kepler's laws. During the first interval of time, an object is in motion from point A to point B.

Undisturbed, it would continue to point c during the second interval. When the object arrives at B , it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S , such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition , these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. At point C , the object receives another impulse toward S , again deflecting its path during the third interval from d to D.

Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve. Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero. Similarly so for each of the triangles.

Leonhard Euler , Daniel Bernoulli , and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity , a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter. In Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them. Bernoulli wrote in a letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In , Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane. Louis Poinsot in began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments". William J.

Rankine's Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:. In an edition of the same book, Rankine stated that "The term angular momentum was introduced by Mr. Hayward," [38] probably referring to R. Hayward's article On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications, [39] which was introduced in , and published in Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries.

Before this, angular momentum was typically referred to as "momentum of rotation" in English. From Wikipedia, the free encyclopedia. Measure of the extent to which an object will continue to rotate in the absence of an applied torque. This gyroscope remains upright while spinning due to the conservation of its angular momentum.

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Second law of motion. History Timeline. Newton's laws of motion. Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton—Jacobi equation Appell's equation of motion Udwadia—Kalaba equation Koopman—von Neumann mechanics. This specific ISBN edition is currently not available.

View all copies of this ISBN edition:. Synopsis The electron density of a non-degenerate ground state system determines essentially all physical properties of the system. Buy New Learn more about this copy. Other Popular Editions of the Same Title. Search for all books with this author and title. Customers who bought this item also bought. Stock Image. New Hardcover Quantity Available: Book Depository hard to find London, United Kingdom. Seller Rating:. Published by Springer New Paperback Quantity Available: Mezey Editor , Beverly E.

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Popular Features. New Releases. Description The electron density of a non-degenerate ground state system determines essentially all physical properties of the system. This statement of the Hohenberg-Kohn theorem of Density Functional Theory plays an exceptionally important role among all the fundamental relations of Molecular Physics. In particular, the electron density distribution and the dynamic properties of this density determine both the local and global reactivities of molecules.

High resolution experimental electron densities are increasingly becoming available for more and more molecules, including macromolecules such as proteins. Furthermore, many of the early difficulties with the determination of electron densities in the vicinity of light nuclei have been overcome.

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These electron densities provide detailed information that gives important insight into the fundamentals of molecular structure and a better understanding of chemical reactions. The results of electron density analysis are used in a variety of applied fields, such as pharmaceutical drug discovery and biotechnology. If the functional form of a molecular electron density is known, then various molecular properties affecting reactivity can be determined by quantum chemical computational techniques or alternative approximate methods.

Product details Format Paperback pages Dimensions x x Other books in this series. Add to basket. Organometallic Ion Chemistry B. Molecular Similarity and Reactivity R. Molecular Theory of Solvation Fumio Hirata.

Molecular Design of Tautomeric Compounds V. Ultrafast Dynamics of Chemical Systems J. Entropy and Entropy Generation J. Table of contents 1.