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The fitting parameters formula_7 and formula_8 are related to material properties, where |
However, the fitting parameter formula_14 does not reproduce the total energy of the free atoms. |
The total energy equation is used to determine elastic and thermal material constants in quantum chemical simulation packages. |
In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the "Nahm transform" – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators. |
Deep study of the Nahm equations was carried out by Nigel Hitchin and Simon Donaldson. Conceptually, the equations arise in the process of infinite-dimensional hyperkähler reduction. Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the moduli space of monopoles; and the existence of hyperkähler structure on coadjoint orbits of complex semisimple Lie groups, proved by Peter Kronheimer, Olivier Biquard, and A.G. Kovalev. |
Let "T"1("z"),"T"2("z"), "T"3("z") be three matrix-valued meromorphic functions of a complex variable "z". The Nahm equations are a system of matrix differential equations |
together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the Levi-Civita symbol, in the form |
More generally, instead of considering "N" by "N" matrices, one can consider Nahm's equations with values in a Lie algebra g. |
The variable "z" is restricted to the open interval (0,2), and the following conditions are imposed: |
The Nahm equations can be written in the Lax form as follows. Set |
then the system of Nahm equations is equivalent to the Lax equation |
As an immediate corollary, we obtain that the spectrum of the matrix "A" does not depend on "z". Therefore, the characteristic equation |
which determines the so-called spectral curve in the twistor space "TP"1, is invariant under the flow in "z". |
In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the "N"-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter "q" approaches 1, the "N"-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics. |
In mathematical physics, the Gordon decomposition (named after Walter Gordon) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation. |
For any solution formula_1 of the massive Dirac equation, |
the Lorentz covariant number-current formula_3 may be expressed as |
is the spinor generator of Lorentz transformations. |
The corresponding momentum-space version for plane wave solutions formula_6 and formula_7 obeying |
One sees that from Dirac's equation that |
and, from the conjugate of Dirac's equation, |
From Dirac algebra, one may show that Dirac matrices satisfy |
which amounts to just the Gordon decomposition, after some algebra. |
The second, spin-dependent, part of the current coupled to the photon field, formula_17 yields, up to an ignorable total divergence, |
that is, an effective Pauli moment term, formula_19. |
This decomposition of the current into a particle number-flux (first term) and bound spin contribution (second term) requires formula_20. |
If one assumed that the given solution has energy formula_21 so that formula_22, one might obtain a decomposition that is valid for both massive and massless cases. |
Using the Dirac equation again, one finds that |
where formula_28 is the vector of Pauli matrices. |
With the particle-number density identified with formula_29, and for a near plane-wave |
solution of finite extent, one may interpret the first term in the decomposition as the current formula_30, due to particles moving at speed formula_31. |
The second term, formula_32 is the current due to the gradients in the intrinsic magnetic moment density. The magnetic moment itself is found by integrating by parts to show that |
For a single massive particle in its rest frame, where formula_34, the magnetic moment reduces to |
where formula_36 and formula_37 is the Dirac value of the gyromagnetic ratio. |
For a single massless particle obeying the right-handed Weyl equation, the spin-1/2 is locked to the direction formula_38 of its kinetic momentum and the magnetic moment becomes |
For the both massive and massless cases, one also has an expression for the momentum density as part of the symmetric Belinfante–Rosenfeld stress–energy tensor |
Using the Dirac equation one may evaluate formula_41 to find the energy density to be formula_42, and the momentum density, |
If one used the non-symmetric canonical energy-momentum tensor |
one would not find the bound spin-momentum contribution. |
By an integration by parts one finds that the spin contribution to the total angular momentum is |
This is what is expected, so the division by 2 in the spin contribution to the momentum density is necessary. The absence of a division by 2 in the formula for the current reflects the formula_37 gyromagnetic ratio of the electron. In other words, a spin-density gradient is twice as effective at making an electric current as it is at contributing to the linear momentum. |
Motivated by the Riemann–Silberstein vector form of Maxwell's equations, Michael Berry uses the Gordon strategy to obtain gauge-invariant expressions for the intrinsic spin angular-momentum density for solutions to Maxwell's equations. |
He assumes that the solutions are monochromatic and uses the phasor expressions formula_47, formula_48. The time average of the Poynting vector momentum density is then given by |
We have used Maxwell's equations in passing from the first to the second and third lines, and in expression such as formula_52 the scalar product is between the fields so that the vector character is determined by the formula_53. |
and for a fluid with intrinsic angular momentum density formula_55 we have |
these identities suggest that the spin density can be identified as either |
The two decompositions coincide when the field is paraxial. They also coincide when the field is a pure helicity state – i.e. when formula_59 where the helicity formula_60 takes the values formula_61 for light that is right or left circularly polarized respectively. In other cases they may differ. |
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. |
The solutions to the equations, universally denoted as or (Greek psi), are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations (see classical field theory for background). |
In the Schrödinger picture, the wave function or field is the solution to the Schrödinger equation; |
one of the postulates of quantum mechanics. All relativistic wave equations can be constructed by specifying various forms of the Hamiltonian operator "Ĥ" describing the quantum system. Alternatively, Feynman's path integral formulation uses a Lagrangian rather than a Hamiltonian operator. |
More generally – the modern formalism behind relativistic wave equations is Lorentz group theory, wherein the spin of the particle has a correspondence with the representations of the Lorentz group. |
Late 1920s: Relativistic quantum mechanics of spin-0 and spin- particles. |
A description of quantum mechanical systems which could account for "relativistic" effects was sought for by many theoretical physicists; from the late 1920s to the mid-1940s. The first basis for relativistic quantum mechanics, i.e. special relativity applied with quantum mechanics together, was found by all those who discovered what is frequently called the Klein–Gordon equation: |
by inserting the energy operator and momentum operator into the relativistic energy–momentum relation: |
The solutions to () are scalar fields. The KG equation is undesirable due to its prediction of "negative" energies and probabilities, as a result of the quadratic nature of () – inevitable in a relativistic theory. This equation was initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called the Schrödinger equation) was still of importance. Nevertheless, – () is applicable to spin-0 bosons. |
Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the fine structure in the Hydrogen spectral series. The mysterious underlying property was "spin". The first two-dimensional "spin matrices" (better known as the Pauli matrices) were introduced by Pauli in the Pauli equation; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in magnetic fields, but this was "phenomenological". Weyl found a relativistic equation in terms of the Pauli matrices; the Weyl equation, for "massless" spin- fermions. The problem was resolved by Dirac in the late 1920s, when he furthered the application of equation () to the electron – by various manipulations he factorized the equation into the form: |
and one of these factors is the Dirac equation (see below), upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matrices and in a relativistic wave equation, and explained the fine structure of hydrogen. The solutions to () are multi-component spinor fields, and each component satisfies (). A remarkable result of spinor solutions is that half of the components describe a particle while the other half describe an antiparticle; in this case the electron and positron. The Dirac equation is now known to apply for all massive spin- fermions. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation. |
Although a landmark in quantum theory, the Dirac equation is only true for spin- fermions, and still predicts negative energy solutions, which caused controversy at the time (in particular – not all physicists were comfortable with the "Dirac sea" of negative energy states). |
1930s–1960s: Relativistic quantum mechanics of higher-spin particles. |
The natural problem became clear: to generalize the Dirac equation to particles with "any spin"; both fermions and bosons, and in the same equations their antiparticles (possible because of the spinor formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus by van der Waerden in 1929), and ideally with positive energy solutions. |
This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of (): |
where is a spinor field now with infinitely many components, irreducible to a finite number of tensors or spinors, to remove the indeterminacy in sign. The matrices and are infinite-dimensional matrices, related to infinitesimal Lorentz transformations. He did not demand that each component of to satisfy equation (), instead he regenerated the equation using a Lorentz-invariant action, via the principle of least action, and application of Lorentz group theory. |
Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16). They were anticipated later (in a more involved way) by de Broglie (1934), and Duffin, Kemmer, and Petiau (around 1938–1939) see Duffin–Kemmer–Petiau algebra. The Dirac–Fierz–Pauli formalism was more sophisticated than Majorana’s, as spinors were new mathematical tools in the early twentieth century, although Majorana’s paper of 1932 was difficult to fully understand; it took Pauli and Wigner some time to understand it, around 1940. |
Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinors and , symmetric in all indices, for a massive particle of spin for integer (see Van der Waerden notation for the meaning of the dotted indices): |
where is the momentum as a covariant spinor operator. For , the equations reduce to the coupled Dirac equations and and together transform as the original Dirac spinor. Eliminating either or shows that and each fulfill (). |
In 1941, Rarita and Schwinger focussed on spin- particles and derived the Rarita–Schwinger equation, including a Lagrangian to generate it, and later generalized the equations analogous to spin for integer . In 1945, Pauli suggested Majorana's 1932 paper to Bhabha, who returned to the general ideas introduced by Majorana in 1932. Bhabha and Lubanski proposed a completely general set of equations by replacing the mass terms in () and () by an arbitrary constant, subject to a set of conditions which the wave functions must obey. |
Finally, in the year 1948 (the same year as Feynman's path integral formulation was cast), Bargmann and Wigner formulated the general equation for massive particles which could have any spin, by considering the Dirac equation with a totally symmetric finite-component spinor, and using Lorentz group theory (as Majorana did): the Bargmann–Wigner equations. In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by H. Joos and Steven Weinberg, the Joos–Weinberg equation. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles. |
The relativistic description of spin particles has been a difficult problem in quantum theory. It is still an area of the present-day research because the problem is only partially solved; including interactions in the equations is problematic, and paradoxical predictions (even from the Dirac equation) are still present. |
The following equations have solutions which satisfy the superposition principle, that is, the wave functions are additive. |
Throughout, the standard conventions of tensor index notation and Feynman slash notation are used, including Greek indices which take the values 1, 2, 3 for the spatial components and 0 for the timelike component of the indexed quantities. The wave functions are denoted ", and are the components of the four-gradient operator. |
In matrix equations, the Pauli matrices are denoted by " in which , where is the identity matrix: |
and the other matrices have their usual representations. The expression |
is a matrix operator which acts on 2-component spinor fields. |
The gamma matrices are denoted by "", in which again , and there are a number of representations to select from. The matrix is "not" necessarily the identity matrix. The expression |
is a matrix operator which acts on 4-component spinor fields. |
Note that terms such as "" scalar multiply an identity matrix of the relevant dimension, the common sizes are or , and are "conventionally" not written for simplicity. |
The Duffin–Kemmer–Petiau equation is an alternative equation for spin-0 and spin-1 particles: |
Start with the standard special relativity (SR) 4-vectors |
Note that each 4-vector is related to another by a Lorentz scalar: |
Now, just apply the standard Lorentz scalar product rule to each one: |
The last equation is a fundamental quantum relation. |
When applied to a Lorentz scalar field formula_21, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations. |
The Schrödinger equation is the low-velocity limiting case ("v" « "c") of the Klein–Gordon equation. |
When the relation is applied to a four-vector field formula_25 instead of a Lorentz scalar field formula_21, then one gets the Proca equation (in Lorenz gauge): |
If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation (in Lorenz gauge) |
Under a proper orthochronous Lorentz transformation in Minkowski space, all one-particle quantum states of spin with spin z-component locally transform under some representation of the Lorentz group: |
where is some finite-dimensional representation, i.e. a matrix. Here is thought of as a column vector containing components with the allowed values of . The quantum numbers and as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of may occur more than once depending on the representation. Representations with several possible values for are considered below. |
The irreducible representations are labeled by a pair of half-integers or integers . From these all other representations can be built up using a variety of standard methods, like taking tensor products and direct sums. In particular, space-time itself constitutes a 4-vector representation so that . To put this into context; Dirac spinors transform under the representation. In general, the representation space has subspaces that under the subgroup of spatial rotations, SO(3), transform irreducibly like objects of spin "j", where each allowed value: |
occurs exactly once. In general, "tensor products of irreducible representations" are reducible; they decompose as direct sums of irreducible representations. |
The representations and can each separately represent particles of spin . A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation. |
There are equations which have solutions that do not satisfy the superposition principle. |
In physics, the acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure formula_1 or particle velocity u as a function of position x and time formula_2. A simplified form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions. |
For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article or the survey paper. |
The wave equation describing sound in one dimension (position formula_3) is |
where formula_1 is the acoustic pressure (the local decoration the from the ambient pressure), and where formula_6 is the speed of sound.same |
Provided that the speed formula_6 is a constant, not dependent on frequency (the dispersionless case), then the most general solution is |
where formula_9 and formula_10 are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (formula_9) travelling up the x-axis and the other (formula_10) down the x-axis at the speed formula_6. The particular case of a sinusoidal wave travelling in one direction is obtained by choosing either formula_9 or formula_10 to be a sinusoid, and the other to be zero, giving |
where formula_17 is the angular frequency of the wave and formula_18 is its wave number. |
The derivation of the wave equation involves three steps: derivation of the equation of state, the linearized one-dimensional continuity equation, and the linearized one-dimensional force equation. |
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