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The transmission and reflection probabilities are in fact oscillating with formula_71. The classical result of perfect transmission without any reflection (formula_72, formula_73) is reproduced not only in the limit of high energy formula_74 but also when the energy and barrier width satisfy formula_75, where formula_76 (see peaks near formula_77 and 1.8 in the above figure). Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height. |
The transmission probability at formula_28 evaluates to |
The calculation presented above may at first seem unrealistic and hardly |
useful. However it has proved to be a suitable model for a variety of real-life |
systems. One such example are interfaces between two conducting materials. In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the kinetic term in the above Hamiltonian with an effective mass formula_6. Often the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a barrier potential as above. Electrons may then tunnel from one material to the other giving rise to a current. |
The operation of a scanning tunneling microscope (STM) relies on this tunneling effect. In that case, the barrier is due to the gap between the tip of the STM and the underlying object. Since the tunnel current depends exponentially on the barrier width, this device is extremely sensitive to height variations on the examined sample. |
The above model is one-dimensional, while space is three-dimensional. One should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others; they are separable. The Schrödinger equation may then be reduced to the case considered here by an ansatz for the wave function of the type: formula_81. |
For another, related model of a barrier, see Delta potential barrier (QM), which can be regarded as a special case of the finite potential barrier. All results from this article immediately apply to the delta potential barrier by taking the limits formula_82 while keeping formula_83 constant. |
In the general theory of relativity the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. |
The equations were first published by Einstein in 1915 in the form of a tensor equation which related the local "" (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). |
Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation. |
As well as implying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light. |
Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves. |
The Einstein field equations (EFE) may be written in the form: |
where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant. |
where is the Ricci curvature tensor, and is the scalar curvature. This is a symmetric second-degree tensor that depends on only the metric tensor and its first- and second derivatives. |
The Einstein gravitational constant is defined as |
where is the Newtonian constant of gravitation and is the speed of light in vacuum. |
The EFE can thus also be written as |
In standard units, each term on the left has units of 1/length2. |
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime. |
These equations, together with the geodesic equation, which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of general relativity. |
The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom, which correspond to the freedom to choose a coordinate system. |
Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when is everywhere zero) define Einstein manifolds. |
The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor , since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations. |
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler (MTW). The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]): |
The third sign above is related to the choice of convention for the Ricci tensor: |
With these definitions Misner, Thorne, and Wheeler classify themselves as , whereas Weinberg (1972) is , Peebles (1980) and Efstathiou et al. (1990) are , Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are . |
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative: |
The sign of the cosmological term would change in both these versions if the metric sign convention is used rather than the MTW metric sign convention adopted here. |
Taking the trace with respect to the metric of both sides of the EFE one gets |
where is the spacetime dimension. Solving for and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form: |
Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace in the expression on the right with the Minkowski metric without significant loss of accuracy). |
the term containing the cosmological constant was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting. This effort was unsuccessful because: |
Einstein then abandoned , remarking to George Gamow "that the introduction of the cosmological term was the biggest blunder of his life". |
The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent astronomical observations have shown an accelerating expansion of the universe, and to explain this a positive value of is needed. The cosmological constant is negligible at the scale of a galaxy or smaller. |
Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor: |
This tensor describes a vacuum state with an energy density and isotropic pressure that are fixed constants and given by |
where it is assumed that has SI unit m and is defined as above. |
The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity. |
General relativity is consistent with the local conservation of energy and momentum expressed as |
which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition. |
The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics, which is linear in the wavefunction. |
The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant appearing in the EFE is determined by making these two approximations. |
If the energy–momentum tensor is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting in the trace-reversed field equations, the vacuum equations can be written as |
In the case of nonzero cosmological constant, the equations are |
The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution. |
Manifolds with a vanishing Ricci tensor, , are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds. |
If the energy–momentum tensor is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor |
is used, then the Einstein field equations are called the "Einstein–Maxwell equations" (with cosmological constant , taken to be zero in conventional relativity theory): |
Additionally, the covariant Maxwell equations are also applicable in free space: |
where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the 2-form is zero, and the second that its exterior derivative is zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential such that |
in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation that may lack a globally defined potential. |
The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions. |
The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe. |
One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum. In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright, self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system. New solutions have been discovered using these methods by LeBlanc and Kohli and Haslam. |
The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of gravitational radiation. |
Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written |
using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as: |
Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields. |
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or forces to strains or deformations. |
Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it is often necessary to account for the directional dependence of the material, and the scalar parameter is generalized to a tensor. Constitutive relations are also modified to account for the rate of response of materials and their non-linear behavior. See the article Linear response function. |
The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law. It deals with the case of linear elastic materials. Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used. Walter Noll advanced the use of constitutive equations, clarifying their classification and the role of invariance requirements, constraints, and definitions of terms |
like "material", "isotropic", "aeolotropic", etc. The class of "constitutive relations" of the form "stress rate = f (velocity gradient, stress, density)" was the subject of Walter Noll's dissertation in 1954 under Clifford Truesdell. |
In modern condensed matter physics, the constitutive equation plays a major role. See Linear constitutive equations and Nonlinear correlation functions. |
Friction is a complicated phenomenon. Macroscopically, the friction force "F" between the interface of two materials can be modelled as proportional to the reaction force "R" at a point of contact between two interfaces through a dimensionless coefficient of friction "μ"f, which depends on the pair of materials: |
This can be applied to static friction (friction preventing two stationary objects from slipping on their own), kinetic friction (friction between two objects scraping/sliding past each other), or rolling (frictional force which prevents slipping but causes a torque to exert on a round object). |
The stress-strain constitutive relation for linear materials is commonly known as Hooke's law. In its simplest form, the law defines the spring constant (or elasticity constant) "k" in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted) displacement "x": |
meaning the material responds linearly. Equivalently, in terms of the stress "σ", Young's modulus "E", and strain "ε" (dimensionless): |
In general, forces which deform solids can be normal to a surface of the material (normal forces), or tangential (shear forces), this can be described mathematically using the stress tensor: |
where "C" is the elasticity tensor and "S" is the compliance tensor. |
Several classes of deformations in elastic materials are the following: |
The relative speed of separation "v"separation of an object A after a collision with another object B is related to the relative speed of approach "v"approach by the coefficient of restitution, defined by Newton's experimental impact law: |
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field or the magnetic field , takes the form: |
is the speed of light (i.e. phase velocity) in a medium with permeability , and permittivity , and is the Laplace operator. In a vacuum, meters per second, a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In most older literature, is called the "magnetic flux density" or "magnetic induction". |
The origin of the electromagnetic wave equation. |
In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, James Clerk Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In "Part VI" of his 1864 paper titled "Electromagnetic Theory of Light", Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented: |
The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws. |
Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction. |
To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are: |
These are the general Maxwell's equations specialized to the case with charge and current both set to zero. |
Taking the curl of the curl equations gives: |
where is any vector function of space. And |
where is a dyadic which when operated on by the divergence operator yields a vector. Since |
then the first term on the right in the identity vanishes and we obtain the wave equations: |
Covariant form of the homogeneous wave equation. |
These relativistic equations can be written in contravariant form as |
The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears. |
where formula_15 is the Ricci curvature tensor and the semicolon indicates covariant differentiation. |
The generalization of the Lorenz gauge condition in curved spacetime is assumed: |
Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous. |
Solutions to the homogeneous electromagnetic wave equation. |
The general solution to the electromagnetic wave equation is a linear superposition of waves of the form |
for virtually "any" well-behaved function of dimensionless argument , where is the angular frequency (in radians per second), and is the wave vector (in radians per meter). |
Although the function can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies. |
In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation: |
where is the wavenumber and is the wavelength. The variable can only be used in this equation when the electromagnetic wave is in a vacuum. |
The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form: |
Consider a plane defined by a unit normal vector |
Then planar traveling wave solutions of the wave equations are |
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