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In order to exhibit the metric it is necessary to pull it back via a suitable "parametrization". A parametrization of a submanifold of is a map whose range is an open subset of . If has the same dimension as , a parametrization is just the inverse of a coordinate map . The parametrization to be used is the inverse of "hyperbolic stereographic projection". This is illustrated in the figure to the left for . It is instructive to compare to stereographic projection for spheres. |
Stereographic projection and its inverse are given by |
where, for simplicity, . The are coordinates on and the are coordinates on . |
The rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. |
The total canonical partition function formula_1 of a system of formula_2 identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions formula_3: |
where formula_6 is the degeneracy of the "j"th quantum level of an individual particle, formula_7 is the Boltzmann constant, and formula_8 is the absolute temperature of system. |
For molecules, under the assumption that total energy levels formula_9 can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom) |
and the number of degenerate states are given as products of the single contributions |
where "trans", "ns", "rot", "vib" and "e" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions |
can be written as a product itself |
Rotational energies are quantized. For a diatomic molecule like CO or HCl or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are |
J is the quantum number for total rotational angular momentum and takes all integer values starting at zero,i.e. formula_15 is the rotational constant, and formula_16 is the moment of inertia. Here we are using "B" in energy units. If it is expressed in frequency units, replace "B" by "hB" in all the expression that follow, where "h" is Planck's constant. If "B" is given in units of formula_17, then replace "B" by "hcB" where c is the speed of light in vacuum. |
For each value of J, we have rotational degeneracy, formula_18 = (2J+1), so the rotational partition function is therefore |
For all but the lightest molecules or the very lowest temperatures we have formula_20. This suggests we can approximate the sum by replacing the sum over J by an integral of J treated as a continuous variable. |
This approximation is known as the high temperature limit. It is also called the classical approximation as this is the result for the canonical partition function for a classical rigid rod. |
Using the Euler–Maclaurin formula an improved estimate can be found |
For the CO molecule at formula_23, the (unit less) contribution formula_24 to formula_3 turns out to be in the range of formula_26. |
The mean thermal rotational energy per molecule can now be computed by taking the derivative of formula_24 with respect to temperature formula_28. In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is formula_29. |
A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written formula_43 and formula_44, which can often be determined by rotational spectroscopy. In terms of these constants, the rotational partition function can be written in the high temperature limit as |
with formula_42 again known as the rotational symmetry number which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. Like in the case of the diatomic treated explicitly above, this factor corrects for the fact that only a fraction of the nuclear spin functions can be used for any given molecular level to construct wavefunctions that overall obey the required exchange symmetries. The expression for formula_47 works for asymmetric, symmetric and spherical top rotors. |
In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential. |
The delta potential well is a limiting case of the finite potential well, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential. |
This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions. |
The time-independent Schrödinger equation for the wave function of a particle in one dimension in a potential is |
where is the reduced Planck constant, and is the energy of the particle. |
It is called a "delta potential well" if is negative, and a "delta potential barrier" if is positive. The delta has been defined to occur at the origin for simplicity; a shift in the delta function's argument does not change any of the proceeding results. |
The potential splits the space in two parts ( < 0 and > 0). In each of these parts the potential energy is zero, and the Schrödinger equation reduces to |
this is a linear differential equation with constant coefficients, whose solutions are linear combinations of and , where the wave number is related to the energy by |
In general, due to the presence of the delta potential in the origin, the coefficients of the solution need not be the same in both half-spaces: |
where, in the case of positive energies (real ), represents a wave traveling to the right, and one traveling to the left. |
One obtains a relation between the coefficients by imposing that the wavefunction be continuous at the origin: |
A second relation can be found by studying the derivative of the wavefunction. Normally, we could also impose differentiability at the origin, but this is not possible because of the delta potential. However, if we integrate the Schrödinger equation around = 0, over an interval [−"ε", +"ε"]: |
In the limit as "ε" → 0, the right-hand side of this equation vanishes; the left-hand side becomes |
Substituting the definition of into this expression yields |
The boundary conditions thus give the following restrictions on the coefficients |
In any one-dimensional attractive potential there will be a bound state. To find its energy, note that for < 0, = = is imaginary, and the wave functions which were oscillating for positive energies in the calculation above are now exponentially increasing or decreasing functions of "x" (see above). Requiring that the wave functions do not diverge at infinity eliminates half of the terms: = = 0. The wave function is then |
From the boundary conditions and normalization conditions, it follows that |
from which it follows that must be negative, that is, the bound state only exists for the well, and not for the barrier. The Fourier transform of this wave function is a Lorentzian function. |
The energy of the bound state is then |
For positive energies, the particle is free to move in either half-space: < 0 or > 0. It may be scattered at the delta-function potential. |
The quantum case can be studied in the following situation: a particle incident on the barrier from the left side . It may be reflected or transmitted . |
To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations = 1 (incoming particle), = (reflection), = 0 (no incoming particle from the right) and = (transmission), and solve for and even though we do not have any equations in . |
Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. The result is that there is a non-zero probability |
for the particle to be reflected. This does not depend on the sign of , that is, a barrier has the same probability of reflecting the particle as a well. This is a significant difference from classical mechanics, where the reflection probability would be 1 for the barrier (the particle simply bounces back), and 0 for the well (the particle passes through the well undisturbed). |
In summary, the probability for transmission is |
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 regards the 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 . 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 local delta-function 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 air between the tip of the STM and the underlying object. The strength of the barrier is related to the separation being stronger the further apart the two are. For a more general model of this situation, see Finite potential barrier (QM). The delta function potential barrier is the limiting case of the model considered there for very high and narrow barriers. |
The above model is one-dimensional while the space around us is three-dimensional. So, in fact, 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. The Schrödinger equation may then be reduced to the case considered here by an Ansatz for the wave function of the type formula_18. |
Alternatively, it is possible to generalize the delta function to exist on the surface of some domain "D" (see Laplacian of the indicator). |
The delta function model is actually a one-dimensional version of the Hydrogen atom according to the "dimensional scaling" method developed by the group of Dudley R. Herschbach |
The delta function model becomes particularly useful with the "double-well" Dirac Delta function model which represents a one-dimensional version of the Hydrogen molecule ion, as shown in the following section. |
The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation: |
where formula_21 is the "internuclear" distance with Dirac delta-function (negative) peaks located at = ±/2 (shown in brown in the diagram). Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use atomic units and set formula_22. Here formula_23 is a formally adjustable parameter. From the single-well case, we can infer the "ansatz" for the solution to be |
Matching of the wavefunction at the Dirac delta-function peaks yields the determinant |
Thus, formula_26 is found to be governed by the "pseudo-quadratic" equation |
which has two solutions formula_28. For the case of equal charges (symmetric homonuclear case), = 1, and the pseudo-quadratic reduces to |
The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram), where = , and is called "gerade". Correspondingly, the "−" case is the wave function that is anti-symmetric about the midpoint, where = −, and is called "ungerade" (shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional <chem>H2^+</chem> and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by |
where "W" is the standard Lambert "W" function. Note that the lowest energy corresponds to the symmetric solution formula_31. In the case of "unequal" charges, and for that matter the three-dimensional molecular problem, the solutions are given by a "generalization" of the Lambert "W" function (see section on generalization of Lambert W function and references herein). |
One of the most interesting cases is when "qR" ≤ 1, which results in formula_32. Thus, one has a non-trivial bound state solution with = 0. For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the transmission coefficient is unity at zero energy. |
The Kundu equation is a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. It was proposed by Anjan Kundu as |
with arbitrary function formula_2 and the subscripts denoting partial derivatives. Equation (1) is shown to be reducible for the choice of formula_3 to an integrable class of mixed nonlinear Schrödinger equation with cubic–quintic nonlinearity, given in a representative form |
Here formula_5 are independent parameters, while formula_6 Equation (1), more specifically equation (2) is known as the Kundu equation. |
The Kundu equation is a completely integrable system, allowing Lax pair representation, exact solutions, and higher conserved quantity. |
Along with its different particular cases, this equation has been investigated for finding its exact travelling wave solutions, exact solitary wave solutions via bilinearization, and Darboux transformation together with the orbital stability for such solitary wave solutions. |
The Kundu equation has been applied to various physical processes such as fluid dynamics, plasma physics, and nonlinear optics. It is linked to the mixed nonlinear Schrödinger equation through a gauge transformation and is reducible to a variety of known integrable equations such as the nonlinear Schrödinger equation (NLSE), derivative NLSE, higher nonlinear derivative NLSE, Chen–Lee–Liu, Gerjikov-Vanov, and Kundu–Eckhaus equations, for different choices of the parameters. |
A generalization of the nonlinear Schrödinger equation with additional quintic nonlinearity and a nonlinear dispersive term was proposed in the form |
which may be obtained from the Kundu Equation (2), when restricted to formula_8. The same equation, limited further to the particular case formula_9 was introduced later as the Eckhaus equation, following which equation (3) is presently known as the Kundu-Ekchaus equation. The Kundu-Ekchaus equation can be reduced to the nonlinear Schrödinger equation through a nonlinear transformation of the field and known therefore to be gauge equivalent integrable systems, since they are equivalent under the gauge transformation. |
The Kundu-Ekchaus equation is associated with a Lax pair, higher conserved quantity, exact soliton solution, rogue wave solution etc. Over the years various aspects of this equation, its generalizations and link with other equations have been studied. In particular, relationship of Kundu-Ekchaus equation with the Johnson's hydrodynamic equation near criticality is established, its discretizations, reduction via Lie symmetry, complex structure via Bernoulli subequation, bright and dark soliton solutions via Bäcklund transformation and Darboux transformation with the associated rogue wave solutions, are studied. |
A multi-component generalisation of the Kundu-Ekchaus equation (3), known as Radhakrishnan, Kundu and Laskshmanan (RKL) equation was proposed in nonlinear optics for fiber optics communication through soliton pulses in a birefringent non-Kerr medium and analysed subsequently for its exact soliton solution and other aspects in a series of papers. |
Though the Kundu-Ekchaus equation (3) is gauge equivalent to the nonlinear Schrödinger equation, they differ with respect to their Hamiltonian structures and field commutation relations. The Hamiltonian operator of the Kundu-Ekchaus equation quantum field model given by |
and defined through the bosonic field operator commutation relation formula_11, is more complicated than the well-known bosonic Hamiltonian of the quantum nonlinear Schrödinger equation. Here formula_12 indicates normal ordering in bosonic operators. This model corresponds to a double formula_13-function interacting Bose gas and is difficult to solve directly. |
However, under a nonlinear transformation of the field below: |
i.e. in the same form as the quantum model of the Nonlinear Schrödinger equation (NLSE), though it differs from the NLSE in its contents, since now the fields involved are no longer bosonic operators but exhibit anion like properties. |
though at the coinciding points the bosonic commutation relation still holds. In analogy with the Lieb Limiger model of formula_19 function bose gas, the quantum Kundu-Ekchaus model in the N-particle sector therefore corresponds to a one-dimensional (1D) anion gas interacting via a formula_20 function interaction. This model of interacting anion gas was proposed and exactly solved by the Bethe ansatz in |
and this basic anion model is studied further for investigating various aspects of the 1D anion gas as well as extended in different directions. |
In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equations within the nonlinear Schrödinger class: |
The equation was independently introduced by Wiktor Eckhaus and by Anjan Kundu to model the propagation of waves in dispersive media. |
The Eckhaus equation can be linearized to the linear Schrödinger equation: |
This linearization also implies that the Eckhaus equation is integrable. |
In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left. |
Although classically a particle behaving as a point mass would be reflected if its energy is less than formula_1, a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated. |
The time-independent Schrödinger equation for the wave function formula_2 reads |
where formula_4 is the Hamiltonian, formula_5 is the (reduced) |
Planck constant, formula_6 is the mass, formula_7 the energy of the particle and |
is the barrier potential with height formula_9 and width formula_10. formula_11 |
The barrier is positioned between formula_13 and formula_14. The barrier can be shifted to any formula_15 position without changing the results. The first term in the Hamiltonian, formula_16 is the kinetic energy. |
The barrier divides the space in three parts (formula_17). In any of these parts, the potential is constant, meaning that the particle is quasi-free, and the solution of the Schrödinger equation can be written as a superposition of left and right moving waves (see free particle). If formula_18 |
where the wave numbers are related to the energy via |
The coefficients formula_29 have to be found from the boundary conditions of the wave function at formula_13 and formula_14. The wave function and its derivative have to be continuous everywhere, so |
Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients |
If the energy equals the barrier height, the second differential of the wavefunction inside the barrier region is 0, and hence the solutions of the Schrödinger equation are not exponentials anymore but linear functions of the space coordinate |
The complete solution of the Schrödinger equation is found in the same way as above by matching wave functions and their derivatives at formula_13 and formula_14. That results in the following restrictions on the coefficients: |
At this point, it is instructive to compare the situation to the classical case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy formula_7 larger than the barrier height formula_1 would "always" pass the barrier, and a classical particle with formula_48 incident on the barrier would "always" get reflected. |
To study the quantum case, consider the following situation: a particle incident on the barrier from the left side (formula_49). It may be reflected (formula_50) or transmitted (formula_51). |
To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations formula_52 (incoming particle), formula_53 (reflection), formula_54=0 (no incoming particle from the right), and formula_55 (transmission). We then eliminate the coefficients formula_56 from the equation and solve for formula_57 and formula_58. |
Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. Note that these expressions hold for any energy formula_61. |
The surprising result is that for energies less than the barrier height, formula_48 there is a non-zero probability |
for the particle to be transmitted through the barrier, with formula_64. This effect, which differs from the classical case, is called quantum tunneling. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector formula_65, whereas within the barrier it is exponentially damped over a distance formula_66. If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed. |
Equally surprising is that for energies larger than the barrier height, formula_18, the particle may be reflected from the barrier with a non-zero probability |
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