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Delta potential well (QM)
Encyclopedia
The delta potential is a potential that gives rise to many interesting results in quantum mechanics. It consists of a time-independent Schrödinger equation
for a particle in a potential well
defined by a Dirac delta function
in one dimension.
For those familiar with the particle in a box
problem, the delta function potential well is a special case of the finite potential well
, and follows as a limit as the depth goes to infinity and the width goes to zero, keeping their product constant.
The time-independent Schrödinger equation
for the wave function ψ(x) of a particle in one dimension in a potential V(x) is
where ħ is the reduced Planck constant
and E is the energy
of the particle.
The delta potential is the potential
where δ(x) is the Dirac delta function
. It is called a delta potential well if λ is negative and a delta potential barrier if λ 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.
this is a linear differential equation
with constant coefficients
whose solutions are linear combination
s of eikx and e−ikx, where the wave number k is related to the energy by k = /ħ. 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 k), eikx represents a wave traveling to the right, and e−ikx one traveling to the left.
Two relations between the coefficients can be found by imposing that the wave function be continuous in the origin (ψ(0) = ψL(0) = ψR(0) = Ar + Al = Br + Bl), and by integrating the Schrödinger equation around x = 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
[ψ′R(0) − ψ′L(0)] + λψ(0) (Because 
). Substituting the definition of ψ into this expression, we get
The boundary conditions thus give the following restrictions on the coefficients
For positive energies, the particle is free to move in either half-space: x < 0 or x > 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 (Ar). It may be reflected (Al) or transmitted (Br).
To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations Ar = 1 (incoming particle), Al = r (reflection), Bl = 0 (no incoming particle from the right) and Br = t (transmission), and solve for r and t. The result is:
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 λ, 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).
Taking this to conclusion, the probability for transmission is:
.
In any one-dimensional attractive potential there will be a bound state
. To find its energy, note that for E < 0, k = i/ħ = iκ is complex 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: Al = Br = 0. The wave function is then
From the boundary conditions and normalization conditions, it follows that
from which it follows that λ must be negative, that is the bound state only exists for the well, and not for the barrier. The energy of the bound state is then
. 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:
.
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 model is described by the corresponding Schrödinger equation:
where the potential is now:
where
is the "internuclear" distance with Dirac delta function (negative) peaks located at 
(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
. Here 
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,
is found to be governed by the pseudo-quadratic equation:
which has two solutions
. For the case of equal charges (symmetric homonuclear case), 
and the pseudo-quadratic reduces to:
The "+" case corresponds to a wave function symmetric about the mid-point (shown in red in the diagram) where
and is called gerade
. Correspondingly, the "-" case is the wave function that is anti-symmetric about the mid-point where
is called ungerade (shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional 
and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for a particle in a potential well
Potential well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy because it is captured in the local minimum of a potential well...
defined by a Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
in one dimension.
For those familiar with the particle in a box
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
problem, the delta function potential well is a special case of the finite potential well
Finite potential well
The finite potential well is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a box, but one which has finite potential walls. Unlike the infinite potential well, there is a probability associated with the particle being found...
, and follows as a limit as the depth goes to infinity and the width goes to zero, keeping their product constant.
Definition
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for the wave function ψ(x) of a particle in one dimension in a potential V(x) is
where ħ is the reduced Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
and E is the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of the particle.
The delta potential is the potential
where δ(x) is the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
. It is called a delta potential well if λ is negative and a delta potential barrier if λ 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.
Derivation
The potential splits the space in two parts (x < 0 and x > 0). In each of these parts the potential energy is zero, and the Schrödinger equation reduces to this is a linear differential equationLinear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
with constant coefficients
Constant coefficients
In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. In other words, it singles out special operators, within the larger class of...
whose solutions are linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
s of eikx and e−ikx, where the wave number k is related to the energy by k = /ħ. 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 k), eikx represents a wave traveling to the right, and e−ikx one traveling to the left.
Two relations between the coefficients can be found by imposing that the wave function be continuous in the origin (ψ(0) = ψL(0) = ψR(0) = Ar + Al = Br + Bl), and by integrating the Schrödinger equation around x = 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
[ψ′R(0) − ψ′L(0)] + λψ(0) (Because ). Substituting the definition of ψ into this expression, we getThe boundary conditions thus give the following restrictions on the coefficients
Transmission and reflection (positive energies)
The quantum case can be studied in the following situation: a particle incident on the barrier from the left side (Ar). It may be reflected (Al) or transmitted (Br).
To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations Ar = 1 (incoming particle), Al = r (reflection), Bl = 0 (no incoming particle from the right) and Br = t (transmission), and solve for r and t. The result is:
Due to the mirror symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
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 λ, 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).
Taking this to conclusion, the probability for transmission is:
.Bound state (negative energy)
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
. To find its energy, note that for E < 0, k = i/ħ = iκ is complex 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: Al = Br = 0. The wave function is then
From the boundary conditions and normalization conditions, it follows that
from which it follows that λ must be negative, that is the bound state only exists for the well, and not for the barrier. The energy of the bound state is then
Remarks and application
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 massEffective mass
In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that electrons and holes in a crystal respond to electric and magnetic fields almost as if they were particles with a mass dependence in their...
. 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
Scanning tunneling microscope
A scanning tunneling microscope is an instrument for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer , the Nobel Prize in Physics in 1986. For an STM, good resolution is considered to be 0.1 nm lateral resolution and...
(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)
Finite potential barrier (QM)
In quantum mechanics, the rectangular potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle...
. 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:
.The delta function model is actually a one-dimensional version of the Hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
according to the dimensional scaling method developed by the group of Dudley R. Herschbach
Dudley R. Herschbach
Dudley Robert Herschbach is an American chemist at Harvard University. He won the 1986 Nobel Prize in Chemistry jointly with Yuan T. Lee and John C...
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.
Double-well Dirac delta function model
where the potential is now:
where
is the "internuclear" distance with Dirac delta function (negative) peaks located at (shown in brown in the diagram). Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use Atomic unitsAtomic units
Atomic units form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article...
and set
. Here is a formally adjustable parameter. From the single well case, we can infer the "ansatzAnsatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
" for the solution to be:
Matching of the wavefunction at the Dirac delta function peaks yields the determinant:
Thus,
is found to be governed by the pseudo-quadratic equation:which has two solutions
. For the case of equal charges (symmetric homonuclear case), and the pseudo-quadratic reduces to:The "+" case corresponds to a wave function symmetric about the mid-point (shown in red in the diagram) where
and is called geradeMolecular term symbol
In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i.e. its electronic quantum state which is an eigenstate of the electronic molecular Hamiltonian. It is the equivalent of the term...
. Correspondingly, the "-" case is the wave function that is anti-symmetric about the mid-point where
is called ungerade (shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional 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
. 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
which results in 
. Thus, we will have a non-trivial bound state solution that has 
. For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the Transmission coefficientTransmission coefficientThe transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered...
is unity at zero energy.
See also
- The free particleFree particleIn physics, a free particle is a particle that, in some sense, is not bound. In classical physics, this means the particle is present in a "field-free" space.-Classical Free Particle:The classical free particle is characterized simply by a fixed velocity...
- The particle in a boxParticle in a boxIn quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
- The finite potential wellFinite potential wellThe finite potential well is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a box, but one which has finite potential walls. Unlike the infinite potential well, there is a probability associated with the particle being found...
- The particle in a ringParticle in a ringIn quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring is...
- The particle in a spherically symmetric potential
- The quantum harmonic oscillatorQuantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...
- The hydrogen atomHydrogen atomA hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
or The hydrogen-like atomHydrogen-like atomA hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself , these ions carry the positive charge e, where Z is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+... - The ring wave guide
- The particle in a one-dimensional lattice (periodic potential)Particle in a one-dimensional lattice (periodic potential)In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside...
- The hydrogen molecular ionHydrogen molecular ionThe hydrogen molecular ion, dihydrogen cation, or H2+, is the simplest molecular ion. It is composed of two positively-charged protons and one negatively-charged electron, and can be formed from ionization of a neutral hydrogen molecule...
- Holstein–Herring method
- The free particle