Physics
Delta Function Potential
The delta function potential is a concept in quantum mechanics that represents a potential energy function as a Dirac delta function. It is often used to model interactions in quantum systems, such as the potential energy experienced by a particle in the vicinity of a localized impurity or barrier. The delta function potential is a key tool for understanding quantum mechanical behavior in various physical systems.
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6 Key excerpts on "Delta Function Potential"
eBook - PDF- Guillaume Merle, Oliver J. Harper, Philippe Ribiere(Authors)
- 2023(Publication Date)
- Wiley-VCH(Publisher)
1.4 Bound State of a Particle in a “Delta Function Potential”, Fourier Analysis 23 Letting a approach 0 and V 0 approach infinity yields: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ A 2 ≃ B 1 2 ( 1 + i 2 √ 2m ℏ 2 V 0 ( 1 + E 2V 0 ) ) ≃ B 1 2 ( 1 + i 2 √ 2m ℏ 2 V 0 ) A ′ 2 ≃ B 1 2 ( 1 − i 2 √ 2m ℏ 2 V 0 ( 1 + E 2V 0 ) ) ≃ B 1 2 ( 1 − i 2 √ 2m ℏ 2 V 0 ) B ′ 3 ≃ B 1 2 √ − V 0 E ( 1 − E 2V 0 ) √ 2m ℏ 2 V 0 ( 1 + E 2V 0 ) ≃ B 1 2 √ − 2m ℏ 2 E and we deduce: |B ′ 3 | 2 ≃ −|B 1 | 2 m 2 2ℏ 2 E On top of this, according to our previous findings: sin ka ≃ √ 2m ℏ 2 V 0 ⇒ sin ka k ≃ √ V 0 (E + V 0 ) = V 0 √ 1 + E V 0 ≃ V 0 = a The wave function normalization condition therefore becomes: (|B 1 | 2 + |B ′ 3 | 2 ) 2 + (|A 2 | 2 + |A ′ 2 | 2 )a + (A 2 A ′∗ 2 + A ∗ 2 A ′ 2 )a ≃ 1 ⇔ |B 1 | 2 2 ( 1 − m 2 2ℏ 2 E ) √ − ℏ 2 2mE ≃ 1 ⇔ |B 1 | 2 ≃ 2 [ ( 1 − m 2 2ℏ 2 E ) √ − ℏ 2 2mE ] −1 The kinetic energy of the particle in the well, according to the previous results, is: E k = − ℏ 2 2m ∫ a 2 − a 2 ∗ (x) d 2 (x) dx 2 dx = ℏ 2 k 2 2m ∫ a 2 − a 2 (A ∗ 2 e −ikx + A ′∗ 2 e ikx )(A 2 e ikx + A ′ 2 e −ikx )dx = (E + V 0 ) ∫ a 2 − a 2 (|A 2 | 2 + |A ′ 2 | 2 + A 2 A ′∗ 2 e 2ikx + A ∗ 2 A ′ 2 e −2ikx )dx = (E + V 0 ) [ (|A 2 | 2 + |A ′ 2 | 2 )a + (A 2 A ′∗ 2 + A ∗ 2 A ′ 2 ) sin ka k ] 24 1 Solutions to the Exercises of Chapter I (Complement K I ). Waves and Particles Letting a approach 0 and V 0 approach infinity, we find: E k ≃ (E + V 0 )(A 2 + A ′ 2 )(A ∗ 2 + A ′∗ 2 )a ≃ |B 1 | 2 (E + V 0 )a ≃ |B 1 | 2 ≃ 2 [ ( 1 − m 2 2ℏ 2 E ) √ − ℏ 2 2mE ] −1 Finally, as a approaches 0 and V 0 approaches infinity with = aV 0 , the square potential well approaches a delta function well whose area equals , and the energy E, therefore, approaches the energy of the particle in the bound state as studied in question a, namely E = − m 2 2ℏ 2 .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
In applied mathematics, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy. Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corres-ponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta function as such was introduced as a convenient notation by Paul Dirac in his influential 1927 book Principles of Quantum Mechanics . He called it the delta function since he used it as a continuous analogue of the discrete Kronecker delta. Definitions The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, and which is also constrained to satisfy the identity ________________________ WORLD TECHNOLOGIES ________________________ This is merely a heuristic definition. The Dirac delta is not a true function, as no function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, sinc( x / a )/ a (where sinc is the sinc function) becomes the delta function in the limit as a → 0, yet this function does not approach zero for values of x outside the origin, rather it oscillates between 1/ x and −1/ x more and more rapidly as a approaches zero.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- White Word Publications(Publisher)
In applied mathematics, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy. Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta function as such was introduced as a convenient notation by Paul Dirac in his influential 1927 book Principles of Quantum Mechanics . He called it the delta function since he used it as a continuous analogue of the discrete Kronecker delta. Definitions The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, and which is also constrained to satisfy the identity ________________________ WORLD TECHNOLOGIES ________________________ This is merely a heuristic definition. The Dirac delta is not a true function, as no function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, sinc( x / a )/ a (where sinc is the sinc function) becomes the delta function in the limit as a → 0, yet this function does not approach zero for values of x outside the origin, rather it oscillates between 1/ x and −1/ x more and more rapidly as a approaches zero.
eBook - PDFQuantum Theory
Elements
- D. R. Bates(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
define a density function p(E) such that p(E)dE is the number of states with energy between Ε and Ε + dE. Now in the space of the vector η there is one state per unit volume, and hence when η is large the number in the spherical shell bounded by radii n, η + dn is ánn^dn or ^n[Lj2n)^k'^ dk. Since 4. THE C O N T I N U U M 151 illustrate some of the properties of these potentials it is sufficient to replace them by ideal rectangular forms as in (d), (e), (f) of the same figure. However it must be remembered that by introducing discontinuities into the potential a sharp break is made with classical mechanics, which corresponds to a Id) (B) (E) (F) F I G . 1. Rectangular potentials. limiting situation in which the de Broglie wavelength of the particle (hip) becomes small by comparison with the distance in which the potential changes appreciably (Chapter 7). With an exactly rectangular potential this limit is artificially excluded. 0 Α F I G . 2. Rectangular potential barrier. Consider first the potential (d), shown again in Fig. 2, and assume that a particle of mass m approaches the barrier from the left with energy Ε < V Q.
- Frank H. Stillinger(Author)
- 2015(Publication Date)
- Princeton University Press(Publisher)
Here is a list of the most basic of those general properties. Some are obvious, others more subtle, but all provide a necessary background for the theoretical develop-ments of the next two sections, I.D and I.E, and of subsequent chapters. (1) For any quantum state, the potential energy function Φ is a single-valued function of the nuclear position coordinates r 1 … r N . (2) Any geometrically isolated set of nuclei in free space (i.e., remote from container walls) can be translated and/or rotated arbitrarily without changing the value of the potential energy function Φ . (3) Permutation of the positions of any pair of nuclei with equal atomic numbers ( Z j = Z l , but possibly different isotopes) leaves the potential energy function Φ unchanged. This invariance remains valid even in the presence of wall forces. (4) The electrostatic charges Z j e carried by the nuclei cause coulombic divergences in the Φ ( r 1 … r N | n ′ ) whenever the separation between two nuclei tends to zero. Specifically, Φ ( r jk → 0 | n ′ ) ∼ Z j Z k / r jk + A ( r 1 … r N | n ′ ), (I.28) where the function A remains finite in that nuclear confluence limit. For virtually all condensed-matter applications of interest, the instances of r jk remain large enough even at their smallest occurrences that these nuclear Coulomb singularities are obscured by powerful (but bounded) electron-cloud overlap repulsions. The latter are implicit in the A function. (5) Aside from rare nuclear configurations producing electronic degeneracies such as “con-ical intersections” [Yarkony, 1996, 2001; Domcke and Yarkony, 2012], Φ is continuous and at least twice differentiable in nuclear position coordinates r 1 … r N away from any nuclear coincidences. In particular, this phenomenon implies that relative or absolute minima of Φ are locally at least quadratic in those nuclear coordinates. Such minima are mechanically stable configurations, that is, forces on all nuclei vanish.
eBook - PDF- Vishnu S. Mathur, Surendra Singh(Authors)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
1 As a consequence of boundary conditions, the bound state spectrum in one dimension is discrete and nondegenerate. This means that each bound state energy eigenvalue belongs to one (and only one) wave function. (ii) The wave function as well as its first derivative must be continuous at each point in space even if the potential has (finite) discontinuity. This is because the wave function must be twice differentiable as it satisfies a second order differential equation. For the wave function’s second derivative to exist, its first derivative must be continuous, which in turn requires the wave function itself to be continuous. When the potential has infinite discontinuity (as in an infinite square well or Dirac Delta Function Potential), the first derivative may be discontinuous and the wave function may have a kink. 1 For scattering solutions, the wave function approaches a constant as | x | → ∞ corresponding to incident or scattered particle flux ∝ | ψ ( x ) | 2 . 89 90 Concepts in Quantum Mechanics 4.1 Motion of a Particle across a Potential Step Consider the motion of a particle of energy E along a single axis [number of degrees of freedom of the system f = 1)] in the presence of a potential step of height V o . In this case, the potential V ( x ) has the form [Fig. 4.1] V ( x ) = ( 0 , x < 0 : Region I V o , x > 0 : Region II . (4.1.1) The particle approaches the potential step from the left. We consider two cases separately: (i) Particle energy greater than the potential step: E > V o Let the quantum state of the particle in regions I and II be represented by ψ 1 ( x ) and ψ 2 ( x ) in the coordinate representation. Then for the potential of Eq. (4.1.1), time-independent Schr¨ odinger equation [Eq. (3.5.4)] takes the following forms for the two regions d 2 ψ 1 ( x ) dx 2 + k 2 ψ 1 ( x ) = 0 x < 0 : Region I (4.1.2) d 2 ψ 2 ( x ) dx 2 + k 0 2 ψ 2 ( x ) = 0 x > 0 : Region II (4.1.3) where k = r 2 mE ~ 2 and k 0 = r 2 m ( E -V o ) ~ 2 .
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