Delta potential

A delta potential is a point interaction imposing continuity of the wavefunction and a jump in its derivative.

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Bound state in a delta potential

An attractive one-dimensional delta potential has one even exponentially decaying bound state.

Multiple delta potential

For delta interactions at points

x_{j}

, a Green-function solution reduces the Schrodinger equation to a finite linear system for the values

ψ (x_{j})

.

Symmetric double-delta potential

The symmetric double-delta potential

V (x) = \frac{ℏ ^{2} U _{0}}{2 m} (δ (x - a) + δ (x + a))

separates into even and odd parity channels. At either interaction, the wavefunction is continuous and its derivative has the jump

ψ^{'} (a^{+}) - ψ^{'} (a^{-}) = U_{0} ψ (a) .

Odd-parity S-matrix of a symmetric double-delta potential ( $S_{--}$ )

For positive wavenumber

k

, the odd channel of the symmetric double-delta potential has

S_{--} (k) = e^{- 2 ika} \frac{k e ^{ika} + U _{0} s i n ( ka )}{k e ^{- ika} + U _{0} s i n ( ka )} .

For real

k

its numerator and denominator are complex conjugates, making the channel unitary. Its poles are the zeros of

k e^{- ika} + U_{0} sin (ka)

.

Odd bound state of a symmetric double-delta potential

For

U_{0} < 0

, an odd bound state has

k = iκ

with

κ > 0

and obeys

2 κ = - U_{0} (1 - e^{- 2 κa}) .

A nonzero positive solution exists exactly when

- U_{0} a > 1

; equality is the zero-energy threshold.

Odd resonance of a strongly repulsive symmetric double-delta potential

For

U_{0} a ≫ 1

, the lowest odd scattering resonance pole lies near the first odd mode trapped between the barriers:

k = \frac{π}{a} (1 - \frac{1}{U _{0} a} + O ((U_{0} a)^{- 2})) - i \frac{π ^{2}}{U _{0}^{2} a ^{3}} (1 + O ((U_{0} a)^{- 1}))

in its real part and to leading nonzero order in its imaginary part. Writing

E = ℏ^{2} k^{2} / (2 m) = E_{R} - i Γ/2

, the survival probability decays as

e^{- Γ t /ℏ}

.

Scattering by a delta potential

For

V (x) = V_{0} δ (x)

,

E = ℏ^{2} k^{2} / (2 m)

, and

γ = m V_{0} / (ℏ^{2} k)

, continuity and the derivative jump give

t = \frac{1}{1 + iγ}, r = \frac{- iγ}{1 + iγ} .

They obey

∣ r ∣^{2} + ∣ t ∣^{2} = 1

and

r^{*} t + t^{*} r = 0

.

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Delta potential by

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Delta potential, often referred to as the Dirac delta potential, is a mathematical construct used in quantum mechanics and quantum field theory. It represents an idealized potential energy function that is localized at a single point in space. The Dirac delta function, denoted as \(\delta(x - x_0)\), is defined such that: 1. \(\delta(x - x_0) = 0\) for all \(x \neq x_0\), 2.

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