Delta function potentials

Problem:

A one-dimensional potential well is given in the form of a delta function at x = 0, V(x) = Cδ(x), C < 0.  A stream of non-relativistic particles of mass m and energy E approaches the origin from one side. 
(a)  Derive an expression for the reflectance R(E).  
(b)  Can you express R(E) in terms of sin2(δ), where δ is the phase sift of the transmitted wave.

Solution:

Problem:

A one-dimensional potential well is given in the form of a delta function at x = 0,
U(x) = Cδ(x), C < 0.
(a)  A non-relativistic particle of mass m and energy E is incident from one side of the well.
Derive an expression for the coefficient of transmission T(E).
(b)  Since a bound state can exist with the attractive potential, find the binding energy of the ground state of the system. 

Solution:

Problem:

Consider the non-relativistic motion in one dimension of a particle outside an infinite barrier at x ≤ 0 with an additional delta function potential at x = a, i.e. U(x) = ∞ for  x ≤ 0,   U(x) = Fδ(x - a) for x > 0, where F is a positive constant.  Derive an analytical expression for the phase shift δ(k) for a particle approaching the origin from x = +∞ with momentum ħk.

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Solution:

Problem:

Consider the scattering of a particle of mass m and total energy  E = ħ2k2/(2m) under the influence of a localized one-dimensional potential.
(a)  Let the potential be a delta function potential well, U(x) = -aU0δ(x) with a > 0 and U0 = ħ2k02/(2m).  What are the asymptotic boundary conditions at x = ∞ and the matching conditions at x = 0 for the wave function?
(b)  Define the transmission coefficient T and the reflection coefficient R and find the relationship between T and R.
(c)  How does the transmission coefficient depend on E?
(d)  Now the potential is replaced by a double delta function potential well.  The delta functions are a distance b apart, i.e. U(x) = -aU0δ(x) - aU0δ(x - b).  By inspecting the matching conditions without solving the algebra equation, explain intuitively the limiting behavior of the transmission coefficient T for E --> 0 and E --> ∞.

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Solution:

Problem:

Consider an electron trapped in a one-dimensional periodic potential with period a. 
The electron's potential energy is given by
U(x) = ∑-∞+∞Aδ(x - na).
where A is a constant.
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The eigenfunctions of the Hamiltonian have the form Φ(x) = exp(ikx)u(x), where u(x) is a periodic function with period a, u(x + a) = u(x) (Bloch's theorem).  Find the allowed energy eigenvalues of the electron.

Solution: