Square potentials, continuum states

Problem:

Consider the one-dimensional potential step defined by U(x) = 0 , x < 0,  U(x) = U0, x > 0.

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Suppose a wave incident from the left has energy E = 4U0.  What is the probability that the wave will be reflected?

Solution:

Problem:

Consider a one-dimensional step potential, of the form
U(x) = 0 for x < 0,
U(x) = U0 for x > 0, with U0 > 0.
A particle with mass m and energy E > U0 is incident on this step from the left.
(a)  Write down the appropriate solutions for the time-independent Schroedinger equation for this particle in the x < 0 region and in the x > 0 region.
(b)  Apply the appropriate boundary conditions at x = 0 to match these solutions.
(c)  Derive expressions for the probabilities that the particle is reflected and transmitted by the step.


Solution:

Problem:

Consider a one-dimensional quantum-mechanical scattering problem, involving a particle of mass m moving through a region with potential energy function
U(x) = U0,  0 ≤ x ≤ L,   U(x) = 0 otherwise.
The particle moves from -∞ to +∞.  Assume that its energy is chosen to be exactly U0.  Find the transmission and the reflection probabilities.

Solution:

Problem:

Let U(x) = 0 for x < 0, U(x) = U0 > 0 for 0 < x < a, and U(x) = 0 for x > a.  A particle of mass m and energy E > U0 is incident on this potential barrier.  Find the energies E for which resonances occur, i.e.  for which the probability of transmission  is 1.

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Solution:

Problem:

A plane wave representing an electron beam with energy E is incident from the negative x direction onto a potential energy step described by the function
U(x) = U2 for x < 0,
U(x) = U1 for 0 < x < a,
U(x) = 0 for x > a,
where U1 = π2ħ2/(8ma2),  E = 2U1, U1 < U2 < E.
Evaluate the transmittance T.  For which value of U2 is T the largest?

Solution:

Problem:

(a)  Calculate the transmission coefficient for a particle with mass m and kinetic energy E passing through the rectangular potential barrier
U{x) = 0 for x < 0, U(x) = U0 for 0 < x < a, U(x) = 0 for x > a,  with E < U0.
(b)  Show that for E << U0 and 2mU0a22 >> 1 the transmittance T can be written as
T ~ (16E/U0)exp[-2(2m(U0 a2)/ħ2)½].

Solution:

Problem:

(a)  Calculate the transmission coefficient for a particle with mass m and kinetic energy E < U0 passing through the rectangular potential barrier
U(x) = 0 for x < 0, U(x) = U0 > 0 for 0 < x < a, and U(x) = 0 for x > a.
(b)  Show that for E << U0 and  2mU0a22 >> 1 the transmission coefficient can be written as  T ≈ (16E/U0)exp[-2(2mU02)½a].
(c)  Many heavy nuclei decay by emitting an alpha particle.  In a simple one-dimensional model, the potential barrier the alpha particles have to penetrate can be approximated by
U(r) = 0 for r < R0,  U(r) = U0R0/r for r > R0,
where R0 is the radius of the nucleus and U0 is the barrier height for r0 = R.  The energy E of the alpha particle can be assumed to be much smaller than U0.  For a non constant potential barrier the expression for the transmission coefficient found in part (b) can be used as a guide.  Assume that for E << U0, we have T ≈ exp[-2∫R1R2 dr (2m(U(r) - E)/ħ2)½].
The integration limits R1 and R2 are determined as solutions to the equation U(r) = E. 
Calculate the alpha transmission coefficient and the decay constant λ, i.e. the decay probability per second.

Solution: