In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, resonance cross section occurs in the context of quantum
scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunli ...
, which deals with studying the scattering of quantum particles from potentials. The scattering problem deals with the calculation of flux distribution of scattered particles/waves as a function of the potential, and of the state (characterized by conservation of
momentum/energy) of the incident particle. For a free quantum particle incident on the potential, the plane wave solution to the time-independent
Schrödinger wave equation is:
:
For one-dimensional problems, the transmission coefficient
is of interest. It is defined as:
:
where
is the probability current density. This gives the fraction of incident beam of particles that makes it through the potential. For three-dimensional problems, one would calculate the
scattering cross-section , which, roughly speaking, is the total area of the incident beam which is scattered. Another quantity of relevance is the partial cross-section,
, which denotes the scattering cross section for a partial wave of a definite angular momentum eigenstate. These quantities naturally depend on
, the wave-vector of the incident wave, which is related to its energy by:
:
The values of these quantities of interest, the
transmission coefficient (in case of one dimensional potentials), and the partial cross-section
show peaks in their variation with the incident energy
. These phenomena are called resonances.
One-dimensional case
Mathematical description
A one-dimensional
finite square potential is given by
:
The sign of
determines whether the square potential is a well or a barrier. To study the phenomena of resonance, the time-independent
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
for a stationary state of a massive particle with energy
is solved:
:
The wave function solutions for the three regions
are
:
Here,
and
are the wave numbers in the potential-free region and within the potential respectively:
:
:
To calculate
, a coefficient in the wave function is set as
, which corresponds to the fact that there is no wave incident on the potential from the right. Imposing the condition that the wave function
and its derivative
should be continuous at the well/barrier boundaries
and
, the relations between the coefficients are found, which allows
to be found as:
:
It follows that the transmission coefficient
reaches its maximum value of 1 when:
:
for any integer value
. This is the resonance condition, which leads to the peaking of
to its maxima, called resonance.
Physical picture (Standing de Broglie Waves and the Fabry-Pérot Etalon)
From the above expression, resonance occurs when the distance covered by the particle in traversing the well and back (
) is an integral multiple of the De Broglie wavelength of a particle inside the potential (
). For
, reflections at potential discontinuities are not accompanied by any phase change.
[Claude Cohen-Tannaoudji, Bernanrd Diu, Frank Laloe.(1992), Quantum Mechanics ( Vol. 1), Wiley-VCH, p.73] Therefore, resonances correspond to the formation of standing waves within the potential barrier/well. At resonance, the waves incident on the potential at
and the waves reflecting between the walls of the potential are in phase, and reinforce each other. Far from resonances, standing waves can't be formed. Then, waves reflecting between both walls of the potential (at
and
) and the wave transmitted through
are out of phase, and destroy each other by interference. The physics is similar to that of transmission in
Fabry–Pérot interferometer
In optics, a Fabry–Pérot interferometer (FPI) or etalon is an optical cavity made from two parallel reflecting surfaces (i.e.: thin mirrors). Optical waves can pass through the optical cavity only when they are in resonance with it. It is ...
in optics, where the resonance condition and functional form of the transmission coefficient are the same.
Nature of resonance curves
The transmission coefficient swings between its maximum of 1 and minimum of
as a function of the length of square well (
) with a period of
. The minima of the transmission tend to
in the limit of large energy
, resulting in more shallow resonances, and inversely tend to
in the limit of low energy