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, q ...
, the case of a particle in a one-dimensional ring is similar to the
particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
. The particle follows the path of a semicircle from
to
where it cannot escape, because the potential from
to
is infinite. Instead there is total reflection, meaning the particle bounces back and forth between
to
. The
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 th ...
for a
free particle
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
which is restricted to a semicircle (technically, whose
configuration space is the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
) is
Wave function
Using
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
on the 1-dimensional semicircle, the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
depends only on the
angular
Angular may refer to:
Anatomy
* Angular artery, the terminal part of the facial artery
* Angular bone, a large bone in the lower jaw of amphibians and reptiles
* Angular incisure, a small anatomical notch on the stomach
* Angular gyrus, a region o ...
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
, and so
Substituting the Laplacian in cylindrical coordinates, the wave function is therefore expressed as
The moment of inertia for a semicircle, best expressed in cylindrical coordinates, is
. Solving the integral, one finds that the moment of inertia of a semicircle is
, exactly the same for a hoop of the same radius. The wave function can now be expressed as
, which is easily solvable.
Since the particle cannot escape the region from
to
, the general solution to this differential equation is
Defining
, we can calculate the energy as
. We then apply the boundary conditions, where
and
are continuous and the wave function is normalizable:
Like the infinite square well, the first boundary condition demands that the wave function equals 0 at both
and
. Basically
Since the wave function
, the coefficient A must equal 0 because
. The wave function also equals 0 at
so we must apply this boundary condition. Discarding the trivial solution where ''B''=0, the wave function
only when ''m'' is an integer since
. This boundary condition quantizes the energy where the energy equals
where ''m'' is any integer. The condition ''m''=0 is ruled out because
everywhere, meaning that the particle is not in the potential at all. Negative integers are also ruled out since they can easily be absorbed in the normalization condition.
We then normalize the wave function, yielding a result where
. The normalized wave function is
The ground state energy of the system is
. Like the particle in a box, there exists nodes in the excited states of the system where both
and
are both 0, which means that the probability of finding the particle at these nodes are 0.
Analysis
Since the wave function is only dependent on the azimuthal angle
, the measurable quantities of the system are the angular position and angular momentum, expressed with the operators
and
respectively.
Using cylindrical coordinates, the operators
and
are expressed as
and
respectively, where these observables play a role similar to position and momentum for the particle in a box. The commutation and uncertainty relations for angular position and angular momentum are given as follows:
Boundary conditions
As with all quantum mechanics problems, if the boundary conditions are changed so does the wave function. If a particle is confined to the motion of an entire ring ranging from 0 to
, the particle is subject only to a periodic boundary condition (see
particle in a ring In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S^1) is
: ...
). If a particle is confined to the motion of
to
, the issue of even and odd parity becomes important.
The wave equation for such a potential is given as:
where
and
are for odd and even ''m'' respectively.
Similarly, if the semicircular potential well is a finite well, the solution will resemble that of the finite potential well where the angular operators
and
replace the linear operators ''x'' and ''p''.
See also
*
Particle in a ring In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S^1) is
: ...
*
Particle in a box
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
*
Finite potential well The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlik ...
*
Delta function potential
In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it t ...
*
Gas in a box
In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other excep ...
*
Particle in a spherically symmetric potential
In the quantum mechanics description of a particle in spherical coordinates, a spherically symmetric potential, is a potential that depends only on the distance between the particle and a defined centre point. One example of a spherical potentia ...
Quantum models
Quantum mechanical potentials