Particle in a ring
Encyclopedia
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
) is
coordinate, and so
Requiring that the wave function be periodic
in
with a period 
(from the demand that the wave functions be single-valued function
s on the circle
), and that they be normalized
leads to the conditions
,
and
Under these conditions, the solution to the Schrödinger equation is given by
eigenvalues
are quantize
d because of the periodic boundary conditions, and they are required to satisfy
, or
The eigenfunction and eigenenergies are
where 
Therefore, there are two degenerate quantum states for every value of
(corresponding to 
). Therefore there are 2n+1 states with energies up to an energy indexed by the number n.
The case of a particle in a one-dimensional ring is an instructive example when studying the quantization
of angular momentum
for, say, an electron
orbiting the nucleus
. The azimuthal wave functions in that case are identical to the energy eigenfunction
s of the particle on a ring.
The statement that any wavefunction for the particle on a ring can be written as a superposition
of energy
eigenfunction
s is exactly identical to the Fourier theorem about the development of any periodic function
in a Fourier series
.
This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.
, aromatic compounds contain atomic rings, such as benzene
rings (the Kekulé
structure) consisting of five or six, usually carbon
, atoms. So does the surface of "buckyballs" (buckminsterfullerene). These molecules are exceptionally stable.
The above explains why the ring behaves like a circular waveguide
, with the valence electrons orbiting in both directions.
To fill all energy levels up to n requires
electrons, as electrons have additionally two possible orientations of their spins.
The rule that
excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is known as the Hückel's rule
.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the case of a particle in a one-dimensional ring is similar to the particle in a box
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
. The Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for a free particle
Free particle
In physics, a free particle is a particle that, in some sense, is not bound. In classical physics, this means the particle is present in a "field-free" space.-Classical Free Particle:The classical free particle is characterized simply by a fixed velocity...
which is restricted to a ring (technically, whose configuration space
Configuration space
- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
is the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
) isWave function
Using polar coordinates on the 1 dimensional ring, the wave function depends only on the angularAngle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
coordinate, and so
Requiring that the wave function be periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
in
with a period (from the demand that the wave functions be single-valued functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s on the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
), and that they be normalized
Normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...
leads to the conditions
,and
Under these conditions, the solution to the Schrödinger equation is given by
Energy eigenvalues
The energyEnergy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
eigenvalues
are quantizeQuantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
d because of the periodic boundary conditions, and they are required to satisfy
, orThe eigenfunction and eigenenergies are
where Therefore, there are two degenerate quantum states for every value of
(corresponding to ). Therefore there are 2n+1 states with energies up to an energy indexed by the number n.The case of a particle in a one-dimensional ring is an instructive example when studying the quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
for, say, an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
orbiting the nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
. The azimuthal wave functions in that case are identical to the energy eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of the particle on a ring.
The statement that any wavefunction for the particle on a ring can be written as a superposition
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...
of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s is exactly identical to the Fourier theorem about the development of any periodic function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
in a Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
.
This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.
Application
In organic chemistryOrganic chemistry
Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, composition, reactions, and preparation of carbon-based compounds, hydrocarbons, and their derivatives...
, aromatic compounds contain atomic rings, such as benzene
Benzene
Benzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....
rings (the Kekulé
Friedrich August Kekulé von Stradonitz
Friedrich August Kekule von Stradonitz was a German organic chemist. From the 1850s until his death, Kekule was one of the most prominent chemists in Europe, especially in theoretical chemistry...
structure) consisting of five or six, usually carbon
Carbon
Carbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds...
, atoms. So does the surface of "buckyballs" (buckminsterfullerene). These molecules are exceptionally stable.
The above explains why the ring behaves like a circular waveguide
Waveguide
A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...
, with the valence electrons orbiting in both directions.
To fill all energy levels up to n requires
electrons, as electrons have additionally two possible orientations of their spins.The rule that
excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is known as the Hückel's ruleHückel's rule
In organic chemistry, Hückel's rule estimates whether a planar ring molecule will have aromatic properties. The quantum mechanical basis for its formulation was first worked out by physical chemist Erich Hückel in 1931...
.
See also
- Angular momentumAngular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
- Harmonic analysisHarmonic analysisHarmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
- One-dimensional periodic case