In
rotordynamics
Rotordynamics, also known as rotor dynamics, is a specialized branch of applied mechanics concerned with the behavior and diagnosis of rotating structures. It is commonly used to analyze the behavior of structures ranging from jet engines and st ...
, the rigid rotor is a mechanical model of
rotating
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
systems. An arbitrary rigid rotor is a 3-dimensional
rigid object, such as a
top
A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect.
Once set in motion, a top will usually wobble for a few ...
. To orient such an object in space requires three angles, known as
Euler angles. A special rigid rotor is the ''linear rotor'' requiring only two angles to describe, for example of a diatomic
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
. More
general molecules are 3-dimensional, such as water (asymmetric rotor),
ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous wa ...
(symmetric rotor), or
methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Ea ...
(spherical rotor).
Linear rotor
The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses and the values of the masses are the only characteristics of the rigid model. However, for many actual diatomics this model is too restrictive since distances are usually not completely fixed. Corrections on the rigid model can be made to compensate for small variations in the distance. Even in such a case the rigid rotor model is a useful point of departure (zeroth-order model).
Classical linear rigid rotor
The classical linear rotor consists of two point masses
and
(with
reduced mass
In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
) at a distance
of each other. The rotor is rigid if
is independent of time. The kinematics of a linear rigid rotor is usually described by means of
spherical polar coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
, which form a coordinate system of R
3. In the physics convention the coordinates are the co-latitude (zenith) angle
, the longitudinal (azimuth) angle
and the distance
. The angles specify the orientation of the rotor in space. The kinetic energy
of the linear rigid rotor is given by
:
where
and
are
scale (or Lamé) factors.
Scale factors are of importance for quantum mechanical applications since they enter the
Laplacian expressed in
curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
. In the case at hand (constant
)
:
The classical Hamiltonian function of the linear rigid rotor is
:
Quantum mechanical linear rigid rotor
The linear rigid rotor model can be used 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 chemistr ...
to predict the rotational energy of a
diatomic molecule. The rotational energy depends on the
moment of inertia for the system,
. In the
center of mass reference frame, the moment of inertia is equal to:
:
where
is the
reduced mass
In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
of the molecule and
is the distance between the two atoms.
According to
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 chemistr ...
, the energy levels of a system can be determined by solving 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 ...
:
:
where
is 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 ...
and
is the energy (
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
) operator. For the rigid rotor in a field-free space, the energy operator corresponds to the
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
of the system:
:
where
is
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
and
is the
Laplacian. The Laplacian is given above in terms of spherical polar coordinates. The energy operator written in terms of these coordinates is:
: