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This space-dependence is called a normal mode. Usually, for problems with continuous dependence on (x, y, z) there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are countably many normal modes (usually numbered n = 1, 2 ...
A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10 13 Hz to approximately 10 14 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm −1 and wavelengths of approximately 30 to 3 μm.
The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Q k. Normal coordinates decouple the classical vibrational motions of the molecule and thus give an easy route to ...
Normal coordinates exist on a normal neighborhood of a point p in M. A normal neighborhood U is an open subset of M such that there is a proper neighborhood V of the origin in the tangent space T p M, and exp p acts as a diffeomorphism between U and V. On a normal neighborhood U of p in M, the chart is given by: :=: The isomorphism E, and ...
The equation for () has solutions which exponentially grow or decay for >, are linear or constant for = and are periodic for <. Physically it is expected that a solution to the problem of a vibrating drum head will be oscillatory in time, and this leaves only the third case, K < 0 , {\displaystyle K<0,} so we choose K = − λ 2 {\displaystyle ...
The number of normal modes is the same as the number of particles. Still, the Fourier space is very useful given the periodicity of the system. A set of N "normal coordinates" Q k may be introduced, defined as the discrete Fourier transforms of the x k and N "conjugate momenta" Π k defined as the Fourier transforms of the p k:
An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations = and = (but these may not hold at other points in the frame). These coordinates are also called normal coordinates.
The Hamiltonian of the particle is: ^ = ^ + ^ = ^ + ^, where m is the particle's mass, k is the force constant, = / is the angular frequency of the oscillator, ^ is the position operator (given by x in the coordinate basis), and ^ is the momentum operator (given by ^ = / in the coordinate basis).