Normal mode

Normal modes in an oscillating system are special solutions where all the parts of the system are oscillating with the same frequency (called normal frequencies or allowed frequencies). The concept of normal modes is of vital importance in wave theory, optics and quantum mechanics.

Finding normal modes in harmonic oscillation utilitizes the strength of linear algebra and linear sets of differential equations. One can present the problem as a matrix-vector equation and then solve for its eigenvectors. After finding them, the normal modes are the eigenvectors where the normal frequencies are the eigenvalues.

Table of contents

1 Example - normal modes of coupled oscillators

2 Standing waves

3 Normal modes in quantum mechanics

4 See also

Example - normal modes of coupled oscillators

Consider two bodies, each of mass M, attached to three springs with stiffness K. They are attached in the following manner:

|---M---M---|

where the edge points are fixed and cannot move. We'll use x₁(t) to denote the displacement of the leftmost mass, and x₂(t) to denote the displacement of the rightmost.

If we denote the second derivative of x(t) with respect to time as x″, the equations of motion are:

And in matrix representation:

For simplicity of writing lets assume that (K/M) = 1. Note that this "1" has a units of square frequency. In a normal mode the solution satisfy:

Or:

Demanding that a non-trivial solution exists, i.e the determinant is equal to 0 leads for solving the characteristic polynomial, we obtain:

The left vector it the eigenvector and the right scalar is the normal frequency.

The first normal mode is:

and the second normal mode is:

The general soultion is a superposition of the normal modes where $c 1, c 2,φ 1,φ 2$ are determined by the initial conditions of the problem.

The process demonstrated here can be generlized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics.

Standing waves

A standing wave is a continuous form of normal mode. In a standing wave, all the space elements (i.e (x,y,z) coordinates) are oscillating in the same frequency and in phase (reaching the equilibrium point together), but each has a different amplitude.

The general form of a standing wave is:

where f(x,y,z) represents the dependence of amplitude on location and the cosine\sine are the oscillations in time.

Physically, standing waves are formed by the interference of waves and their reflections (although one also say that a moving wave is a superposition of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the f(x,y,z) form of the standing wave. This space-dependence is called a normal mode.

Usually, for a problems with continuous dependence on (x,y,z) there is no single or finite number of normal mode, but there are infinitely normal modes. If the problem is bounded (i.e it is defined on a finite section of space) there are countably many (a discrete infinity of ) normal modes (usually numbered n = 1,2,3,...). If the problem is not bounded, there is a continuous spectrum of normal modes.

The allowed frequencies are dependent in the normal modes, as well in physical constants of the problem (density, tension, pressure etc.) which sets the phase velocity of the wave. The range of all possible normal frequencies is called frequency spectrum. Usually, each frequency is modulated by the amplitude in which it was arisen, creating then a grpah of the power spectrum of the oscliations.

When relating to music, normal modes of a vibrating instruments (strings, air pipes, drums etc) are called "harmonies".

Normal modes in quantum mechanics

In quantum mechanics, a state of a system is described by a wavefunction |ψ> of (x, t) which solves the Schrödinger equation. The square of the abselute value of |ψ>, i.e.

P (x, t) = | ψ(x, t) | 2

is the probability (density) to measure the particle in place x at time t.

Usually, when involving some sort of potential, the wavefunction is decomposed into a superposition of energy eigenstates, each oscillating with frequency of . Thus, we may write

The eigenstates have a physical meaning farer than an orthonormal basis. When a measurement takes place, the wavefunction collapses into one of its eigenstates and henceforth the particle wavefunction is described only by the eigenstate corresponding to the measured energy.