applet-magic.com Thayer Watkins
Silicon Valley & Tornado Alley
U.S.A.

A Property of Solutions to a Generalized Helmholtz Equation

The Helmholtz equation is of the form

∇²φ = −Kφ

Where k is a real positive constant. For the one dimensional case this reduces to

(d²φ/dx²) = −Kφ

Consider a family of solutions to this equation with the same boundary conditions but different values of the parameter k.
The boundary conditions are φ(0)=0 and (dφ/dx) at x=0 equals 1. Such solutions are of the form

φ(x) = sin(kx)/k

Obviously φ(x) is equal to 0 at x=0. The first derivative is

(dφ/dx) = k·cos(kx)/k = cos(kx)
and hence at x=0
(dφ/dx) = 1

Note that the second derivative is

(d²φ/dx²) = −k·sin(k)

which is just the original equation.

The Wavelength of the Solutions

The variable φ(x) is equal to zero when x=0 and when kx=π. Let X=π/k. This is one half of the usual notion of wavelength.

Probability Density

Equations such as the Helmholtz equation arise in quantum theory where the probability density P(x) is proportional to φ²(x).
Let

P(x) = φ²(x)/T_{k}

where T_{k} is the normalization factor that makes values of P(x) sum to unity.

Now consider the probability that the system is to be found in the interval [0, X]; i.e.,

The dependence of the average probability density upon the parameter k is not certain because it depends the normalization factor
as well as the explicit dependence shown above.

Generalization

A generalized Helmholtz equation of one dimension is of the form

d²φ/dx² = −f(x)φ

where f(x) is a positive real function. If f(x) is constant then the solution is a sinusoidal function as displayed above.