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.,

∫₀^XP(x)dx = (1/T_k)∫₀^Xφ²(x)dx
= (1/T_k)∫₀^X[sin(kx)²(x)/k²]dx
= (1/(T_kk²))∫₀^Xsin(kx)²dx

The integral ∫₀^Xsin(kx)²(x)dx is evaluated by changing the variable of integration from x to z where z=kx. Consequently

∫₀^Xsin(kx)²dx = (1/k)∫₀^πsin(z)²dz = (π/2)/k

Thus the probability of the system being in the interval [0, X] is

∫₀^XP(x)dx = (1/(T_kk²))(π/2)/k = π/(2T_kk³)

The average probability density in the interval [0, X] is then

[∫₀^Xsin(kx)²dx]/X = [π/(2T_kk³)]/(π/k) = 1/(2T_kk²)

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.

(To be continued.)