﻿ A Property of Solutions to a Generalized Helmholtz Equation
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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)/Tk

where Tk 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.,

#### ∫0XP(x)dx = (1/Tk)∫0Xφ²(x)dx = (1/Tk)∫0X[sin(kx)²(x)/k²]dx = (1/(Tkk²))∫0Xsin(kx)²dx

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

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

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

#### ∫0XP(x)dx = (1/(Tkk²))(π/2)/k = π/(2Tkk³)

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

#### [∫0Xsin(kx)²dx]/X = [π/(2Tkk³)]/(π/k) = 1/(2Tkk²)

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.)