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A relationship for the Spatial Average of the Probability
Density Function Derived from the Solution to a
Generalized Helmholtz Equation of One Dimension

The equation under consideration is

(d²φ/dx²) = −k²(x)φ(x)

where k(x) is a positive but declining function of x. This can be termed a generalized Helmholtz equation.

Now consider the points of relative maxima and minima of φ², labeled as x_{i} and y_{i}, respectively.
These are points such that dφ²/dx=2φ(dφ/dx) equals zero; (dφ/dx)=0 for the maxima and φ=0 for the minima.
Let the labeling be such that x_{i} > y_{i}.

Multiply the equation by (dφ/dx). to get

(dφ/dx)(d²φ/dx²) = −k²(x)φ(x)(dφ/dx)
which is the same as
½(d((dφ/dx)²)/dx) = −½k²(x)(d(φ²)/dx)

Now
integrate the above equation from
x=y_{i} to x=x_{i}.
The result may be represented as

The average value of φ² over the interval [y_{i}, x_{i}] is essentially one half of the maximum value
for that interval, which is its value at x_{i}. The maximum value of (dφ/dx)² occurs where d(dφ/dx)/dx is equal
to zero. Since d(dφ/dx)/dx=−k²(x)φ(x) that maximum occurs where φ(x)=0. Thus the average of (dφ/dx)²
over the interval [y_{i}, x_{i}] is also essentially one half of its maximum value of that interval, which is its value
at y_{i}. The value of k²(z_{i}) is also its average value over the interval [y_{i}, x_{i}].
Let the average value of a variable be denoted by the underscore of its symbol. Thus