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Density Function Derived from the Solution to a Generalized Helmholtz Equation of One Dimension |
The equation under consideration is
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
Now integrate the above equation from x=y_{i} to x=x_{i}. The result may be represented as
At x_{i} (dφ/dx) is equal to 0. By the Generalized Mean Value Theorem the RHS of the above equation may be expressed as
where z_{i} is some point between y_{i} and x_{i}.
But φ² is equal to zero at y_{i}.
Therefore the previous equation reduces to
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
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