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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 xi and yi, 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 xi > yi.

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=yi to x=xi. The result may be represented as

½[((dφ/dx)²]yixi = −½∫yixik²(x)(φ²(x)dx

At xi (dφ/dx) is equal to 0. By the Generalized Mean Value Theorem the RHS of the above equation may be expressed as

= −½∫yixik²(x)(φ²(x)dx = −½k²(zi)∫yixi(φ²(x)dx
which reduces to

where zi is some point between yi and xi.

But φ² is equal to zero at yi.

Therefore the previous equation reduces to

−½(dφ/dx)²yi = −½k²(zi)φ²(xi)
or, equivalently
φ²(xi) = (dφ/dx)²yi/k²(zi)

The average value of φ² over the interval [yi, xi] is essentially one half of the maximum value for that interval, which is its value at xi. 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 [yi, xi] is also essentially one half of its maximum value of that interval, which is its value at yi. The value of k²(zi) is also its average value over the interval [yi, xi]. Let the average value of a variable be denoted by the underscore of its symbol. Thus

φ² = (dφ/dx)²/

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