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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 two adjacent points of relative maxima of φ², a and b. These are points such that dφ²/dx equal zero.

Multiply the equation by φ. and integrate the resulting equation from x=a to x=b. The result may be represented as

abφ(d²φ/dx²)dx = −∫abk²(x)(φ²(x)dx

The LHS may be integrated by parts to yield

[φ(dφ/dx)]ab − ∫ab(dφ/dx)²dx = −∫abk²(x)(φ²(x)dx

The quantity φ(dφ/dx) is equal to zero at both x=a and x=b. Therefore

abk²(x)(φ²(x)dx = ∫ab(dφ/dx)²dx

Let K equal ∫abk²(x)dx. Then dividing the above equation by K gives

abk²(x)(φ²(x)dx/K = ∫ab(dφ/dx)²dx/K

Now divide the numerator and denominator on the RHS by (b−a) to obtain

abk²(x)(φ²(x)dx/K = (∫ab(dφ/dx)²dx)/(b−a)/(K/(b−a))

The expression on the LHS of the above is a weighted average of φ². The expression on the RHS is the ratio of two "unweighted" (equal weighted) averages.

Thus the above equation can be represented as

φ² = (dφ/dx)²/

Another Relationship

Consider again the generalized Helmholtz equation

(d²φ/dx²) = −k²(x)φ(x)

Now multiply both sides of this equation by (dφ/dx) to obtain

(dφ/dx)(d²φ/dx²) = −k²(x)φ(x)(dφ/dx)
which can be represented as
d(½(dφ/dx)²/dx) = −k²d(½φ²)/dx)

Now integrate the above equation from a to b to obtain

[½(dφ/dx)²]ab = −½∫abk²(x)(φ(x)²(x)dx

The expression on the LHS if zero because (dφ/dx) is zero both for x=a and x=b. The expression on the RHS of the above can be integrated by parts to obtain

0 = −[k²(x)φ²(x)]∫ab + ∫abφ(x)²(dk²/dx)dx

This above equation can be expressed in the form

k²(b)φ²(b) = k²(a)φ²(a) + ∫abφ(x)²(dk²/dx)dx

If Φ(x) is defined as k(x)²φ(x)² then the above equation takes the simple form

Φ(b) = Φ(a) + ∫abΦ(x)[(dk²/dx)/k²]dx
or, equivalently
Φ(b) = Φ(a) + ∫abΦ(x)(d(ln(k²))/dx)dx

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