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A Property of Solutions to a Generalized Helmholtz Equation |
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The Helmholtz equation is of the form
Where k is a real positive constant. For the one dimensional case this reduces to
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
Obviously φ(x) is equal to 0 at x=0. The first derivative is
Note that the second derivative is
which is just the original equation.
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.
Equations such as the Helmholtz equation arise in quantum theory where the probability density P(x) is proportional to φ²(x). Let
where T_{k} 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.,
The integral ∫_{0}^{X}sin(kx)²(x)dx is evaluated by changing the variable of integration from x to z where z=kx. Consequently
Thus the probability of the system being in the interval [0, X] is
The average probability density in the interval [0, X] is then
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.
A generalized Helmholtz equation of one dimension is of the form
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.)
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