Schumann resonances in the Earth-ionosphere cavity by Bliokh P.V., Nikolaenko A.P., Filippov Yu.F.

By Bliokh P.V., Nikolaenko A.P., Filippov Yu.F.

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As with the particle in the 1-dimensional box, the energies are quantized, depend on the square of the quantum number, depend inversely on the mass, and inversely on the square of the dimension of the box, here defined by the fixed radius of the ring. The comparison with the 1-dimensional box with infinite walls is logical if one imagines the ring as arising from the line by bending it around so that the two edges meet. The analogy is complete if we consider the length of the box to be equal to the circumference of the ring, and if we recognize that only the even wave functions of the linear box will satisfy the continuity requirements on the circle (see problem assignment).

6) Likewise one can calculate the expected value of the variance of each parameter, for example the variance of the momentum is ∆p2 = (p − p )2 or 2 ψ ∗ (x) (p − p ) ψ(x) dx 2 ∆p = . 7) ψ ∗ (x)ψ(x)dx It is possible to show (see for example Herbert L. 8) or that 1 . 9) 2 This tells us that, in general, the product of the uncertainties in momentum and position is always greater than a finite quantity and can never be zero. This is one manifestation of the Heisenberg Uncertainty Principle that tells us that if the momentum is known with great precision, the position is very uncertain and vice versa, if the position is known with great precision, then the momentum is very uncertain.

The one used here follows that used by McQuarrie and Simon in their classical physical chemistry text. , it is continuous,4 bounded,5 single-valued,6 and has continuous partial derivatives, and the following integral exists, is not equal to zero, and has a finite value. Ψ∗ Ψdτ = Ψ∗ (x, y, z; t)Ψ(x, y, z; t)dxdydz = 0. 1) b: The probability that the particle is in the volume element dτ = dxdydz is given by Ψ∗ Ψdτ = |Ψ|2 dτ . 7 1 State: the property or properties of a system at any given time. (wave) function: a mathematical description of the property of the system.

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